Finite Group Databasebeta

← #33102516

Subgroups — #33102516

Order 18688.

1292 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic146C211yesmaximal, characteristic
2C2×#17988716C212yesmaximal, complemented
3#22610870C212yesmaximal, complemented
4#22610869C212yesmaximal, complemented
5#22610865C212yesmaximal, complemented
6#22609909C211yesmaximal, characteristic
7#22609909C211yesmaximal, characteristic
8#22609695C211yesmaximal, characteristic
9#22609693C211yesmaximal, characteristic
10C22⋊C8×Dic73C211yesmaximal, characteristic
11C73×#1797C211yesmaximal, characteristic
12C23×Dic146C411yescharacteristic
13C2×C4×Dic146—28no—
14C2×C4×Dic146C411yescharacteristic
15C2×C4×Dic146C411yescharacteristic
16C2×C4×Dic146C411yescharacteristic
17C2×C4×Dic146C411yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21C2×C4×Dic146C2211yescharacteristic
22C22×C4⋊Dic73C411yescharacteristic
23C22×#11924007C411yescharacteristic
24C22×#11924007C411yescharacteristic
25C22×C4×Dic73C2211yescharacteristic
26C22×C4×Dic73C2211yescharacteristic
27#17988716—28no—
28C2×#14556520C2212yes—
29C2×#14556520C2212yes—
30C2×C22⋊#10596831C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32C2×#14556502C2212yes—
33C2×#14556501C2212yes—
34C2×C8×Dic73C2212yes—
35#17988342C2212yes—
36#17988341C2212yes—
37#17988341C2212yes—
38#17988341C2212yes—
39#17988341C2212yes—
40C22×C42×C73C2211yescharacteristic
41C2×C4⋊C8×C73C2212yes—
42C2×C22⋊C8×C73C2211yescharacteristic
43C2×C22⋊C8×C73C2211yescharacteristic
44#336×C73C2212yes—
45C22×Dic146—24no—
46C22×Dic146—24no—
47C22×Dic146—24no—
48C22×Dic146C2×C411yescharacteristic
49C22×Dic146C2×C411yescharacteristic
50C22×Dic146C2×C411yescharacteristic
51C22×Dic146C2×C411yescharacteristic
52C23×Dic73C2×C411yescharacteristic
53C23×Dic73C2×C411yescharacteristic
54C23×Dic73C2×C411yescharacteristic
55C23×Dic73C2×C411yescharacteristic
56C22×#10596831C2312yes—
57C22×#10596831C2312yes—
58C4×Dic146—24no—
59C4×Dic146—24no—
60C4×Dic146—24no—
61C4×Dic146—24no—
62C4×Dic146—24no—
63C4×Dic146—24no—
64C4×Dic146—24no—
65C4×Dic146D418yes—
66C2×C4⋊Dic73—24no—
67C2×C4⋊Dic73C2×C411yescharacteristic
68C2×C4⋊Dic73C2×C411yescharacteristic
69C2×C4⋊Dic73C2×C411yescharacteristic
70C2×C4⋊Dic73C2×C411yescharacteristic
71C2×#11924007—24no—
72C2×#11924007—24no—
73C2×#11924007C2×C411yescharacteristic
74C2×#11924007C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2×C411yescharacteristic
79C2×#11924007C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×C4×Dic73—24no—
82C2×C4×Dic73—24no—
83C2×C4×Dic73C2×C411yescharacteristic
84C2×C4×Dic73C2×C411yescharacteristic
85C2×C4×Dic73C2×C411yescharacteristic
86C2×C4×Dic73C2×C411yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88C2×C4×Dic73C2311yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556520—24no—
92#14556520—24no—
93C22⋊#10596831—22no—
94C22⋊#10596831Q814yescomplemented
95#14556502—24no—
96#14556501—24no—
97C8×Dic73—24no—
98C23×C4×C73C2×C411yescharacteristic
99C23×C4×C73C2×C411yescharacteristic
100C23×C4×C73C2311yescharacteristic
101C22×C8×C73C2312yes—
