Finite Group Databasebeta

← #46847903

Subgroups — #46847903

Order 34688.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576184C211yesmaximal, characteristic, complemented
2C2×#22576181C211yesmaximal, characteristic, complemented
3#31774567C211yesmaximal, characteristic, complemented
4#31774540C211yesmaximal, characteristic, complemented
5#31774140C211yesmaximal, characteristic, complemented
6#31774130C211yesmaximal, characteristic, complemented
7#31774116C211yesmaximal, characteristic, complemented
8#31774089C211yesmaximal, characteristic, complemented
9#31774056C211yesmaximal, characteristic, complemented
10#31774041C211yesmaximal, characteristic, complemented
11#31774039C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13#31773999C211yesmaximal, characteristic, complemented
14#31773970C211yesmaximal, characteristic, complemented
15#496×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×#16864653C2211yescharacteristic, complemented
19C2×D4×D271C2211yescharacteristic
20#22576184—24no—
21#22576181—24no—
22C2×#16864644C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864639C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic, complemented
29#22576156—21no—
30#22576142—21no—
31#22576136C2211yescharacteristic
32#22576133—21no—
33#22576132—21no—
34#22576132—21no—
35#22576131C2211yescharacteristic
36#22576131C2211yescharacteristic
37#22576131C2211yescharacteristic, complemented
38#22576130—21no—
39C4×Dic542C2211yescharacteristic, complemented
40C4×D1084C2211yescharacteristic
41C4×D1084C2211yescharacteristic
42D4×Dic271—21no—
43C22⋊C4×D271—21no—
44#22576094C2211yescharacteristic, complemented
45C4⋊(C271⋊C8)C2211yescharacteristic, complemented
46#22576085C2211yescharacteristic, complemented
47#22576084C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic
49#22576081C2211yescharacteristic
50#22576081C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic
53(C2×D271)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56C2×D8×C271C2211yescharacteristic
57C2×D8×C271C2211yescharacteristic
58#127×C271C2211yescharacteristic, complemented
59#124×C271C2211yescharacteristic
60#97×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864655—22no—
63#16864655—41no—
64#16864653—22no—
65#16864653—22no—
66#16864653—41no—
67#16864653—41no—
68C2×Dic542C2311yescharacteristic
69C2×D1084—21no—
70C2×D1084D411yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
78D4×D271—22no—
79D4×D271—41no—
80C22×Dic271—21no—
81C22×Dic271—21no—
82C2×C4×D271—21no—
83C2×C4×D271—21no—
84C2×C4×D271C2311yescharacteristic
85C2×C4×D271C2311yescharacteristic
86#16864644—22no—
87#16864643D414yescomplemented
88#16864642—22no—
89#16864639—22no—
90#16864639—22no—
91#16864637—22no—
92C2×C271⋊C8C2311yescharacteristic
93C2×C271⋊C8C2311yescharacteristic
94C8×D271—22no—
95C4⋊Dic271D411yescharacteristic
96C4⋊Dic271C2311yescharacteristic
97C4⋊Dic271C2311yescharacteristic
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4D411yescharacteristic
106(C2×D271)⋊C4D411yescharacteristic
107C4×Dic271—21no—
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271—21no—
110C2×D4×C271—21no—
111C2×D4×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113C2×D4×C271C2311yescharacteristic
114SD16×C271—22no—
115SD16×C271—41no—
116D8×C271—22no—
117D8×C271—41no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C22⋊C4×C271—21no—
