Finite Group Databasebeta

← #33127262

Subgroups — #33127262

Order 18688.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988894C211yesmaximal, characteristic, complemented
2C2×#17988644C211yesmaximal, characteristic
3#22618241C211yesmaximal, characteristic
4#22611368C211yesmaximal, characteristic
5#22611362C211yesmaximal, characteristic
6#22611131C211yesmaximal, characteristic
7#22611124C211yesmaximal, characteristic
8#22610888C211yesmaximal, characteristic, complemented
9#22610699C211yesmaximal, characteristic, complemented
10#22610671C211yesmaximal, characteristic, complemented
11#22610666C211yesmaximal, characteristic, complemented
12#22610624C211yesmaximal, characteristic, complemented
13#22610624C211yesmaximal, characteristic, complemented
14#22609946C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C73C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17#17988894—24no—
18#17988644—24no—
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988471—21no—
36#17988471—21no—
37#17988460—21no—
38#17988460—21no—
39#17988453C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×C22⋊Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C22×C42×C73C2211yescharacteristic
53#391×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#387×C73—21no—
56#385×C73—21no—
57#380×C73C2211yescharacteristic
58C4×C4⋊C4×C73—21no—
59C4×C4⋊C4×C73C2211yescharacteristic
60C4×C22⋊C4×C73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556583—22no—
66#14556583—22no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77#14556524—22no—
78#14556524—22no—
79#14556522—22no—
80#14556522—22no—
81C4⋊#10596831—22no—
82C4⋊#10596831D414yes—
83#14556515—22no—
84#14556515—22no—
85#14556515—22no—
86#14556515—22no—
87C22⋊#10596831—22no—
88C22⋊#10596831—22no—
89#14556502—22no—
90#14556502—22no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×Q8×C73C2311yescharacteristic
98C23×C4×C73D411yescharacteristic
99C23×C4×C73D411yescharacteristic
100C23×C4×C73C2311yescharacteristic
101C22⋊Q8×C73—22no—
102C22⋊Q8×C73—22no—
103C4×Q8×C73—22no—
104C4×Q8×C73—22no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C42×C73—22no—
114C2×C42×C73—22no—
115C2×C42×C73—21no—
116C2×C42×C73—21no—
117C2×C42×C73D411yescharacteristic
118C2×C42×C73D411yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831C2×D412yes—
141C2×#10596831C2×D412yes—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150#11924007—42no—
151#11924007—42no—
152#11924007—42no—
153#11924007—42no—
154C22⋊Dic73—42no—
155C22⋊Dic73—42no—
156C24×C73C2×D411yescharacteristic
157C2×Q8×C73—21no—
158C2×Q8×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73#4112yes—
163C22×C4×C73—22no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—41no—
177C22×C4×C73—41no—
178C22×C4×C73—41no—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C4⋊C4×C73—24no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—42no—
192C4⋊C4×C73—42no—
193C4⋊C4×C73#4112yes—
194C22⋊C4×C73—24no—
195C22⋊C4×C73—42no—
196C42×C73—22no—
197C42×C73—22no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—42no—
201C42×C73—42no—
202C42×C73C2×D412yes—
203C42×C73#4112yes—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73#13712yes—
211C23×C73C22×D411yescharacteristic
212Q8×C73—22no—
213Q8×C73—22no—
214Q8×C73—41no—
215Q8×C73—41no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—44no—
232C2×C4×C73—44no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73#12112yes—
248C2×C4×C73#12112yes—
249C2×C4×C73C22×D411yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C732−1+411yescharacteristic
254C2×C4×C732−1+411yescharacteristic
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267#10596831—22no—
268#10596831—22no—
269#10596831—41no—
270#10596831—41no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—22no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73C2×#12111yescharacteristic
279C22×C73#52511yescharacteristic
280C22×C73#53611yescharacteristic
281C22×C73C2×#13711yescharacteristic
282C22×C73#57511yescharacteristic
283C22×C73#57511yescharacteristic
284C22×C73#57511yescharacteristic
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—42no—
288C4×C73—41no—
289C4×C73—42no—
290C4×C73—41no—
291C4×C73—41no—
292C4×C73—81no—
293C4×C73—81no—
294C4×C73#54711yescharacteristic
295C4×C73#54811yescharacteristic
296C4×C73#55211yescharacteristic
297C4×C73#55611yescharacteristic
298Dic73—81no—
299Dic73—81no—
300Dic73—81no—
301Dic73—81no—
302#24707—731nomaximal, Sylow
303C2×C73—42no—
304C2×C73#241511yescharacteristic
305C2×C73#305411yescharacteristic
306C2×C73#305611yescharacteristic
307C2×C73#311111yescharacteristic
308C2×C73#311711yescharacteristic
