Finite Group Databasebeta

← #33007960

Subgroups — #33007960

Order 18688.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556550C211yesmaximal, characteristic
2C2×#17988457C211yesmaximal, characteristic, complemented
3C2×#17988453C211yesmaximal, characteristic, complemented
4C2×#17988447C211yesmaximal, characteristic
5#22608176C214yesmaximal
6#22608165C214yesmaximal, complemented
7#22608161C211yesmaximal, characteristic, complemented
8#22608161C211yesmaximal, characteristic, complemented
9C2×C4×C22⋊C4×C73C211yesmaximal, characteristic
10C2×#14556550—24no—
11C2×#14556550—24no—
12C2×#14556550—24no—
13C22×C4⋊Dic73C2211yescharacteristic
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C4⋊Dic73C2214yes—
16C22×C4⋊Dic73C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18C22×C22⋊Dic73C2212yes—
19C22×C22⋊Dic73C2212yes—
20C22×C4×Dic73C2212yes—
21#17988457—24no—
22#17988453—24no—
23#17988447—24no—
24C2×#14556498C2214yes—
25C2×#14556498C2212yes—
26C2×#14556498C2214yes—
27C2×#14556498C2214yes—
28C24×C4×C73C2211yescharacteristic
29C22×C22⋊C4×C73C2212yes—
30C22×C42×C73C2211yescharacteristic
31C22×C42×C73C2211yescharacteristic
32C4×C22⋊C4×C73C418yescomplemented
33C2×#95×C73C2212yes—
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C23×Dic73C2312yes—
38#14556550—232no—
39C2×C4⋊Dic73—28no—
40C2×C4⋊Dic73—22no—
41C2×C4⋊Dic73—22no—
42C2×C4⋊Dic73—22no—
43C2×C4⋊Dic73—22no—
44C2×C4⋊Dic73—22no—
45C2×C4⋊Dic73—22no—
46C2×C4⋊Dic73—22no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73—22no—
50C2×C4⋊Dic73—28no—
51C2×C4⋊Dic73—44no—
52C2×C4⋊Dic73D414yes—
53C2×C4⋊Dic73D414yes—
54C2×C22⋊Dic73—24no—
55C2×C22⋊Dic73—24no—
56C2×C22⋊Dic73—24no—
57C2×C22⋊Dic73—24no—
58C2×C22⋊Dic73—44no—
59C2×C4×Dic73—24no—
60C2×C4×Dic73—24no—
61C2×C4×Dic73—24no—
62#14556498—28no—
63#14556498—24no—
64#14556498—28no—
65#14556498—44no—
66C25×C73C2311yescharacteristic
67C23×C4×C73—22no—
68C23×C4×C73—22no—
69C23×C4×C73—22no—
70C23×C4×C73—22no—
71C23×C4×C73C2×C414yes—
72C23×C4×C73D414yes—
73C23×C4×C73Q814yes—
74C23×C4×C73C2312yes—
75C23×C4×C73C2311yescharacteristic
76C23×C4×C73C2312yes—
77C23×C4×C73C2311yescharacteristic
78C2×C22⋊C4×C73—24no—
79C2×C22⋊C4×C73—24no—
80C2×C22⋊C4×C73C2×C418yes—
81C2×C42×C73—22no—
82C2×C42×C73—22no—
83C2×C42×C73—22no—
84C2×C42×C73—22no—
85C2×C42×C73C2×C414yes—
86C2×C42×C73C2×C414yes—
87#95×C73C2×C418yes—
88C22×Dic73—22no—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—24no—
93C22×Dic73—24no—
94C22×Dic73—24no—
95C22×Dic73—24no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—24no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—24no—
104C22×Dic73—24no—
105C22×Dic73—24no—
106C22×Dic73C2×D414yes—
107C22×Dic73C2×D414yes—
108C4⋊Dic73—28no—
109C4⋊Dic73—28no—
110C4⋊Dic73—416no—
111C4⋊Dic73—44no—
112C4⋊Dic73—44no—
113C22⋊Dic73—216no—
114C22⋊Dic73—48no—
115C4×Dic73—216no—
116C24×C73—21no—
117C24×C73—21no—
118C24×C73—22no—
119C24×C73—22no—
120C24×C73—21no—
121C24×C73—21no—
122C24×C73—22no—
123C24×C73—22no—
124C24×C73C22×C412yes—
125C24×C73C2×D412yes—
126C24×C73C2×Q812yes—
127C24×C73C2411yescharacteristic
128C22×C4×C73—22no—
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73—22no—
