Finite Group Databasebeta

← #33094958

Subgroups — #33094958

Order 18688.

1426 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556570C211yesmaximal, characteristic
2C2×#17988551C211yesmaximal, characteristic
3#22610399C214yesmaximal, complemented
4#22610397C214yesmaximal, complemented
5#22610009C212yesmaximal, complemented
6#22609854C212yesmaximal, complemented
7C2×#407×C73C211yesmaximal, characteristic
8C23×Dic146C2211yescharacteristic
9C22×#11924012C2212yes—
10C2×#14556570—21no—
11C2×#14556570—21no—
12C2×#14556570—21no—
13C2×#14556570—21no—
14C2×#14556570—21no—
15C2×#14556570—21no—
16C2×#14556570—21no—
17C2×#14556570—21no—
18C2×#14556570C412yes—
19C2×#14556570C412yes—
20C2×#14556570C2212yes—
21C2×#14556570C2212yes—
22C22×C4⋊Dic73C2211yescharacteristic
23C22×#11924007C412yes—
24C22×C22⋊Dic73C412yes—
25C2×#14556509C2218yescomplemented
26#17988551—21no—
27#17988551—21no—
28#17988551C2212yes—
29#17988551C2212yes—
30C2×C22⋊#10596831—21no—
31C2×C22⋊#10596831—21no—
32#17988355C2218yescomplemented
33C24×C4×C73C2211yescharacteristic
34C22×M16×C73C2212yes—
35#407×C73—21no—
36#407×C73—21no—
37#407×C73C2212yes—
38#407×C73C2212yes—
39C2×C22⋊C8×C73—21no—
40C2×C22⋊C8×C73—21no—
41C22×Dic146—21no—
42C22×Dic146—21no—
43C22×Dic146—21no—
44C22×Dic146—21no—
45C22×Dic146—44no—
46C22×Dic146C2×C412yes—
47C22×Dic146C2312yes—
48C23×Dic73C2×C412yes—
49C23×Dic73C2×C412yes—
50C2×#11924012—22no—
51C2×#11924012—44no—
52C2×#11924012C2314yes—
53C22×#10596831—22no—
54#14556570—22no—
55#14556570—22no—
56#14556570—22no—
57#14556570—22no—
58#14556570—22no—
59#14556570—22no—
60#14556570—22no—
61#14556570—22no—
62#14556570—22no—
63#14556570—22no—
64#14556570—22no—
65#14556570—22no—
66#14556570—22no—
67#14556570—22no—
68#14556570—22no—
69#14556570—22no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2×C412yes—
75C2×C4⋊Dic73C2312yes—
76C2×#11924007—22no—
77C2×#11924007—22no—
78C2×#11924007—22no—
79C2×#11924007—22no—
80C2×#11924007C2×C414yes—
81C2×#11924007C2×C414yes—
82C2×C22⋊Dic73—22no—
83C2×C22⋊Dic73—22no—
84C2×C22⋊Dic73—22no—
85C2×C22⋊Dic73—22no—
86C2×C22⋊Dic73C2×C414yes—
87C2×C22⋊Dic73C2×C414yes—
88#14556509—48no—
89C22⋊#10596831—22no—
90C22⋊#10596831—22no—
91C22⋊#10596831—22no—
92C22⋊#10596831—22no—
93C25×C73C2×C411yescharacteristic
94C23×C4×C73—44no—
95C23×C4×C73C2×C411yescharacteristic
96C23×C4×C73C2×C411yescharacteristic
97C23×C4×C73C2×C411yescharacteristic
98C23×C4×C73D411yescharacteristic
99C23×C4×C73D411yescharacteristic
100C23×C4×C73D411yescharacteristic
101C23×C4×C73D411yescharacteristic
102C23×C4×C73D411yescharacteristic
103C23×C4×C73D411yescharacteristic
104C23×C4×C73D411yescharacteristic
105C23×C4×C73D411yescharacteristic
106C23×C4×C73C2311yescharacteristic
