Finite Group Databasebeta

← #34034838

Subgroups — #34034838

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22627182C211yesmaximal, characteristic, complemented
2C2×#17989708C211yesmaximal, characteristic, complemented
3#22627153C212yesmaximal, complemented
4#22627151C212yesmaximal, complemented
5#22627149C212yesmaximal, complemented
6#22627148C212yesmaximal, complemented
7#22627127C211yesmaximal, characteristic, complemented
8#22627114C211yesmaximal, characteristic, complemented
9#22627113C211yesmaximal, characteristic, complemented
10#22627094C211yesmaximal, characteristic, complemented
11#22626703C211yesmaximal, characteristic
12#17989709C2211yescharacteristic, complemented
13#17989709C2211yescharacteristic, complemented
14#17989708—21no—
15#17989708—21no—
16#17989708—21no—
17#17989708—21no—
18#17989708C2211yescharacteristic, complemented
19#17989708C2212yescomplemented
20#17989708C2212yescomplemented
21#17989708C2211yescharacteristic, complemented
22C2×#14556688C2211yescharacteristic
23#17989706—21no—
24#17989706C2211yescharacteristic
25#17989706C2211yescharacteristic
26C2×#14556687—21no—
27C2×#14556686—21no—
28C2×#14556686C2211yescharacteristic
29C22×#11924040—21no—
30#17989692C2212yescomplemented
31#17989691C2212yescomplemented
32#17989688C2212yescomplemented
33#17989687C2212yescomplemented
34#17989687C2212yescomplemented
35#17989687C2212yescomplemented
36#17989682C2212yescomplemented
37#64408⋊C16C2212yescomplemented
38D73⋊#131C2211yescharacteristic
39C2×#14556678C2211yescharacteristic
40#17989634C412yes—
41#17989632C412yes—
42#17989630C2212yes—
43#17989629C2212yes—
44#17989602C411yescharacteristic
45#17989585C411yescharacteristic
46#17989584C411yescharacteristic
47#17989553C2211yescharacteristic
48#17989383C411yescharacteristic
49#14556688—21no—
50#14556688—21no—
51#14556688—21no—
52#14556688—21no—
53#14556688C2311yescharacteristic
54#14556688C2311yescharacteristic
55#14556688C2312yes—
56#14556687—22no—
57#14556687—22no—
58#14556687—21no—
59#14556687—21no—
60#14556687—21no—
61#14556687—21no—
62#14556686—21no—
63#14556686—21no—
64#14556686—21no—
65#14556686D411yescharacteristic
66#14556686D411yescharacteristic
67#14556686C2312yes—
68#14556686C2311yescharacteristic
69#14556686C2311yescharacteristic
70C2×#11924040—22no—
71C2×#11924040—22no—
72C2×#11924040—21no—
73C2×#11924040—21no—
74C2×#11924040—21no—
75C2×#11924040D411yescharacteristic
76C2×#11924040D411yescharacteristic
77#14556679C2×C411yescharacteristic
78#14556679C2311yescharacteristic
79#14556678—21no—
80#14556678—21no—
81#14556678—21no—
82#14556678—21no—
83#14556678C2×C412yes—
84#14556678C2×C411yescharacteristic
85#14556678C2312yes—
86#14556678C2311yescharacteristic
87C2×#11924038C2×C411yescharacteristic
88D73⋊M16—21no—
89D73⋊M16C2×C411yescharacteristic
90C2×#11924037—21no—
91C2×#11924036—21no—
92C2×#11924036C2311yescharacteristic
93C22×#10596833—21no—
94#14556670C2×C411yescharacteristic
95#14556662C2×C412yes—
96#14556661C2×C412yes—
97#14556656C2×C412yes—
98#14556655C2×C412yes—
99#14556655C2×C412yes—
