Finite Group Databasebeta

← #33144837

Subgroups — #33144837

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988628C211yesmaximal, characteristic
2#22613640C211yesmaximal, characteristic
3#22613629C211yesmaximal, characteristic
4#22611410C211yesmaximal, characteristic, complemented
5#22611405C211yesmaximal, characteristic, complemented
6#22611266C211yesmaximal, characteristic, complemented
7#22611261C211yesmaximal, characteristic, complemented
8#22610669C211yesmaximal, characteristic, complemented
9#22610665C211yesmaximal, characteristic, complemented
10#22610127C211yesmaximal, characteristic
11#22610122C211yesmaximal, characteristic
12#22610086C211yesmaximal, characteristic, complemented
13#22609961C211yesmaximal, characteristic
14#22609716C211yesmaximal, characteristic
15C2×#501×C73C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×#14556567—21no—
18C2×#14556549C2211yescharacteristic
19C2×C4⋊C4×D73C2211yescharacteristic
20#17988628—24no—
21C2×#14556525C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988522—21no—
30#17988520—21no—
31#17988509—21no—
32#17988506—21no—
33#17988486C2211yescharacteristic
34#17988486C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988483—21no—
38#17988468—21no—
39#17988458—21no—
40#17988456C2211yescharacteristic
41#17988456C2211yescharacteristic
42#17988444C2211yescharacteristic
43#17988444C2211yescharacteristic
44C4×#11924005C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988342C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4⋊Q8×C73C2211yescharacteristic
54C2×#125×C73C2211yescharacteristic
55#501×C73—24no—
56C2×#107×C73C2211yescharacteristic
57C2×#107×C73C2211yescharacteristic
58C2×#106×C73C2211yescharacteristic
59C2×#106×C73C2211yescharacteristic
60C2×#97×C73C2211yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73C2311yescharacteristic
63C2×#11924011—22no—
64C2×#11924011—22no—
65C2×#11924011Q814yescomplemented
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556567—44no—
69#14556549—22no—
70#14556549—41no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C4⋊C4×D73—22no—
85C4⋊C4×D73—41no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×#11924005D411yescharacteristic
90C2×#11924005D411yescharacteristic
91C2×C4×Dic73—21no—
92C2×C4×Dic73C2311yescharacteristic
93#14556525—22no—
94#14556524—22no—
95#14556523—22no—
96#14556522—22no—
97#14556522—41no—
98#14556522—41no—
99#14556503Q814yescomplemented
100C8×Dic73—22no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×Q8×C73—21no—
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C4⋊Q8×C73—22no—
113C4⋊Q8×C73—41no—
114#125×C73—22no—
115#125×C73—41no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#107×C73—22no—
125#107×C73D414yes—
126#106×C73—22no—
127#106×C73—22no—
128#97×C73—22no—
129C23×D73C2×D411yescharacteristic
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C2×C4×D73—21no—
138C2×C4×D73—21no—
139C2×C4×D73—21no—
140C2×C4×D73—21no—
141C2×C4×D73—42no—
142C2×C4×D73—42no—
143C2×C4×D73—81no—
144C2×C4×D73C2×Q812yes—
145C2×C4×D73C2×Q812yes—
146#11924011—28no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831C2×Q812yes—
152C2×#10596831C2×Q812yes—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—42no—
162C4⋊Dic73—42no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167#11924007—81no—
168#11924007—81no—
169#11924007—81no—
170#11924005—22no—
171#11924005—22no—
172#11924005—42no—
173#11924005—42no—
174#11924005—81no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—42no—
178C2×Q8×C73—42no—
179C2×Q8×C73—42no—
180C2×Q8×C73—81no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2411yescharacteristic
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73C2×D412yes—