107C23×C4×C73C2311yescharacteristic
108C23×C4×C73C2311yescharacteristic
109C2×M16×C73—22no—
110C2×M16×C73—44no—
111C2×M16×C73C2314yes—
112C22×C8×C73—22no—
113C22⋊C8×C73—22no—
114C22⋊C8×C73—22no—
115C22⋊C8×C73—22no—
116C22⋊C8×C73—22no—
117C2×Dic146—22no—
118C2×Dic146—22no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146—28no—
122C2×Dic146—44no—
123C2×Dic146—44no—
124C2×Dic146—44no—
125C2×Dic146—44no—
126C2×Dic146—44no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—44no—
132C22×Dic73—44no—
133C22×Dic73C22×C414yes—
134C22×Dic73C22×C414yes—
135#11924012—28no—
136#11924012—44no—
137C2×#10596831—24no—
138C2×#10596831—44no—
139C4⋊Dic73—22no—
140C4⋊Dic73—22no—
141C4⋊Dic73—22no—
142C4⋊Dic73—22no—
143#11924007—24no—
144#11924007—24no—
145#11924007—24no—
146#11924007—24no—
147C22⋊Dic73—24no—
148C22⋊Dic73—24no—
149C22⋊Dic73—24no—
150C22⋊Dic73—24no—
151C24×C73—24no—
152C24×C73—42no—
153C24×C73—42no—
154C24×C73C22⋊C411yescharacteristic
155C24×C73C22⋊C411yescharacteristic
156C24×C73C22⋊C411yescharacteristic
157C24×C73C22⋊C411yescharacteristic
158C24×C73C22×C411yescharacteristic
159C24×C73C22×C411yescharacteristic
160C24×C73C22×C411yescharacteristic
161C22×C4×C73—24no—
162C22×C4×C73—24no—
163C22×C4×C73—24no—
164C22×C4×C73—42no—
165C22×C4×C73—42no—
166C22×C4×C73—42no—
167C22×C4×C73—42no—
168C22×C4×C73—42no—
169C22×C4×C73—42no—
170C22×C4×C73—42no—
171C22×C4×C73—42no—
172C22×C4×C73—42no—
173C22×C4×C73—42no—
174C22×C4×C73—42no—
175C22×C4×C73C22⋊C411yescharacteristic
176C22×C4×C73C22⋊C411yescharacteristic
177C22×C4×C73C22⋊C411yescharacteristic
178C22×C4×C73C22⋊C411yescharacteristic
179C22×C4×C73C22⋊C411yescharacteristic
180C22×C4×C73C22⋊C411yescharacteristic
181C22×C4×C73C22⋊C411yescharacteristic
182C22×C4×C73C22⋊C411yescharacteristic
183C22×C4×C73C22⋊C411yescharacteristic
184C22×C4×C73C22⋊C411yescharacteristic
185C22×C4×C73C22⋊C411yescharacteristic
186C22×C4×C73C22⋊C411yescharacteristic
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2411yescharacteristic
203M16×C73—28no—
204M16×C73—44no—
205C2×C8×C73—24no—
206C2×C8×C73—44no—
207C23×C73—22no—
208C23×C73—22no—
209C23×C73—22no—
210C23×C73—22no—
211C23×C73—22no—
212C23×C73—22no—
213C23×C73—22no—
214C23×C73—22no—
215C23×C73—22no—
216C23×C73—22no—
217C23×C73—22no—
218C23×C73—22no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—416no—
228C23×C73#10114yescomplemented
229C23×C73C2×C22⋊C411yescharacteristic
230C23×C73C2×C22⋊C411yescharacteristic
231C23×C73C2×C22⋊C411yescharacteristic
232C23×C73C2×C22⋊C411yescharacteristic
233C23×C73C2×C22⋊C411yescharacteristic
234C23×C73C2×C22⋊C411yescharacteristic
235C23×C73C23×C411yescharacteristic
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73C2×C22⋊C411yescharacteristic