100#14556655C2×C412yes—
101#14556645C2×C412yes—
102C8×#64408C2312yes—
103C2×#11924027C2×C411yescharacteristic
104#41×D73C2×C411yescharacteristic
105#14556624C811yescharacteristic
106#14556621C811yescharacteristic
107#14556620C812yes—
108#14556617C811yescharacteristic
109#14556615C812yes—
110#14556612C2×C412yes—
111#14556610C2×C412yes—
112M16×D73C2×C411yescharacteristic
113AGL(1,ℤ/8)×C73C811yescharacteristic
114#11924040—22no—
115#11924040—21no—
116#11924040—21no—
117#11924040—22no—
118#11924040C2×D411yescharacteristic
119#11924040C2×D411yescharacteristic
120#11924040C2×D411yescharacteristic
121#11924040C2×D411yescharacteristic
122#11924038—21no—
123#11924038—21no—
124#11924038—21no—
125#11924038C22×C411yescharacteristic
126#11924038C22×C411yescharacteristic
127#11924038C22×C412yes—
128#11924037—21no—
129#11924037—22no—
130#11924037—21no—
131#11924037—22no—
132#11924037—21no—
133#11924036—21no—
134#11924036—21no—
135#11924036—21no—
136#11924036C22×C411yescharacteristic
137#11924036C2×D411yescharacteristic
138#11924036#4111yescharacteristic
139#11924036C2411yescharacteristic
140C2×#10596833—21no—
141C2×#10596833—22no—
142C2×#10596833—21no—
143C2×#10596833—22no—
144C2×#10596833—21no—
145C2×#10596833C2×D411yescharacteristic
146C2×#10596833#4111yescharacteristic
147C4⋊#64408—21no—
148C22⋊#64408—21no—
149C4×#64408C22×C412yes—
150#11924029C22×C411yescharacteristic
151#11924028—21no—
152#11924028C2×C811yescharacteristic
153#11924027—21no—
154#11924027—21no—
155#11924027—21no—
156#11924027C2×C812yes—
157#11924027C22×C411yescharacteristic
158#11924027C22×C412yes—
159C2×Dic146C2×C811yescharacteristic
160Q8×D73C2×C811yescharacteristic
161C2×C22⋊D73—21no—
162D4×D73—21no—
163D4×D73C2×C811yescharacteristic
164C22×Dic73—21no—
165C2×C4×D73—21no—
166C2×C4×D73C22×C411yescharacteristic
167Dic292C2×C812yes—
168#11924018C2×C812yes—
169#11924017C2×C812yes—
170#11924016C2×C812yes—
171#11924016C2×C812yes—
172#11924013C2×C812yes—
173#11924012C2×C811yescharacteristic
174#11924011C2×C812yes—
175C8×D73C22×C412yes—
176#41×C73C2×C811yescharacteristic
177C2×D4×C73C2×C811yescharacteristic
178SD16×C73C2×C812yes—
179D8×C73C2×C812yes—
180M16×C73C2×C811yescharacteristic
181C23×C73—21no—
182C22×D73—21no—
183C22×D73#13111yescharacteristic
184C2×#64408—22no—
185Q8×C73C22×C811yescharacteristic
186D4×C73—21no—
187D4×C73—21no—
188D4×C73C22×C811yescharacteristic
189D4×C73C22×C812yes—
190C2×C4×C73—21no—
191C2×C4×C73C22×C811yescharacteristic
192C22⋊D73—21no—
193C22⋊D73—21no—
194C22⋊D73—22no—
195C22⋊D73—22no—
196C22⋊D73—21no—
197C2×Dic73—21no—
198C2×Dic73—21no—
199C2×Dic73—22no—
200C2×Dic73—22no—
201C2×Dic73C4×D411yescharacteristic
202D292—21no—
203D292C22×C811yescharacteristic
204C4×D73—21no—
205C4×D73—21no—
206C4×D73—21no—
207C4×D73C4×D411yescharacteristic
208C4×D73C22×C811yescharacteristic
209C4×D73#13111yescharacteristic
210C4×D73C23×C411yescharacteristic
211Dic146—21no—
212Dic146—21no—
213Dic146C22×C811yescharacteristic
214Dic146C22×C812yes—
215#10596833—22no—
216#10596833—21no—
217#10596833—21no—
