Finite Group Databasebeta

← #33056580

Subgroups — #33056580

Order 18688.

594 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620535C211yesmaximal, characteristic
2#22618908C211yesmaximal, characteristic
3#22618860C211yesmaximal, characteristic, complemented
4#22608840C211yesmaximal, characteristic, complemented
5#22608837C211yesmaximal, characteristic, complemented
6#22608836C211yesmaximal, characteristic, complemented
7#22608832C211yesmaximal, characteristic, complemented
8C8×#11924018C211yesmaximal, characteristic, complemented
9C8×#11924017C211yesmaximal, characteristic, complemented
10C8×#11924015C211yesmaximal, characteristic, complemented
11#22608697C211yesmaximal, characteristic, complemented
12#22608402C211yesmaximal, characteristic, complemented
13#22608395C211yesmaximal, characteristic
14C8×#11924011C211yesmaximal, characteristic, complemented
15C8×Q16×C73C211yesmaximal, characteristic, complemented
16#17989141C2211yescharacteristic
17#17989124C2211yescharacteristic
18#17988995C411yescharacteristic
19#17988980C411yescharacteristic
20#17988954C411yescharacteristic
21#17988948C411yescharacteristic
22#17988844C411yescharacteristic
23#17988842C411yescharacteristic
24#17988836C411yescharacteristic
25#17988834C411yescharacteristic
26#17988718C2211yescharacteristic
27#17988718C2211yescharacteristic
28#17988716—21no—
29#17988714C2211yescharacteristic
30C8×Dic146—21no—
31C8×Dic146C2211yescharacteristic
32C4×#11924018C2211yescharacteristic
33C4×#11924017C2211yescharacteristic
34C4×#11924015C2211yescharacteristic
35#17988664C2211yescharacteristic
36#17988657—21no—
37C8×D292—21no—
38C8×D292C2211yescharacteristic
39#17988647—21no—
40#17988643C2211yescharacteristic
41#17988638C2211yescharacteristic
42#17988612—21no—
43#17988609C2211yescharacteristic
44#17988541—21no—
45#17988541—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988338C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52Q8⋊#10596831C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54#17988317C2211yescharacteristic
55#17988316C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988308C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60C8×Q8×C73C2211yescharacteristic
61C8×Q8×C73C2211yescharacteristic
62C4×Q16×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65Q8⋊C8×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic
67C2×#11924018C2×C411yescharacteristic
68C2×#11924017C2×C411yescharacteristic
69C2×#11924015C2×C411yescharacteristic
70C2×#11924011—21no—
71C2×#11924011—21no—
72C2×#11924011C2×C411yescharacteristic
73C2×#11924011C2×C411yescharacteristic
74C2×C8×D73—21no—
75C2×C8×D73D411yescharacteristic
76C2×C8×D73D411yescharacteristic
77#14556590C2×C411yescharacteristic
78#14556589C2×C411yescharacteristic
79#14556586—21no—
80#14556582C2×C411yescharacteristic
81#14556581—21no—
82#14556580C2×C411yescharacteristic
83#14556576—21no—
84#14556571C2×C411yescharacteristic
85#14556569—21no—
86#14556567C2×C411yescharacteristic
87#14556567C2×C411yescharacteristic
88#14556566C2×C411yescharacteristic
89C4×Dic146—21no—
90C4×Dic146C2311yescharacteristic
91C4×D292—21no—
92C4×D292C2311yescharacteristic
93#14556548—21no—
94#14556548—21no—
95C42×D73C2311yescharacteristic
96#14556524C2×C411yescharacteristic
97#14556523C2×C411yescharacteristic