264C2×C4×C73C2×C22⋊C411yescharacteristic
265C2×C4×C73C2×C22⋊C411yescharacteristic
266C2×C4×C73C2×C22⋊C411yescharacteristic
267C2×C4×C73C2×C22⋊C411yescharacteristic
268C2×C4×C73C2×C22⋊C411yescharacteristic
269C2×C4×C73C22×D411yescharacteristic
270C2×C4×C73C22×D411yescharacteristic
271C2×C4×C73H(5,𝔽2)14yes—
272C2×Dic73—24no—
273C2×Dic73—24no—
274C2×Dic73—28no—
275C2×Dic73—44no—
276C2×Dic73—44no—
277C2×Dic73—44no—
278C2×Dic73—44no—
279C2×Dic73—44no—
280Dic146—44no—
281Dic146—44no—
282Dic146—44no—
283Dic146—44no—
284C8×C73—44no—
285#10596831—44no—
286C22×C73—28no—
287C22×C73—21no—
288C22×C73—21no—
289C22×C73—21no—
290C22×C73—21no—
291C22×C73—21no—
292C22×C73—21no—
293C22×C73—21no—
294C22×C73—21no—
295C22×C73—21no—
296C22×C73—21no—
297C22×C73—21no—
298C22×C73—21no—
299C22×C73—21no—
300C22×C73—21no—
301C22×C73—21no—
302C22×C73—21no—
303C22×C73—44no—
304C22×C73—44no—
305C22×C73—44no—
306C22×C73—44no—
307C22×C73—44no—
308C22×C73—44no—
309C22×C73C2×#10112yes—
310C22×C73C2×#10112yes—
311C22×C73C2×#10112yes—
312C22×C73#41312yes—
313C22×C73C22×C22⋊C411yescharacteristic
314C22×C73#51812yes—
315C4×C73—21no—
316C4×C73—21no—
317C4×C73—21no—
318C4×C73—21no—
319C4×C73—21no—
320C4×C73—21no—
321C4×C73#53412yes—
322C4×C73#53412yes—
323Dic73—44no—
324Dic73—44no—
325#21753—731nomaximal, Sylow
326C2×C73—22no—
327C2×C73—22no—
328C2×C73—22no—
329C2×C73—22no—
330C2×C73—22no—
331C2×C73—41no—
332C2×C73—41no—
333C2×C73#230811yescharacteristic
334C2×C73C22×#10111yescharacteristic
335C2×C73C2×#41311yescharacteristic
336C22×C22⋊Q8—731no—
337C2×#407—731no—
338C2×#407D7311yescharacteristic
339#1901—734no—
340#1901—734no—
341#1821—732no—
342#1784—732no—
343C73#2175311yescharacteristic, complemented, Sylow
344C23×Q8—731no—
345C24×C4C2×D7311yescharacteristic
346C22×M16—732no—
347C22×M16C2×D7312yes—
348C2×C22⋊Q8—732no—
349C2×C22⋊Q8—732no—
350C2×C22⋊Q8—732no—
351C2×C22⋊Q8—732no—
352C2×C22⋊Q8—1461no—
353C2×C22⋊Q8—1461no—
354C2×C22⋊Q8—1461no—
355C2×C22⋊Q8—1461no—
356C2×C22⋊Q8—1461no—
357C2×C22⋊Q8—1461no—
358C2×C22⋊Q8—1461no—
359C2×C22⋊Q8—1461no—
360C22×C4⋊C4—732no—
361C22×C4⋊C4—731no—
362C22×C22⋊C4—732no—
363C2×#101—738no—
364#407—21no—
365#407—21no—
366#407—732no—
367#407—732no—
368#407—1461no—
369#407—1461no—
370#407C2×D7312yes—
371#407C2×D7312yes—
372C2×C22⋊C8—21no—
373C2×C22⋊C8—21no—
374C2×C22⋊C8—1461no—
375C2×C22⋊C8—1461no—
376#343—738no—
377C25C4×D7311yescharacteristic
378C22×Q8—2924no—
379C22×Q8—732no—
380C22×Q8—732no—
381C22×Q8—1461no—
382C22×Q8—1461no—
383C22×Q8—1461no—
384C22×Q8—1461no—
385C23×C4—44no—
386C23×C4—732no—
387C23×C4—732no—
388C23×C4C4×D7311yescharacteristic
389C23×C4C4×D7311yescharacteristic
390C23×C4C4×D7311yescharacteristic
391C23×C4D29211yescharacteristic
392C23×C4D29211yescharacteristic