218#10596833C4×D411yescharacteristic
219#10596833C4×D411yescharacteristic
220#10596833C22×D411yescharacteristic
221#10596833C2×#4111yescharacteristic
222C8×C73C22×C812yes—
223#10596831C22×C812yes—
224C22×C73—22no—
225C22×C73—21no—
226C22×C73—42no—
227C22×C73C8×D411yescharacteristic
228C2×D73—21no—
229C2×D73—41no—
230C2×D73—42no—
231C2×D73C8×D411yescharacteristic
232C2×D73C2×#13111yescharacteristic
233#64408—42no—
234C4×C73—21no—
235C4×C73C8×D411yescharacteristic
236C4×C73C23×C811yescharacteristic
237Dic73—21no—
238Dic73—22no—
239Dic73C8×D411yescharacteristic
240Dic73C2×C4×D411yescharacteristic
241#33441—731nomaximal, Sylow
242C2×C73—21no—
243C2×C73—41no—
244C2×C73—42no—
245C2×C73C2×C8×D411yescharacteristic
246D73—21no—
247D73—21no—
248D73—41no—
249C2×#504—731no—
250C2×#504—731no—
251#2955—731no—
252#2171—732no—
253#2171—732no—
254#2170—732no—
255#2170—732no—
256#2153—731no—
257#2117—731no—
258#2117—731no—
259#2108—731no—
260C73#3344111yescharacteristic, complemented, Sylow
261#575—731no—
262C2×#131—731no—
263C2×#131—731no—
264#504—731no—
265#504—732no—
266#504—732no—
267#504—731no—
268#504—731no—
269#504—731no—
270#504—1461no—
271#504—1461no—
272#504—1461no—
273#504—1461no—
274C2×M32—731no—
275C2×M32—731no—
276C2×M32—731no—
277C2×M32—1461no—
278C2×M32—1461no—
279C22×C16—731no—
280C22×C16—1461no—
281C22×C16—1461no—
282#444—732no—
283#444—732no—
284#443—732no—
285#443—732no—
286#430—731no—
287#421—731no—
288#421—731no—
289#405—731no—
290#364—732no—
291#364—732no—
292#350—732no—
293#350—732no—
294#350—732no—
295#350—732no—
296#346—732no—
297C4×C16—732no—
298C2×#41—731no—
299C2×#41—731no—
300#137—731no—
301#137—732no—
302#137—731no—
303AGL(1,ℤ/8)—731no—
304AGL(1,ℤ/8)—732no—
305AGL(1,ℤ/8)#1059684211yescharacteristic
306#135—732no—
307#135—732no—
308#131—732no—
309#131—731no—
310#131—732no—
311#131—731no—
312#131—731no—
313#131—731no—
314#131—1461no—
315#131—1461no—
316#131—1461no—
317#131—1461no—
318C2×M16—731no—
319C2×M16—731no—
320C2×M16—731no—
321C2×M16—1461no—
322C2×M16—1461no—
323C22×C8—731no—
324C22×C8—1461no—
325C22×C8—1461no—
326#117—731no—
327M32—731no—
328M32—731no—
329M32—732no—
330M32—1462no—
331M32—1461no—
332M32—1462no—
333M32—1461no—
334M32—1461no—
335M32—1461no—
336M32—1461no—
337M32—1461no—
338M32—1461no—
339M32—1461no—
340C2×C16—731no—
341C2×C16—731no—
342C2×C16—731no—
343C2×C16—731no—
344C2×C16—731no—
345C2×C16—731no—
346C2×C16—732no—
347C2×C16—1462no—
348C2×C16—1461no—
349C2×C16—1462no—
350C2×C16—1461no—
351C2×C16—1461no—
352C2×C16—1461no—
353C2×C16—1461no—
354C2×C16—1461no—
355#108—732no—
356#108—732no—
357#104—732no—
358#104—732no—
359#104—732no—
360#104—732no—
361#97—732no—
362C4×C8—732no—
363#41—731no—
364#41—732no—
365#41—732no—
366#41—731no—
367#41—731no—
368#41—1461no—
369#41—1461no—
370#41—1461no—
371#41—1461no—
372#41C2×#1059684211yescharacteristic
373C2×Q8—731no—
374C2×Q8—731no—
375C2×D4—731no—
376C2×D4—1461no—
377C2×D4—1461no—
378C2×D4C2×#1059684211yescharacteristic
379C22×C4—731no—
380C22×C4—1461no—
381C22×C4—1461no—
382Q16—732no—
383Q16—732no—
384SD16—732no—