98#14556522C2×C411yescharacteristic
99C4⋊#10596831—21no—
100C4⋊#10596831—21no—
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556516C2×C411yescharacteristic
108#14556515C2×C411yescharacteristic
109#14556515C2×C411yescharacteristic
110#14556514C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556511C2×C411yescharacteristic
113#14556503—21no—
114#14556503—21no—
115#14556503—21no—
116#14556503—21no—
117#14556502—21no—
118#14556502—21no—
119#14556501—21no—
120#14556501—21no—
121#14556501C2×C411yescharacteristic
122#14556501C2×C411yescharacteristic
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C8×Dic73—21no—
127C8×Dic73D411yescharacteristic
128C8×Dic73D411yescharacteristic
129C2×Q16×C73C2×C411yescharacteristic
130C4×Q8×C73C2311yescharacteristic
131C4×Q8×C73C2311yescharacteristic
132#107×C73C2×C411yescharacteristic
133C4⋊C8×C73—21no—
134C4⋊C8×C73—21no—
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#103×C73C2×C411yescharacteristic
138#103×C73C2×C411yescharacteristic
139C4×C8×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×Dic146—21no—
145C2×Dic146C22×C411yescharacteristic
146C2×D292—21no—
147C2×D292C22×C411yescharacteristic
148C2×C4×D73—21no—
149C2×C4×D73C22×C411yescharacteristic
150C2×C4×D73C2×D411yescharacteristic
151C2×C4×D73#4111yescharacteristic
152#11924018—22no—
153#11924017—22no—
154#11924015—22no—
155#11924011—22no—
156#11924011—22no—
157#11924011—42no—
158#11924011—42no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831C22×C411yescharacteristic
163C2×#10596831C22×C411yescharacteristic
164C2×#10596831C2×D411yescharacteristic
165C2×#10596831C2×D411yescharacteristic
166C8×D73—22no—
167C8×D73—41no—
168C8×D73—42no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73C22×C411yescharacteristic
172C4⋊Dic73C22×C411yescharacteristic
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180#11924005—21no—
181#11924005—21no—
182#11924005—21no—
183#11924005—21no—
184#11924005—21no—
185#11924005—21no—
186#11924005—21no—
187C4×Dic73—21no—
188C4×Dic73C22×C411yescharacteristic
189C4×Dic73C2×D411yescharacteristic
190C4×Dic73#4111yescharacteristic
191C2×Q8×C73C22×C411yescharacteristic
192C2×Q8×C73C22×C411yescharacteristic
193Q16×C73—22no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73C22×C411yescharacteristic
198C2×C8×C73C22×C411yescharacteristic
199C2×C8×C73C2×D411yescharacteristic
200C2×C8×C73C2×D411yescharacteristic
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73C22×C411yescharacteristic
204C4⋊C4×C73C22×C411yescharacteristic
205C42×C73—21no—
206C42×C73—21no—
207C42×C73C2411yescharacteristic
208C22×D73—21no—
209C22×D73C4×D411yescharacteristic
210Q8×C73—21no—
211Q8×C73—21no—
212Q8×C73—21no—
213Q8×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73C23×C411yescharacteristic
219C2×C4×C73C22×D411yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73C4×D411yescharacteristic
225C2×Dic73C4×D411yescharacteristic
226C2×Dic73C4×D411yescharacteristic
227D292—21no—
228D292—21no—
229D292—42no—
230C4×D73—21no—
231C4×D73—21no—
232C4×D73—21no—
233C4×D73—21no—
234C4×D73—41no—
235C4×D73—41no—
236C4×D73—41no—
237Dic146—21no—
238Dic146—21no—
239Dic146—42no—
240C8×C73—21no—
241C8×C73—21no—
242C8×C73—41no—
243C8×C73—41no—
244C8×C73—41no—
245C8×C73#13112yes—
246C8×C73#13712yes—
247#10596831—21no—
248#10596831—21no—