132C22×C4×C73—21no—
133C22×C4×C73—21no—
134C22×C4×C73—22no—
135C22×C4×C73—22no—
136C22×C4×C73—28no—
137C22×C4×C73—28no—
138C22×C4×C73—28no—
139C22×C4×C73—28no—
140C22×C4×C73—28no—
141C22×C4×C73—28no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73C4⋊C4116yescomplemented
149C22×C4×C73C22×C414yes—
150C22×C4×C73C22×C412yes—
151C22×C4×C73C22×C414yes—
152C22×C4×C73C22×C412yes—
153C22×C4×C73C2×D412yes—
154C22×C4×C73C2×D412yes—
155C22×C4×C73C2×D412yes—
156C22×C4×C73C2×D412yes—
157C22×C4×C73C2×Q812yes—
158C22×C4×C73C2×Q812yes—
159C22×C4×C73#4112yes—
160C22×C4×C73#4112yes—
161C22×C4×C73#4114yes—
162C22×C4×C73#4114yes—
163C22⋊C4×C73—216no—
164C42×C73—28no—
165C42×C73—28no—
166C23×C73—24no—
167C23×C73—22no—
168C23×C73—22no—
169C23×C73—24no—
170C23×C73—22no—
171C23×C73—22no—
172C23×C73—22no—
173C23×C73—22no—
174C23×C73—22no—
175C23×C73—22no—
176C23×C73—24no—
177C23×C73—22no—
178C23×C73—22no—
179C23×C73—24no—
180C23×C73—42no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—42no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73C2×C4⋊C414yes—
189C23×C73C23×C411yescharacteristic
190C23×C73C22×D411yescharacteristic
191C23×C73C22×D411yescharacteristic
192C23×C73C22×D411yescharacteristic
193C23×C73C22×Q811yescharacteristic
194C23×C73C2×#4111yescharacteristic
195C23×C73C2×#4111yescharacteristic
196C23×C73C2×#4112yes—
197C23×C73C2×#4112yes—
198C23×C73H(5,𝔽2)12yes—
199C23×C732−1+412yes—
200C2×C4×C73—24no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—24no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—24no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—24no—
212C2×C4×C73—232no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—42no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73C2×C4⋊C414yes—
221C2×C4×C73C2×C4⋊C414yes—
222C2×C4×C73#11718yescomplemented
223C2×C4×C73C4×D414yes—
224C2×C4×C73C4×D414yes—
225C2×C4×C73#12014yes—
226C2×C4×C73C22⋊Q814yes—
227C2×C4×C73C22⋊Q814yes—
228C2×C4×C73#12314yes—
229C2×Dic73—24no—
230C2×Dic73—24no—
231C2×Dic73—28no—
232C2×Dic73—28no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—44no—
238C2×Dic73—44no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—44no—
246C2×Dic73—44no—
247C2×Dic73—44no—
248C2×Dic73—44no—
249C22×C73—24no—
250C22×C73—24no—
251C22×C73—24no—
252C22×C73—44no—
253C22×C73—44no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—44no—
257C22×C73—44no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73C22×C4⋊C411yescharacteristic
261C22×C73C2×#11712yes—
262C22×C73C2×C4×D411yescharacteristic
263C22×C73C2×C4×D411yescharacteristic
264C22×C73#52012yes—
265C22×C73C2×#12011yescharacteristic
266C22×C73C2×C22⋊Q811yescharacteristic
267C22×C73C2×C22⋊Q811yescharacteristic
268C22×C73C2×#12311yescharacteristic
269C22×C73#53211yescharacteristic
270C22×C73#53211yescharacteristic
271C22×C73#53312yes—
272C22×C73#53511yescharacteristic
273C22×C73#53611yescharacteristic
274C22×C73#54412yes—
275C22×C73#54612yes—
276C22×C73#54712yes—
277C22×C73#54812yes—
278C22×C73#55012yes—
279C22×C73#55212yes—
280C22×C73#55312yes—
281C22×C73#55512yes—
282C22×C73#55612yes—
283C4×C73—28no—
284C4×C73—24no—
285C4×C73—44no—
286C4×C73#38518yes—