102C22×C8×C73C2312yes—
103C2×C42×C73—24no—
104C2×C42×C73C2×C411yescharacteristic
105C2×C42×C73C2×C411yescharacteristic
106C2×C42×C73C2311yescharacteristic
107C2×C42×C73C2311yescharacteristic
108C4⋊C8×C73—24no—
109C22⋊C8×C73—22no—
110C22⋊C8×C73—22no—
111C2×Dic146—232no—
112C2×Dic146—22no—
113C2×Dic146—22no—
114C2×Dic146—22no—
115C2×Dic146—22no—
116C2×Dic146—22no—
117C2×Dic146—22no—
118C2×Dic146—22no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146—22no—
122C2×Dic146C22⋊C414yes—
123C2×Dic146M1614yes—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73—41no—
134C22×Dic73—41no—
135C22×Dic73M1614yes—
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C22×Dic73C22×C411yescharacteristic
143C22×Dic73C22×C411yescharacteristic
144C2×#10596831—22no—
145C2×#10596831—22no—
146C2×#10596831—22no—
147C2×#10596831—22no—
148C2×#10596831—22no—
149C2×#10596831C2×Q814yescomplemented
150C4⋊Dic73—22no—
151C4⋊Dic73—22no—
152C4⋊Dic73—22no—
153C4⋊Dic73C22⋊C414yes—
154#11924007—22no—
155#11924007—22no—
156#11924007—22no—
157#11924007—22no—
158#11924007—22no—
159#11924007—22no—
160#11924007C22⋊C414yes—
161#11924007C22⋊C414yes—
162C4×Dic73—22no—
163C4×Dic73—22no—
164C4×Dic73—22no—
165C4×Dic73—22no—
166C4×Dic73—22no—
167C4×Dic73—22no—
168C4×Dic73C2×D414yes—
169C4×Dic73C2×D414yes—
170C24×C73C22×C411yescharacteristic
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—41no—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C411yescharacteristic
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C22×C4×C73C22×C411yescharacteristic
184C22×C4×C73C2×Q812yes—
185C22×C4×C73#4112yes—
186C22×C4×C73C2411yescharacteristic
187C2×C8×C73—22no—
188C2×C8×C73—22no—
189C2×C8×C73—22no—
190C2×C8×C73—22no—
191C2×C8×C73—22no—
192C2×C8×C73—22no—
193C42×C73—22no—
194C42×C73—22no—
195C42×C73—22no—
196C42×C73C2×D414yes—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73C4×Q812yes—
202C23×C73C2×M1612yes—
203C23×C73#13112yes—
204C23×C73C23×C411yescharacteristic
205C2×C4×C73—28no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—44no—
220C2×C4×C73C2×C22⋊C412yes—
221C2×C4×C73C2×C22⋊C412yes—
222C2×C4×C73C4×Q812yes—
223C2×C4×C73C2×M1612yes—
224C2×C4×C73C22×D412yes—
225C2×C4×C73C22×Q811yescharacteristic
226C2×C4×C73C2×#4111yescharacteristic
227C2×C4×C732−1+412yes—
228C2×Dic73—28no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—28no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—44no—
247C2×Dic73—44no—
248C2×Dic73C2×C22⋊C412yes—
249C2×Dic73C2×C22⋊C412yes—
250C2×Dic73C2×C22⋊C412yes—
251C2×Dic73C2×C22⋊C412yes—
252C2×Dic73C2×M1612yes—
253C2×Dic73C2×M1612yes—
254C2×Dic73C2×M1612yes—
255C2×Dic73C2×M1612yes—
256Dic146—28no—
257Dic146—28no—
258Dic146—28no—
259Dic146—28no—
260C8×C73—42no—
261C8×C73—42no—
262#10596831—24no—
263#10596831—42no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73#44614yescomplemented
271C22×C73C22×C22⋊C411yescharacteristic
272C22×C73C2×C4×Q811yescharacteristic
273C22×C73#51911yescharacteristic
274C22×C73C22×M1611yescharacteristic
275C22×C73C22×M1611yescharacteristic