191C2×C8×C73C2×Q812yes—
192C2×C8×C73C2×Q812yes—
193C2×C8×C73#4112yes—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73—42no—
203C4⋊C4×C73C2×D412yes—
204C4⋊C4×C73C2×D412yes—
205C42×C73—21no—
206C42×C73—21no—
207C23×C73C22×D411yescharacteristic
208C22×D73—21no—
209C22×D73—21no—
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—21no—
213C22×D73—21no—
214C22×D73C4⋊Q812yes—
215Q8×C73—84no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C22×Q811yescharacteristic
236C2×C4×C73C22×Q811yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73C4⋊Q812yes—
259C2×Dic73AGL(1,ℤ/8)12yes—
260C2×Dic73#13712yes—
261C4×D73—22no—
262C4×D73—22no—
263C4×D73—22no—
264C4×D73—22no—
265C4×D73—84no—
266C8×C73—22no—
267C8×C73C22⋊Q814yescomplemented
268#10596831—22no—
269#10596831—22no—
270C22×C73C2×C4⋊Q811yescharacteristic
271C22×C73D4211yescharacteristic
272C22×C73#54711yescharacteristic
273C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C73C2×#13711yescharacteristic
275C22×C73#57711yescharacteristic
276C22×C73#57711yescharacteristic
277C2×D73—22no—
278C2×D73—22no—
279C2×D73—22no—
280C2×D73—41no—
281C2×D73—41no—
282C2×D73—41no—
283C2×D73—41no—
284C4×C73—42no—
285C4×C73—41no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—81no—
289C4×C73—81no—
290C4×C73C2×C22⋊Q811yescharacteristic
291C4×C73#54411yescharacteristic
292C4×C73D4×Q811yescharacteristic
293C4×C73#55411yescharacteristic
294Dic73—22no—
295Dic73—22no—
296Dic73—22no—
297Dic73—81no—
298Dic73—81no—
299Dic73—81no—
300Dic73—81no—
301#26078—731nomaximal, Sylow
302C2×C73#244011yescharacteristic
303C2×C73#316311yescharacteristic
304C2×C73#316511yescharacteristic
305C2×C73#327211yescharacteristic
306C2×C73#327811yescharacteristic
307C2×C73#334411yescharacteristic
308C2×C73#334511yescharacteristic
309D73—42no—
310C2×#501D7311yescharacteristic, complemented
311C2×#428—731no—
312#2441—731no—
313#2440—731no—
314#2084—731no—
315#2083—731no—
316#2059—731no—
317#2058—731no—
318#1943—731no—
319#1942—731no—
320#1854—731no—
321#1853—731no—
322#1844—731no—
323#1809—731no—
324#1753—731no—
325C73#2607811yescharacteristic, complemented, Sylow
326C22×M16—731no—
327C2×C4⋊Q8C2×D7311yescharacteristic
328C2×#125C2×D7311yescharacteristic
329C2×C22⋊Q8—1461no—
330C2×#117—731no—
331C22×C4⋊C4—731no—
332#501—24no—
333#428—1464no—
334C2×#107—731no—
335C2×#107—731no—
336C2×#107C2×D7311yescharacteristic
337C2×#107C2×D7311yescharacteristic
338C2×#106—731no—
339C2×#106—731no—
340C2×#106—731no—
341C2×#106—731no—
342C2×#106C2×D7311yescharacteristic
343C2×#106C2×D7311yescharacteristic
344C2×C22⋊C8—731no—
345C2×#97C2×D7311yescharacteristic
346C2×C4×C8—731no—
347#399—1461no—
348#398—1461no—
349#396—1461no—
350#395—1461no—
351#389—731no—
352#389—731no—
353#389—731no—
354#389—731no—
355#389—1461no—
356#387—1461no—
357#385—1461no—
358#384—731no—
359#384—731no—
360#380—731no—
361#380—731no—
362C4×C22⋊C4—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#336—731no—
370#336—731no—
371C22×Q8—21no—
372C23×C4—731no—
373C23×C4—1461no—
374C2×M16—734no—
375C2×M16—1462no—
376C2×M16—1462no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8C22×D7311yescharacteristic
380C22×C8C22×D7311yescharacteristic
381C4⋊Q8—22no—
382C4⋊Q8—41no—
383#125—22no—
384#125—41no—
385C22⋊Q8—2924no—
386#117—2921no—
387#117—1462no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—2921no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1462no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C4⋊C4C22×D7311yescharacteristic
410C2×C22⋊C4—731no—
411C2×C22⋊C4—731no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4—1461no—
414C2×C22⋊C4—1461no—
415C2×C42—731no—
416C2×C42—1461no—
417C2×C42C22×D7311yescharacteristic