287Dic73—44no—
288Dic73—44no—
289Dic73—82no—
290Dic73—82no—
291#13879—731nomaximal, Sylow
292C2×C73—44no—
293C2×C73#228511yescharacteristic
294C2×C73C2×#38511yescharacteristic
295C2×C73#231711yescharacteristic
296C2×C73#235212yes—
297C2×C73#235412yes—
298C2×C73#243111yescharacteristic
299C2×C73#243211yescharacteristic
300C2×C73#243912yes—
301C2×C73#244112yes—
302C2×C73#249911yescharacteristic
303C2×C73#250011yescharacteristic
304C22×#117—731no—
305C2×#384—731no—
306C2×#382—731no—
307C2×#380—731no—
308C2×C4×C22⋊C4D7311yescharacteristic
309#1439—734no—
310#1436—734no—
311C4⋊#95—731no—
312C4⋊#95—731no—
313C73#1387911yescharacteristic, complemented, Sylow
314C24×C4C2×D7311yescharacteristic
315C2×#117—1464no—
316C2×#117—1464no—
317C2×#117—1464no—
318C22×C4⋊C4—734no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—731no—
322C22×C4⋊C4—731no—
323C22×C22⋊C4—732no—
324C22×C22⋊C4—732no—
325C22×C22⋊C4C2×D7312yes—
326C22×C42—732no—
327C22×C42C2×D7311yescharacteristic
328C22×C42C2×D7311yescharacteristic
329#384—1464no—
330#382—1464no—
331#380—1464no—
332C4×C22⋊C4Dic7318yescomplemented
333C2×#95—734no—
334C2×#95—734no—
335C2×#95—732no—
336C2×#95—734no—
337C2×#95C2×D7312yes—
338C25C22×D7311yescharacteristic
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—732no—
344C23×C4—732no—
345C23×C4—732no—
346C23×C4—732no—
347C23×C4Dic14614yes—
348C23×C4D29214yes—
349C23×C4C2×Dic7314yes—
350C23×C4C22×D7312yes—
351C23×C4C22×D7311yescharacteristic
352C23×C4C22×D7312yes—
353C23×C4C22×D7311yescharacteristic
354#117—14632no—
355C2×C4⋊C4—2924no—
356C2×C4⋊C4—734no—
357C2×C4⋊C4—734no—
358C2×C4⋊C4—1468no—
359C2×C4⋊C4—1462no—
360C2×C4⋊C4—1462no—
361C2×C4⋊C4—1462no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1462no—
364C2×C4⋊C4—1462no—
365C2×C4⋊C4—1468no—
366C2×C4⋊C4—1462no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1462no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—2924no—
373C2×C22⋊C4—1464no—
374C2×C22⋊C4—1464no—
375C2×C22⋊C4—1464no—
376C2×C22⋊C4—1464no—
377C2×C22⋊C4C2×Dic7318yes—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—1464no—
383C2×C42—1464no—
384C2×C42—1464no—
385C2×C42C2×Dic7314yes—
386C2×C42C2×Dic7314yes—
387#95—2924no—
388#95—1468no—
389#95—1464no—
390#95—1468no—
391#95C2×Dic7318yes—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—21no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24C22×Dic7312yes—
401C24C2×D29212yes—
402C24C2×Dic14612yes—
403C24C23×D7311yescharacteristic
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—734no—
425C22×C4—734no—
426C22×C4—1462no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4—1464no—
433C22×C4—1464no—
434C22×C4—1464no—
435C22×C4—1464no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1464no—
441C22×C4—1464no—
442C22×C4—1464no—
443C22×C4—1464no—
444C22×C4C4⋊Dic73116yescomplemented
445C22×C4C22×Dic7314yes—
446C22×C4C22×Dic7312yes—
447C22×C4C22×Dic7314yes—
448C22×C4C22×Dic7312yes—
449C22×C4D4×D7312yes—
450C22×C4D4×D7312yes—
451C22×C4C2×D29212yes—
452C22×C4C2×D29212yes—
453C22×C4C2×Dic14612yes—
454C22×C4C2×Dic14612yes—
455C22×C4#1192402712yes—
456C22×C4#1192402712yes—
457C22×C4#1192402814yes—
458C22×C4#1192402814yes—