385SD16—732no—
386SD16—732no—
387SD16C2×#1059684212yes—
388D8—732no—
389D8C2×#1059684212yes—
390M16—731no—
391M16—732no—
392M16—731no—
393M16—731no—
394M16—732no—
395M16—1462no—
396M16—1461no—
397M16—1462no—
398M16—1461no—
399M16—1461no—
400M16—1461no—
401M16—1461no—
402M16—1461no—
403M16C2×#1059684211yescharacteristic
404C2×C8—731no—
405C2×C8—731no—
406C2×C8—731no—
407C2×C8—731no—
408C2×C8—731no—
409C2×C8—731no—
410C2×C8—732no—
411C2×C8—1462no—
412C2×C8—1461no—
413C2×C8—1462no—
414C2×C8—1461no—
415C2×C8—1461no—
416C2×C8—1461no—
417C2×C8—1461no—
418C2×C8—1461no—
419C4⋊C4—1461no—
420C22⋊C4—1461no—
421C42—732no—
422C16—731no—
423C16—731no—
424C16—731no—
425C16—731no—
426C16—1462no—
427C16—1462no—
428C16—1461no—
429C16—1461no—
430C23—21no—
431C23—731no—
432C23—1461no—
433Q8—731no—
434Q8—732no—
435Q8—1461no—
436Q8—1461no—
437Q8C22×#1059684211yescharacteristic
438D4—21no—
439D4—21no—
440D4—731no—
441D4—1461no—
442D4—1461no—
443D4—1461no—
444D4—1462no—
445D4—1462no—
446D4—1461no—
447D4C22×#1059684212yes—
448D4C22×#1059684211yescharacteristic
449C2×C4—21no—
450C2×C4—731no—
451C2×C4—731no—
452C2×C4—731no—
453C2×C4—731no—
454C2×C4—731no—
455C2×C4—1461no—
456C2×C4—1461no—
457C2×C4—1462no—
458C2×C4—1461no—
459C2×C4—1462no—
460C2×C4—1462no—
461C2×C4—1461no—
462C2×C4—1461no—
463C2×C4C22×#1059684211yescharacteristic
464C8—731no—
465C8—731no—
466C8—731no—
467C8—731no—
468C8—732no—
469C8—1462no—
470C8—1461no—
471C8—1461no—
472C8C22×#1059684212yes—
473C22—21no—
474C22—22no—
475C22—42no—
476C22—2921no—
477C22—2922no—
478C22—731no—
479C22—731no—
480C22—1461no—
481C22D4×#1059684211yescharacteristic
482C4—21no—
483C4—2922no—
484C4—731no—
485C4—731no—
486C4—1461no—
487C4—1462no—
488C4D4×#1059684211yescharacteristic
489C4C23×#1059684211yescharacteristic
490C2—21no—
491C2—42no—
492C2—41no—
493C2—2921no—
494C2—1461no—
495C2—1461no—
496C2C2×D4×#1059684211yescharacteristic
subgroupG/H
centerC2C2×D4×#10596842
commutatorC4×C73C23×C8
FrattiniC4C23×#10596842
FittingAGL(1,ℤ/8)×C73C8
socleC2×C73C2×C8×D4
prime radical#17989383C4
cosocle#11924036C24
hypercenterAGL(1,ℤ/8)#10596842
hypocenterC73#33441
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22627182C2semidirect
2. C2×#17989708C2semidirect
3. #22627153C2semidirect
4. #22627151C2semidirect
5. #22627149C2semidirect
6. #22627148C2semidirect
7. #22627127C2semidirect
8. #22627114C2semidirect
9. #22627113C2semidirect
10. #22627094C2semidirect
12. #17989709C22semidirect
13. #17989709C22semidirect
18. #17989708C22semidirect
19. #17989708C22semidirect
20. #17989708C22semidirect
21. #17989708C22semidirect
30. #17989692C22semidirect
31. #17989691C22semidirect
32. #17989688C22semidirect
33. #17989687C22semidirect
34. #17989687C22semidirect
35. #17989687C22semidirect
36. #17989682C22semidirect
37. #64408⋊C16C22semidirect
260. C73#33441semidirect

Maximal subgroups

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

subgroupindex
#226271822
C2×#179897082
#226271532
#226271512
#226271492
#226271482
#226271272
#226271142
#226271132
#226270942
#226267032
#3344173