249#10596831—21no—
250#10596831—41no—
251#10596831—41no—
252#10596831—41no—
253#10596831#13112yes—
254C22×C73C2×C4×D411yescharacteristic
255C2×D73—21no—
256C2×D73—21no—
257C2×D73—21no—
258C2×D73—41no—
259C2×D73—41no—
260C2×D73—41no—
261C4×C73—21no—
262C4×C73—41no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73C2×#13111yescharacteristic
267C4×C73C2×#13111yescharacteristic
268C4×C73C2×#13711yescharacteristic
269C4×C73#57511yescharacteristic
270Dic73—21no—
271Dic73—21no—
272Dic73—21no—
273Dic73—41no—
274Dic73—41no—
275Dic73—41no—
276#17965—731nomaximal, Sylow
277C2×C73#293111yescharacteristic
278C2×C73C4×#13711yescharacteristic
279C2×C73#295511yescharacteristic
280D73—41no—
281D73—81no—
282#3172—731no—
283#2966—731no—
284#2960—731no—
285#1593—731no—
286#1593—731no—
287#1592—731no—
288#1592—731no—
289C8×Q16—731no—
290C8×Q16D7311yescharacteristic, complemented
291C8×SD16—731no—
292C8×SD16—731no—
293#1569—731no—
294#1497—731no—
295#1495—731no—
296C8×M16—731no—
297C73#1796511yescharacteristic, complemented, Sylow
298#533—731no—
299#529—731no—
300#501—731no—
301#497—731no—
302#491—731no—
303#489—731no—
304#470—731no—
305#470—731no—
306#468—731no—
307#468—731no—
308#446—731no—
309#446—731no—
310#446—731no—
311#446—1461no—
312C8×Q8—731no—
313C8×Q8—1461no—
314C8×Q8C2×D7311yescharacteristic
315C8×Q8C2×D7311yescharacteristic
316C4×Q16—731no—
317C4×Q16C2×D7311yescharacteristic
318C4×SD16—731no—
319C4×SD16—731no—
320#436—731no—
321#435—1461no—
322C8×D4—731no—
323C8×D4—1461no—
324#433—1461no—
325#432—731no—
326#431—731no—
327#424—1461no—
328#423—731no—
329#405—1461no—
330#405—1461no—
331C4×M16—731no—
332C2×C4×C8—731no—
333#335—731no—
334#335C2×D7311yescharacteristic
335#334—731no—
336#334—731no—
337Q8⋊C8—731no—
338Q8⋊C8—731no—
339Q8⋊C8—731no—
340Q8⋊C8—731no—
341Q8⋊C8C2×D7311yescharacteristic
342Q8⋊C8C2×D7311yescharacteristic
343D4⋊C8—731no—
344D4⋊C8—731no—
345#322—731no—
346#322—731no—
347C82—731no—
348C82C2×D7311yescharacteristic
349C2×Q16—731no—
350C2×Q16C4×D7311yescharacteristic
351C2×SD16—731no—
352C2×SD16—731no—
353C2×M16—731no—
354C2×M16—731no—
355C2×M16—1461no—
356C2×M16—1461no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8—1461no—
360C4⋊Q8—731no—
361C4⋊Q8—731no—
362#126—1461no—
363#125—731no—
364#125—1461no—
365#124—731no—
366#123—1461no—
367C22⋊Q8—731no—
368C22⋊Q8—731no—
369C22⋊Q8—731no—
370C22⋊Q8—1461no—
371#121—731no—
372C4×Q8—731no—
373C4×Q8—1461no—
374C4×Q8C22×D7311yescharacteristic
375C4×Q8C22×D7311yescharacteristic
376C4×D4—731no—
377C4×D4—1461no—
378#117—1461no—
379#117—1461no—
380C2×C42—731no—
381#107—731no—
382#107C4×D7311yescharacteristic
383#106—731no—
384#106—731no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—1461no—
390C4⋊C8—1461no—
391C4⋊C8—1461no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8C22×D7311yescharacteristic
396C4⋊C8C22×D7311yescharacteristic
397#103—731no—
398#103—731no—
399#103—731no—
400#103—731no—
401#103C4×D7311yescharacteristic
402#103C4×D7311yescharacteristic
403#102—731no—
404#102—731no—
405C22⋊C8—1461no—
406C22⋊C8—1461no—
407C22⋊C8—1461no—
408C22⋊C8—1461no—
409#97—731no—
410#97—731no—
411#97—1461no—
412#97—1461no—
413#97—1461no—
414#97—1461no—
415C4×C8—21no—
416C4×C8—21no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—1461no—
423C4×C8C22×D7311yescharacteristic