459C4⋊C4—29216no—
460C4⋊C4—2924no—
461C4⋊C4—2924no—
462C4⋊C4—1468no—
463C4⋊C4—1468no—
464C22⋊C4—216no—
465C22⋊C4—2928no—
466C22⋊C4—14616no—
467C42—28no—
468C42—28no—
469C42—14616no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×C4⋊Dic7314yes—
493C23C23×Dic7311yescharacteristic
494C23C2×D4×D7311yescharacteristic
495C23C2×D4×D7311yescharacteristic
496C23C22×D29211yescharacteristic
497C23C22×Dic14611yescharacteristic
498C23C2×#1192402711yescharacteristic
499C23C2×#1192402711yescharacteristic
500C23C2×#1192402812yes—
501C23C2×#1192402812yes—
502C23#1455663712yes—
503C23#1455663812yes—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—232no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—2922no—
525C2×C4—2922no—
526C2×C4—2922no—
527C2×C4—2922no—
528C2×C4—2924no—
529C2×C4—2924no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2924no—
535C2×C4—2924no—
536C2×C4—2924no—
537C2×C4—2924no—
538C2×C4—2922no—
539C2×C4—2922no—
540C2×C4—1468no—
541C2×C4—1468no—
542C2×C4—1464no—
543C2×C4—1464no—
544C2×C4C2×C4⋊Dic7314yes—
545C2×C4C2×C4⋊Dic7314yes—
546C2×C4#1455655018yescomplemented
547C2×C4D4×Dic7314yes—
548C2×C4D4×Dic7314yes—
549C2×C4#1455656114yes—
550C2×C4#1455656814yes—
551C2×C4#1455656814yes—
552C2×C4#1455657514yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22C22×C4⋊Dic7311yescharacteristic
565C22C2×#1455655012yes—
566C22C2×D4×Dic7311yescharacteristic
567C22C2×D4×Dic7311yescharacteristic
568C22#1798907012yes—
569C22C2×#1455656111yescharacteristic
570C22C2×#1455656811yescharacteristic
571C22C2×#1455656811yescharacteristic
572C22C2×#1455657511yescharacteristic
573C22#1798913711yescharacteristic
574C22#1798913711yescharacteristic
575C22#1798914412yes—
576C22#1798915611yescharacteristic
577C22#1798916311yescharacteristic
578C22#1798921512yes—
579C22#1798923312yes—
580C22#1798924212yes—
581C22#1798924612yes—
582C22#1798925812yes—
583C22#1798927712yes—
584C22#1798928312yes—
585C22#1798930212yes—
586C22#1798930712yes—
587C4—24no—
588C4—28no—
589C4—44no—
590C4—2924no—
591C4—2924no—
592C4—5842no—
593C4—5842no—
594C4#1798846218yes—
595C2—44no—
596C2#2261233811yescharacteristic
597C2C2×#1798846211yescharacteristic
598C2#2261255411yescharacteristic
599C2#2261286012yes—
600C2#2261288012yes—
601C2#2261355011yescharacteristic
602C2#2261355911yescharacteristic
603C2#2261362212yes—
604C2#2261364412yes—
605C2#2261438411yescharacteristic
606C2#2261439311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×C4×C22⋊C4×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×C4×C22⋊C4D73
hypocenterC73#13879
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17988457C2semidirect
3. C2×#17988453C2semidirect
6. #22608165C2semidirect
7. #22608161C2semidirect
8. #22608161C2semidirect
32. C4×C22⋊C4×C73C4semidirect
148. C22×C4×C73C4⋊C4semidirect
222. C2×C4×C73#117semidirect
313. C73#13879semidirect
332. C4×C22⋊C4Dic73semidirect
444. C22×C4C4⋊Dic73semidirect
546. C2×C4#14556550semidirect

Maximal subgroups

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

subgroupindex
C22×#145565502
C2×#179884572
C2×#179884532
C2×#179884472
#226081762
#226081652
#226081612
C2×C4×C22⋊C4×C732
#1387973