276C22×C73C2×#13111yescharacteristic
277C22×C73#56811yescharacteristic
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—22no—
281C4×C73—22no—
282C4×C73—41no—
283C4×C73#54812yes—
284C4×C73D4×Q812yes—
285Dic73—22no—
286Dic73—22no—
287Dic73—22no—
288Dic73—22no—
289Dic73—22no—
290Dic73—41no—
291Dic73—41no—
292Dic73#40714yes—
293#22510—731nomaximal, Sylow
294C2×C73—24no—
295C2×C73Q8×C22⋊C411yescharacteristic
296C2×C73C2×#40711yescharacteristic
297C2×C73#288011yescharacteristic
298C2×C73C2×#44611yescharacteristic
299C2×C73Q8×M1611yescharacteristic
300C2×C73#298311yescharacteristic
301C2×C73#298411yescharacteristic
302C22×C4×Q8—731no—
303C2×#446—732no—
304#1985—732no—
305#1985—732no—
306#1984—732no—
307#1797—731no—
308#1797—731no—
309#1797D7311yescharacteristic
310#1749—731no—
311#1749—731no—
312C4×C22⋊C8—731no—
313C73#2251011yescharacteristic, complemented, Sylow
314C23×Q8—731no—
315C2×C4×Q8—731no—
316C2×C4×Q8—731no—
317C2×C4×Q8—731no—
318C2×C4×Q8—731no—
319C2×C4×Q8—731no—
320C2×C4×Q8—731no—
321C2×C4×Q8—731no—
322C2×C4×Q8—731no—
323C2×C4×Q8—1468no—
324C22×C4⋊C4—731no—
325C22×C4⋊C4—731no—
326C22×C4⋊C4—731no—
327C22×C42—731no—
328C22×C42—731no—
329C22×C42C2×D7311yescharacteristic
330#446—1468no—
331C2×C4⋊C8—732no—
332C2×C4⋊C8—732no—
333C2×C4⋊C8C2×D7312yes—
334C2×C22⋊C8—731no—
335C2×C22⋊C8—731no—
336C2×C22⋊C8C2×D7311yescharacteristic
337C2×C22⋊C8C2×D7311yescharacteristic
338C2×#97—732no—
339C2×#97—732no—
340C2×C4×C8—732no—
341#336—732no—
342#336—732no—
343#336—732no—
344#336—732no—
345#336—732no—
346#336C2×D7312yes—
347C22×Q8—731no—
348C22×Q8—731no—
349C22×Q8—731no—
350C22×Q8—731no—
351C22×Q8—1464no—
352C22×Q8—1464no—
353C22×Q8—1464no—
354C23×C4—731no—
355C23×C4—731no—
356C23×C4—731no—
357C23×C4—731no—
358C23×C4C4×D7311yescharacteristic
359C23×C4C4×D7311yescharacteristic
360C23×C4C22×D7311yescharacteristic
361C22×C8—732no—
362C22×C8—732no—
363C22×C8C22×D7312yes—
364C22×C8C22×D7312yes—
365C4×Q8—738no—
366C4×Q8—1464no—
367C4×Q8—1464no—
368C4×Q8—1464no—
369C4×Q8—1464no—
370C4×Q8—1464no—
371C4×Q8—1464no—
372C4×Q8—1464no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—1464no—
386C2×C4⋊C4—1464no—
387C2×C4⋊C4—1464no—
388C2×C42—24no—
389C2×C42—731no—
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—731no—
393C2×C42—731no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—1464no—
398C2×C42—1464no—
399C2×C42C4×D7311yescharacteristic
400C2×C42C4×D7311yescharacteristic
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7311yescharacteristic
403C4⋊C8—24no—
404C4⋊C8—1464no—
405C4⋊C8—1464no—
406C22⋊C8—22no—
407C22⋊C8—22no—
408C22⋊C8—734no—
409C22⋊C8—1462no—
410#97—1464no—
411#97—1464no—
412C4×C8—1464no—
413C24C2×C4×D7311yescharacteristic
414C2×Q8—734no—
415C2×Q8—734no—
416C2×Q8—1462no—
417C2×Q8—1462no—
418C2×Q8—1462no—
419C2×Q8—1462no—
420C2×Q8—1462no—
421C2×Q8—1462no—
422C2×Q8—1462no—
423C2×Q8—1462no—
424C2×Q8—14632no—
425C2×Q8—1462no—
426C2×Q8—1462no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—2921no—
436C22×C4—2921no—
437C22×C4—731no—
438C22×C4—734no—