121C22⋊C4×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—42no—
125C22×D271—42no—
126C22×D271—81no—
127C22×D271C2×D411yescharacteristic
128C22×D271C2×D411yescharacteristic
129Dic542—21no—
130Dic542C2×D412yes—
131D1084—21no—
132D1084—41no—
133D1084—42no—
134D1084C2×D412yes—
135C22⋊D271—42no—
136C22⋊D271—42no—
137C22⋊D271—42no—
138C22⋊D271—81no—
139C22⋊D271—81no—
140C22⋊D271—81no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—42no—
145C2×Dic271—42no—
146C2×Dic271C2×D411yescharacteristic
147C2×Dic271C2×D411yescharacteristic
148C4×D271—21no—
149C4×D271—21no—
150C4×D271—21no—
151C4×D271—21no—
152C4×D271—42no—
153C4×D271—81no—
154C271⋊C8—21no—
155C271⋊C8—21no—
156C23×C271—21no—
157C23×C271—21no—
158C23×C271—21no—
159Q8×C271—21no—
160Q8×C271—21no—
161D4×C271—21no—
162D4×C271—21no—
163D4×C271—21no—
164D4×C271—21no—
165D4×C271—21no—
166D4×C271—21no—
167D4×C271—42no—
168D4×C271—81no—
169C2×C4×C271—21no—
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2411yescharacteristic
173C8×C271—21no—
174C8×C271C2×D412yes—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271C22×D411yescharacteristic
185C2×D271—21no—
186C2×D271—21no—
187C2×D271—22no—
188C2×D271—21no—
189C2×D271—21no—
190C2×D271—21no—
191C2×D271—21no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—84no—
195C4×C271—21no—
196C4×C271—21no—
197C4×C271—41no—
198C4×C271C22×D411yescharacteristic
199C4×C271H(5,𝔽2)11yescharacteristic
200Dic271—21no—
201Dic271—21no—
202Dic271—22no—
203Dic271—41no—
204Dic271—41no—
205C2×C271—81no—
206C2×C271—81no—
207C2×C271—81no—
208C2×C271D4211yescharacteristic
209C2×C271#57511yescharacteristic
210C2×C271#57611yescharacteristic
211D271—41no—
212D271—41no—
213D271—42no—
214C271#326811yescharacteristic, complemented, Sylow
215#3268—2711nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2711no—
217C2×#135—2711no—
218#550—2711no—
219#546—2711no—
220#496D27111yescharacteristic, complemented
221#468—2711no—
222#466—2711no—
223#464—2711no—
224#459—2711no—
225#452—2711no—
226#449—2711no—
227#449—2711no—
228#447—2711no—
229#440—2711no—
230#435—2711no—
231C2×#41—2711no—
232C2×#41—2711no—
233C2×#41—2711no—
234C22×D4—2711no—
235AGL(1,ℤ/8)—5424no—
236#135—5424no—
237C2×Q16—2711no—
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16C2×D27111yescharacteristic, complemented
241C2×SD16C2×D27111yescharacteristic, complemented
242C2×D8—2711no—
243C2×D8—2711no—
244C2×D8C2×D27111yescharacteristic
245C2×D8C2×D27111yescharacteristic
246C2×M16—2711no—
247C22×C8—2711no—
248#127—5421no—
249#127C2×D27111yescharacteristic, complemented
250#124C2×D27111yescharacteristic
251#123—5421no—
252C22⋊Q8—2711no—
253#121—2711no—
254#121—2711no—
255#121—2711no—
256#121—5421no—
257#121—5421no—
258#121—5421no—
259#120—5421no—
260C4×Q8—2711no—
261C4×D4—2711no—
262C4×D4—2711no—
263C4×D4—5421no—
264C2×C22⋊C4—5421no—
265#107—2711no—
266C4⋊C8—2711no—
267#103—2711no—
268#103—2711no—
269#102—2711no—
270#102—2711no—
271#102—2711no—
272#102—2711no—
273C22⋊C8—2711no—
274C22⋊C8—2711no—
275#97C2×D27111yescharacteristic, complemented
276C24—5421no—
277#41—5422no—
278#41—5422no—
279#41—5422no—
280#41—10841no—
281#41—10841no—
282#41—10841no—
283C2×Q8—2711no—
284C2×Q8C22×D27111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2711no—
288C2×D4—2711no—
289C2×D4—5421no—
290C2×D4—5421no—
291C2×D4—5422no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—10841no—
298C2×D4C22×D27111yescharacteristic