424C4×C8C22×D7311yescharacteristic
425C4×C8C22×D7311yescharacteristic
426C2×Q8—731no—
427C2×Q8—1461no—
428C2×Q8C2×C4×D7311yescharacteristic
429C2×Q8C2×C4×D7311yescharacteristic
430C2×D4—731no—
431C2×D4—1461no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—731no—
435C22×C4—1461no—
436Q16—22no—
437Q16—1462no—
438SD16—1462no—
439SD16—1462no—
440M16—2922no—
441M16—2922no—
442M16—1462no—
443M16—1462no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—2921no—
448C2×C8—2922no—
449C2×C8—731no—
450C2×C8—731no—
451C2×C8—731no—
452C2×C8—731no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1462no—
457C2×C8C2×C4×D7311yescharacteristic
458C2×C8C2×C4×D7311yescharacteristic
459C2×C8D4×D7311yescharacteristic
460C2×C8D4×D7311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—731no—
464C4⋊C4—731no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4C2×C4×D7311yescharacteristic
475C4⋊C4C2×C4×D7311yescharacteristic
476C22⋊C4—1461no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C22⋊C4—1461no—
482C22⋊C4—1461no—
483C42—21no—
484C42—21no—
485C42—731no—
486C42—731no—
487C42—731no—
488C42—1461no—
489C42C23×D7311yescharacteristic
490C23—731no—
491C23—1461no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—2922no—
497Q8—1461no—
498Q8—1461no—
499D4—2922no—
500D4—1461no—
501D4—1461no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—731no—
510C2×C4—731no—
511C2×C4—731no—
512C2×C4—1461no—
513C2×C4—1461no—
514C2×C4—1461no—
515C2×C4—1461no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4C22×C4×D7311yescharacteristic
520C2×C4C2×D4×D7311yescharacteristic
521C2×C4#41×D7311yescharacteristic
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—2921no—
528C8—2921no—
529C8—2921no—
530C8—732no—
531C8—1461no—
532C8—1461no—
533C8—1461no—
534C8#1455659712yes—
535C8#1455662312yes—
536C22—2921no—
537C22—2921no—
538C22—2921no—
539C22—1461no—
540C22—1461no—
541C22—1461no—
542C22C4×D4×D7311yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—2921no—
549C4—2921no—
550C4—2921no—
551C4—1461no—
552C4—1461no—
553C4—1461no—
554C4#131×D7311yescharacteristic
555C4C2×#1455659711yescharacteristic
556C4C2×#1455662311yescharacteristic
557C4#1798938411yescharacteristic
558C2—2921no—
559C2—5841no—
560C2#2261863811yescharacteristic
561C2C4×#1455662311yescharacteristic
562C2#2261882911yescharacteristic
subgroupG/H
centerC2×C8D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×Q16×C73C2
socleC22×C73C2×C4×D4
prime radical#14556566C2×C4
cosocleC42×C73C24
hypercenterC8×Q16D73
hypocenterC73#17965
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×Q16

Complements

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

NH ≅ G/Ntype
3. #22618860C2semidirect
4. #22608840C2semidirect
5. #22608837C2semidirect
6. #22608836C2semidirect
7. #22608832C2semidirect
8. C8×#11924018C2semidirect
9. C8×#11924017C2semidirect
10. C8×#11924015C2semidirect
11. #22608697C2semidirect
12. #22608402C2semidirect
14. C8×#11924011C2semidirect
15. C8×Q16×C73C2semidirect
290. C8×Q16D73semidirect
297. C73#17965semidirect

Maximal subgroups

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

subgroupindex
#226205352
#226189082
#226188602
#226088402
#226088372
#226088362
#226088322
C8×#119240182
C8×#119240172
C8×#119240152
#226086972
#226084022
#226083952
C8×#119240112
C8×Q16×C732
#1796573