439C22×C4—731no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4C2×C4×D7311yescharacteristic
456C22×C4C2×C4×D7311yescharacteristic
457C22×C4C2×C4×D7311yescharacteristic
458C22×C4C2×C4×D7311yescharacteristic
459C22×C4C2×C4×D7311yescharacteristic
460C22×C4Q8×D7312yes—
461C22×C4#1192402712yes—
462C22×C4C23×D7311yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—734no—
470C2×C8—1462no—
471C2×C8—1462no—
472C2×C8—1462no—
473C2×C8—1462no—
474C2×C8—1462no—
475C4⋊C4—734no—
476C4⋊C4—734no—
477C4⋊C4—734no—
478C4⋊C4—1462no—
479C4⋊C4—1462no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1462no—
483C4⋊C4—1462no—
484C4⋊C4—1462no—
485C4⋊C4—1462no—
486C4⋊C4—1462no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—734no—
491C42—734no—
492C42—1462no—
493C42—1462no—
494C42—1462no—
495C42—1462no—
496C42—1462no—
497C42—1462no—
498C42D4×D7314yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23#1455655612yes—
504C23M16×D7312yes—
505C23#1455659712yes—
506C23C22×C4×D7311yescharacteristic
507Q8—1468no—
508Q8—1468no—
509Q8—1468no—
510Q8—1468no—
511C2×C4—28no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—44no—
526C2×C4—2924no—
527C2×C4—2924no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—732no—
531C2×C4—732no—
532C2×C4—732no—
533C2×C4—732no—
534C2×C4—732no—
535C2×C4—732no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1468no—
543C2×C4—1468no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4C22⋊C4×D7312yes—
555C2×C4C22⋊C4×D7312yes—
556C2×C4#1455655612yes—
557C2×C4M16×D7312yes—
558C2×C4C2×D4×D7312yes—
559C2×C4C2×Q8×D7311yescharacteristic
560C2×C4C2×#1192402711yescharacteristic
561C2×C4#1455663812yes—
562C8—42no—
563C8—42no—
564C8—2922no—
565C8—1464no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22#1798871614yescomplemented
573C22C2×C22⋊C4×D7311yescharacteristic
574C22C2×#1455655611yescharacteristic
575C22#1798906411yescharacteristic
576C22C2×M16×D7311yescharacteristic
577C22C2×M16×D7311yescharacteristic
578C22C2×#1455659711yescharacteristic
579C22#1798935311yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—2921no—
586C4—2921no—
587C4—734no—
588C4—1462no—
589C4—1462no—
590C4—1462no—
591C4—1462no—
592C4—1462no—
593C4#1798924312yes—
594C4#1798925012yes—
595C2—24no—
596C2#2261276311yescharacteristic
597C2#407×D7311yescharacteristic
598C2#2261837711yescharacteristic
599C2C2×#1798871611yescharacteristic
600C2#2261888211yescharacteristic
601C2#2261901511yescharacteristic
602C2#2261901811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1797C2
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenter#1797D73
hypocenterC73#22510
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#17988716C2semidirect
3. #22610870C2semidirect
4. #22610869C2semidirect
5. #22610865C2semidirect
94. C22⋊#10596831Q8semidirect
149. C2×#10596831C2×Q8semidirect
270. C22×C73#446semidirect
313. C73#22510semidirect
572. C22#17988716semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1462
C2×#179887162
#226108702
#226108692
#226108652
#226099092
#226096952
#226096932
C22⋊C8×Dic732
C73×#17972
#2251073