299C2×D4C22×D27111yescharacteristic
300C2×D4C22×D27111yescharacteristic
301C22×C4—2711no—
302C22×C4—2711no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4—5421no—
306C22×C4—5421no—
307Q16—5422no—
308SD16—22no—
309SD16—41no—
310SD16—2714no—
311SD16—5422no—
312D8—22no—
313D8—41no—
314D8—5422no—
315D8—5422no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8—5422no—
320C2×C8C22×D27111yescharacteristic
321C2×C8C22×D27111yescharacteristic
322C4⋊C4—2711no—
323C4⋊C4—2711no—
324C4⋊C4—2711no—
325C4⋊C4—5421no—
326C4⋊C4—5421no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2711no—
330C22⋊C4—2711no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C42—5421no—
337C42C22×D27111yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2711no—
342C23—2711no—
343C23—5421no—
344C23—10842no—
345C23—10842no—
346C23—21681no—
347Q8—21no—
348Q8—21no—
349Q8—2712no—
350Q8—5421no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—2712no—
360D4—5421no—
361D4—10841no—
362D4—10842no—
363D4—10842no—
364D4—10842no—
365D4—10842no—
366D4—21681no—
367D4—21681no—
368D4—21681no—
369C2×C4—21no—
370C2×C4—2711no—
371C2×C4—2711no—
372C2×C4—5421no—
373C2×C4—5421no—
374C2×C4—5421no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—10842no—
380C2×C4—10842no—
381C2×C4—10842no—
382C2×C4—21681no—
383C2×C4D4×D27111yescharacteristic
384C2×C4D4×D27111yescharacteristic
385C2×C4C23×D27111yescharacteristic
386C8—21no—
387C8—5421no—
388C8—5421no—
389C8D4×D27112yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5421no—
400C22—5421no—
401C22—5422no—
402C22—5421no—
403C22—5421no—
404C22—5421no—
405C22—5421no—
406C22—10841no—
407C22—10841no—
408C22—21684no—
409C22C2×D4×D27111yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5421no—
414C4—5421no—
415C4—5422no—
416C4—10841no—
417C4—10841no—
418C4C2×D4×D27111yescharacteristic
419C4#2257620511yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—10841no—
424C2—10841no—
425C2—10842no—
426C2#3177453211yescharacteristic
427C2#3177469311yescharacteristic
428C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#496×C271C2
socleC22×C271C22×D4
hypercenter#496D271
hypocenterC271#3268
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. C2×#22576181C2semidirect
3. #31774567C2semidirect
4. #31774540C2semidirect
5. #31774140C2semidirect
6. #31774130C2semidirect
7. #31774116C2semidirect
8. #31774089C2semidirect
9. #31774056C2semidirect
10. #31774041C2semidirect
11. #31774039C2semidirect
12. #31774029C2semidirect
13. #31773999C2semidirect
14. #31773970C2semidirect
15. #496×C271C2semidirect
18. C2×#16864653C22semidirect
22. C2×#16864644C22semidirect
28. C2×C8×D271C22semidirect
37. #22576131C22semidirect
39. C4×Dic542C22semidirect
44. #22576094C22semidirect
45. C4⋊(C271⋊C8)C22semidirect
46. #22576085C22semidirect
47. #22576084C22semidirect
50. #22576081C22semidirect
51. #22576081C22semidirect
53. (C2×D271)⋊C8C22semidirect
54. C2×SD16×C271C22semidirect
55. C2×SD16×C271C22semidirect
58. #127×C271C22semidirect
60. #97×C271C22semidirect
87. #16864643D4semidirect
214. C271#3268semidirect
220. #496D271semidirect
240. C2×SD16C2×D271semidirect
241. C2×SD16C2×D271semidirect
249. #127C2×D271semidirect
275. #97C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
C2×#225761812
#317745672
#317745402
#317741402
#317741302
#317741162
#317740892
#317740562
#317740412
#317740392
#317740292
#317739992
#317739702
#496×C2712
#3268271