393C23×C4D29211yescharacteristic
394C23×C4D29211yescharacteristic
395C23×C4C22⋊D7311yescharacteristic
396C23×C4C22⋊D7311yescharacteristic
397C23×C4C22⋊D7311yescharacteristic
398C23×C4C22⋊D7311yescharacteristic
399C23×C4C22×D7311yescharacteristic
400C23×C4C22×D7311yescharacteristic
401C23×C4C22×D7311yescharacteristic
402C2×M16—22no—
403C2×M16—44no—
404C2×M16—2924no—
405C2×M16—734no—
406C2×M16—1462no—
407C2×M16C22×D7314yes—
408C22×C8—22no—
409C22×C8—1462no—
410C22⋊Q8—1462no—
411C22⋊Q8—1462no—
412C22⋊Q8—1462no—
413C22⋊Q8—1462no—
414C22⋊Q8—1462no—
415C22⋊Q8—1462no—
416C22⋊Q8—1462no—
417C22⋊Q8—1462no—
418C22⋊Q8—1462no—
419C22⋊Q8—1462no—
420C22⋊Q8—1462no—
421C22⋊Q8—1462no—
422C22⋊Q8—1462no—
423C22⋊Q8—1462no—
424C22⋊Q8—1462no—
425C22⋊Q8—1462no—
426C2×C4⋊C4—734no—
427C2×C4⋊C4—734no—
428C2×C4⋊C4—732no—
429C2×C4⋊C4—732no—
430C2×C4⋊C4—1462no—
431C2×C4⋊C4—1462no—
432C2×C4⋊C4—1462no—
433C2×C4⋊C4—1462no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C22⋊C4—734no—
439C2×C22⋊C4—734no—
440C2×C22⋊C4—1462no—
441C2×C22⋊C4—1462no—
442C2×C22⋊C4—1462no—
443C2×C22⋊C4—1462no—
444#101—2928no—
445C22⋊C8—22no—
446C22⋊C8—22no—
447C22⋊C8—22no—
448C22⋊C8—22no—
449C22⋊C8—1462no—
450C22⋊C8—1462no—
451C22⋊C8—1462no—
452C22⋊C8—1462no—
453C24—24no—
454C24—42no—
455C24—42no—
456C24#1192400511yescharacteristic
457C24#1192400511yescharacteristic
458C24#1192400511yescharacteristic
459C24#1192400511yescharacteristic
460C24C2×C4×D7311yescharacteristic
461C24C2×C4×D7311yescharacteristic
462C24C2×C4×D7311yescharacteristic
463C2×Q8—2924no—
464C2×Q8—2924no—
465C2×Q8—2924no—
466C2×Q8—2924no—
467C2×Q8—2924no—
468C2×Q8—1462no—
469C2×Q8—1462no—
470C2×Q8—1462no—
471C2×Q8—1462no—
472C2×Q8—1468no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—2924no—
488C22×C4—2924no—
489C22×C4—734no—
490C22×C4—734no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4#1192400511yescharacteristic
496C22×C4#1192400511yescharacteristic
497C22×C4#1192400511yescharacteristic
498C22×C4#1192400511yescharacteristic
499C22×C4#1192400511yescharacteristic
500C22×C4#1192400511yescharacteristic
501C22×C4#1192400511yescharacteristic
502C22×C4#1192400511yescharacteristic
503C22×C4#1192400511yescharacteristic
504C22×C4#1192400511yescharacteristic
505C22×C4#1192400511yescharacteristic
506C22×C4#1192400511yescharacteristic
507C22×C4C2×C4×D7311yescharacteristic
508C22×C4C2×C4×D7311yescharacteristic
509C22×C4C2×C4×D7311yescharacteristic
510C22×C4C2×C22⋊D7311yescharacteristic
511C22×C4C2×C22⋊D7311yescharacteristic
512C22×C4C2×C22⋊D7311yescharacteristic
513C22×C4C2×C22⋊D7311yescharacteristic
514C22×C4C2×C22⋊D7311yescharacteristic
515C22×C4C2×C22⋊D7311yescharacteristic
516C22×C4C2×D29211yescharacteristic
517C22×C4C2×D29211yescharacteristic