418#107—22no—
419#107—1462no—
420#107—1462no—
421#107C22⋊D7314yes—
422#106—22no—
423#106—22no—
424#106—2921no—
425#106—2921no—
426#106—1462no—
427#106—1462no—
428C22⋊C8—734no—
429#97—22no—
430C4×C8—1462no—
431#95—731no—
432#95—731no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439C24—731no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—81no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—2922no—
448C22×C4—2922no—
449C22×C4—5841no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—732no—
454C22×C4—732no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4D4×D7311yescharacteristic
464C22×C4D4×D7311yescharacteristic
465C22×C4C23×D7311yescharacteristic
466M16—1468no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—732no—
470C2×C8—732no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8C2×C22⋊D7312yes—
476C2×C8Q8×D7312yes—
477C2×C8Q8×D7312yes—
478C2×C8#1192402912yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—2922no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—2922no—
494C4⋊C4—5841no—
495C4⋊C4—5841no—
496C4⋊C4—5841no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4C2×C22⋊D7312yes—
506C4⋊C4C2×C22⋊D7312yes—
507C22⋊C4—2922no—
508C22⋊C4—2922no—
509C22⋊C4—5841no—
510C22⋊C4—1462no—
511C22⋊C4—1462no—
512C42—21no—
513C42—21no—
514C42—2922no—
515C42—1461no—
516C42—1461no—
517C23—732no—
518C23—1461no—
519C23—1461no—
520C23—1461no—
521C23—1461no—
522C23—1461no—
523C23—1461no—
524C23C2×D4×D7311yescharacteristic
525Q8—84no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—5844no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4—1462no—
569C2×C4—1462no—
570C2×C4C22×C22⋊D7311yescharacteristic
571C2×C4C2×Q8×D7311yescharacteristic
572C2×C4C2×Q8×D7311yescharacteristic
573C2×C4C2×#1192402911yescharacteristic
574C2×C4#1455663611yescharacteristic
575C2×C4#1455663911yescharacteristic
576C8—22no—
577C8—1462no—
578C8—1462no—
579C8#1455656714yescomplemented
580C22—2921no—
581C22—2921no—
582C22—2921no—
583C22—2921no—
584C22—1462no—
585C22—1462no—
586C22—1462no—
587C22C4⋊Q8×D7311yescharacteristic
588C22D4×C22⋊D7311yescharacteristic
589C22#1798924011yescharacteristic
590C22AGL(1,ℤ/8)×D7311yescharacteristic
591C22#137×D7311yescharacteristic
592C22#1798939611yescharacteristic
593C22#1798939711yescharacteristic
594C4—41no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—5841no—
601C4—5841no—
602C4—5841no—
603C4—5841no—
604C4—1462no—
605C4—1462no—
606C4—1462no—
607C4C2×#1455656711yescharacteristic
608C4#1798921411yescharacteristic
609C4Q8×C22⋊D7311yescharacteristic
610C4#1798929811yescharacteristic
611C2—2922no—
612C2#2261362911yescharacteristic
613C2#2262045811yescharacteristic
614C2#2262047611yescharacteristic
615C2#2262152511yescharacteristic
616C2#2262159111yescharacteristic
617C2#2262238511yescharacteristic
618C2#2262240011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#501×C73C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
hypercenterC2×#501D73
hypocenterC73#26078
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
4. #22611410C2semidirect
5. #22611405C2semidirect
6. #22611266C2semidirect
7. #22611261C2semidirect
8. #22610669C2semidirect
9. #22610665C2semidirect
12. #22610086C2semidirect
15. C2×#501×C73C2semidirect
65. C2×#11924011Q8semidirect
99. #14556503Q8semidirect
267. C8×C73C22⋊Q8semidirect
310. C2×#501D73semidirect
325. C73#26078semidirect
579. C8#14556567semidirect

Maximal subgroups

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

subgroupindex
C2×#179886282
#226136402
#226136292
#226114102
#226114052
#226112662
#226112612
#226106692
#226106652
#226101272
#226101222
#226100862
#226099612
#226097162
C2×#501×C732
#2607873