309C2×C73#323611yescharacteristic
310C2×C73#324011yescharacteristic
311C2×#479—731no—
312C2×#432—731no—
313#2851—731no—
314C4×C22⋊Q8D7311yescharacteristic
315#2077—731no—
316#2076—731no—
317#2036—731no—
318#2035—731no—
319#1989—731no—
320#1949—731no—
321#1944—731no—
322#1943—731no—
323#1935—731no—
324#1935—731no—
325#1805—731no—
326C73#2470711yescharacteristic, complemented, Sylow
327C2×#125—731no—
328C2×C22⋊Q8C2×D7311yescharacteristic
329C2×C4×Q8C2×D7311yescharacteristic
330C22×C42C2×D7311yescharacteristic
331#479—1464no—
332#432—1464no—
333C2×#107—731no—
334C2×#107—731no—
335C2×#106—731no—
336C2×#106—731no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×#103—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#103—731no—
343C2×C22⋊C8—731no—
344C2×C22⋊C8—731no—
345C2×#97—731no—
346C2×#97—731no—
347#391C2×D7311yescharacteristic
348#389—731no—
349#389—731no—
350#389C2×D7311yescharacteristic
351#387—21no—
352#387—1461no—
353#387—1461no—
354#385—21no—
355#385—1461no—
356#385—1461no—
357#382—731no—
358#380—731no—
359#380—731no—
360#380C2×D7311yescharacteristic
361C4×C4⋊C4—21no—
362C4×C4⋊C4C2×D7311yescharacteristic
363C4×C22⋊C4—21no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372C22×Q8C22×D7311yescharacteristic
373C23×C4C22⋊D7311yescharacteristic
374C23×C4C22⋊D7311yescharacteristic
375C23×C4C22×D7311yescharacteristic
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—731no—
380#125—1462no—
381#125—1462no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C4×Q8—22no—
385C4×Q8—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4—1461no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42C22⋊D7311yescharacteristic
409C2×C42C22⋊D7311yescharacteristic
410C2×C42C22×D7311yescharacteristic
411C2×C42C22×D7311yescharacteristic
412#107—1462no—
413#107—1462no—
414#106—1462no—
415#106—1462no—
416C4⋊C8—734no—
417C4⋊C8—1462no—
418#103—1462no—
419#103—1462no—
420#103—1462no—
421#103—1462no—
422C22⋊C8—1462no—
423C22⋊C8—1462no—
424#97—1462no—
425#97—1462no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437C24C2×C22⋊D7311yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8#1192402812yes—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4C2×C22⋊D7311yescharacteristic
465C22×C4C2×C22⋊D7311yescharacteristic
466C22×C4C2×C22⋊D7311yescharacteristic
467C22×C4C2×C22⋊D7311yescharacteristic
468C22×C4C2×C22⋊D7311yescharacteristic
469C22×C4C23×D7311yescharacteristic
470C2×C8—732no—
471C2×C8—732no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—2922no—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—2922no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4#1192402812yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—2922no—
506C22⋊C4—2922no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42D4×D7312yes—
514C42#1192402712yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1455662212yes—
522C23C22×C22⋊D7311yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4#1455656312yes—
571C2×C4#1455656312yes—
572C2×C4C2×D4×D7311yescharacteristic
573C2×C4C2×#1192402711yescharacteristic
574C2×C4C2×#1192402811yescharacteristic
575C2×C4C2×#1192402811yescharacteristic
576C2×C4#1455663811yescharacteristic
577C2×C4#1455663811yescharacteristic
578C8—2921no—
579C8—2921no—
580C8—1462no—
581C8—1462no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1455656311yescharacteristic
590C22#1798909811yescharacteristic
591C22#1798915811yescharacteristic
592C22C2×#1455662211yescharacteristic
593C22#1798938511yescharacteristic
594C22#1798938511yescharacteristic
595C22#1798938511yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—5841no—
606C4—5841no—
607C4—5841no—
608C4—5841no—
609C4#1798924211yescharacteristic
610C4#1798924611yescharacteristic
611C4#1798927711yescharacteristic
612C4#1798930711yescharacteristic
613C2—42no—
614C2#2261336011yescharacteristic
615C2#2261953711yescharacteristic
616C2#2261955711yescharacteristic
617C2#2262004011yescharacteristic
618C2#2262010211yescharacteristic
619C2#2262113311yescharacteristic
620C2#2262116711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×C22⋊Q8×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenterC4×C22⋊Q8D73
hypocenterC73#24707
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17988894C2semidirect
8. #22610888C2semidirect
9. #22610699C2semidirect
10. #22610671C2semidirect
11. #22610666C2semidirect
12. #22610624C2semidirect
13. #22610624C2semidirect
14. #22609946C2semidirect
326. C73#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#179888942
C2×#179886442
#226182412
#226113682
#226113622
#226111312
#226111242
#226108882
#226106992
#226106712
#226106662
#226106242
#226099462
C4×C22⋊Q8×C732
#2470773