518C22×C4C2×D29211yescharacteristic
519C22×C4C2×D29211yescharacteristic
520C22×C4C2×D29211yescharacteristic
521C22×C4C2×D29211yescharacteristic
522C22×C4C23×D7311yescharacteristic
523M16—28no—
524M16—44no—
525M16—2924no—
526M16—1468no—
527C2×C8—24no—
528C2×C8—44no—
529C2×C8—2924no—
530C2×C8—1464no—
531C4⋊C4—1464no—
532C4⋊C4—1464no—
533C4⋊C4—1464no—
534C4⋊C4—1464no—
535C4⋊C4—1462no—
536C4⋊C4—1462no—
537C4⋊C4—1462no—
538C4⋊C4—1462no—
539C22⋊C4—1464no—
540C22⋊C4—1464no—
541C22⋊C4—1464no—
542C22⋊C4—1464no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—416no—
564C23#1455650914yescomplemented
565C23C2×#1192400511yescharacteristic
566C23C2×#1192400511yescharacteristic
567C23C2×#1192400511yescharacteristic
568C23C2×#1192400511yescharacteristic
569C23C2×#1192400511yescharacteristic
570C23C2×#1192400511yescharacteristic
571C23C22×C4×D7311yescharacteristic
572Q8—2924no—
573Q8—2924no—
574Q8—2924no—
575Q8—2924no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—2924no—
604C2×C4—2924no—
605C2×C4—2924no—
606C2×C4—2924no—
607C2×C4—2924no—
608C2×C4—1464no—
609C2×C4—1464no—
610C2×C4—1468no—
611C2×C4C2×#1192400511yescharacteristic
612C2×C4C2×#1192400511yescharacteristic
613C2×C4C2×#1192400511yescharacteristic
614C2×C4C2×#1192400511yescharacteristic
615C2×C4C2×#1192400511yescharacteristic
616C2×C4C2×#1192400511yescharacteristic
617C2×C4C22×C22⋊D7311yescharacteristic
618C2×C4C22×D29211yescharacteristic
619C2×C4#1455663614yes—
620C8—44no—
621C8—2924no—
622C22—28no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22C2×#1455650912yes—
646C22C2×#1455650912yes—
647C22C2×#1455650912yes—
648C22#1798856912yes—
649C22C22×#1192400511yescharacteristic
650C22#1798906012yes—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—21no—
657C4—2924no—
658C4—2924no—
659C4#1798914612yes—
660C4#1798914912yes—
661C2—22no—
662C2—22no—
663C2—22no—
664C2—22no—
665C2—22no—
666C2—41no—
667C2—41no—
668C2#2261250611yescharacteristic
669C2C22×#1455650911yescharacteristic
670C2C2×#1798856911yescharacteristic
subgroupG/H
centerC22C22×#11924005
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#407×C73C2
socleC22×C73C22×C22⋊C4
prime radicalC25×C73C2×C4
cosocleC22×C4×C73C24
hypercenterC2×#407D73
hypocenterC73#21753
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24×C4 ◁ C2×#407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22610399C2semidirect
4. #22610397C2semidirect
5. #22610009C2semidirect
6. #22609854C2semidirect
25. C2×#14556509C22semidirect
32. #17988355C22semidirect
228. C23×C73#101semidirect
343. C73#21753semidirect
564. C23#14556509semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145565702
C2×#179885512
#226103992
#226103972
#226100092
#226098542
C2×#407×C732
#2175373