Finite Group Databasebeta

← #46847909

Subgroups — #46847909

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×#22576188C211yesmaximal, characteristic, complemented
2C2×#22576182C211yesmaximal, characteristic, complemented
3#31774568C211yesmaximal, characteristic, complemented
4#31774543C211yesmaximal, characteristic, complemented
5#31774286C211yesmaximal, characteristic, complemented
6#31774143C211yesmaximal, characteristic, complemented
7#31774133C211yesmaximal, characteristic, complemented
8#31774120C211yesmaximal, characteristic, complemented
9#31774093C211yesmaximal, characteristic, complemented
10#31774061C211yesmaximal, characteristic, complemented
11#31774044C211yesmaximal, characteristic, complemented
12#31774044C211yesmaximal, characteristic, complemented
13#31774033C211yesmaximal, characteristic, complemented
14#31773999C211yesmaximal, characteristic, complemented
15#435×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×#16864654C2211yescharacteristic, complemented
19C22×D1084C2211yescharacteristic
20#22576188—24no—
21#22576182—24no—
22C2×Dic1084C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic
24C2×#16864643C2211yescharacteristic
25C2×#16864643C2211yescharacteristic, complemented
26C2×#16864643C2211yescharacteristic, complemented
27C2×D2168C2211yescharacteristic
28C2×D2168C2211yescharacteristic
29C2×D2168C2211yescharacteristic
30C2×D2168C2211yescharacteristic
31#22576157C2211yescharacteristic, complemented
32#22576156—21no—
33#22576148C2211yescharacteristic
34#22576145—21no—
35#22576139C2211yescharacteristic
36#22576134C2211yescharacteristic
37#22576134C2211yescharacteristic
38#22576134C2211yescharacteristic, complemented
39#22576133—21no—
40#22576133—21no—
41#22576133—21no—
42#22576128—21no—
43C4×Dic542C2211yescharacteristic, complemented
44C4×D1084C2211yescharacteristic
45#22576122—21no—
46C22⋊C4×D271—21no—
47#22576094C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576085C2211yescharacteristic, complemented
50#22576082C2211yescharacteristic
51#22576082C2211yescharacteristic
52#22576082C2211yescharacteristic, complemented
53#22576082C2211yescharacteristic, complemented
54C2×M16×C271C2211yescharacteristic
55C22×C8×C271C2211yescharacteristic, complemented
56C4×D4×C271C2211yescharacteristic
57C4⋊C8×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic
59C22⋊C8×C271C2211yescharacteristic, complemented
60#97×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864654—22no—
63#16864654—22no—
64#16864654—22no—
65#16864654—41no—
66#16864654—41no—
67#16864654—41no—
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70C2×D1084—22no—
71C2×D1084—21no—
72C2×D1084—21no—
73C2×D1084—21no—
74C2×D1084—41no—
75C2×D1084C2311yescharacteristic
76C2×D1084C2311yescharacteristic
77C2×D1084C2311yescharacteristic
78C2×D1084C2311yescharacteristic
79C2×C22⋊D271—21no—
80C2×C22⋊D271—21no—
81C2×C22⋊D271—21no—
82C2×C22⋊D271—21no—
83C2×C22⋊D271—21no—
84C2×C22⋊D271—21no—
85C2×C4×D271—21no—
86C2×C4×D271—21no—
87C2×C4×D271—21no—
88C2×C4×D271—21no—
89Dic1084—22no—
90#16864643—22no—
91#16864643—22no—
92#16864643—41no—
93#16864643D414yescomplemented
94D2168—22no—
95D2168—22no—
96D2168—22no—
97D2168—41no—
98C4⋊Dic271C2311yescharacteristic
99C4⋊Dic271C2311yescharacteristic
100Dic271⋊C4—21no—
101Dic271⋊C4—21no—
102C22⋊Dic271—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4—21no—
109C4×Dic271—21no—
110C2×D4×C271D411yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C22×C4×C271C2311yescharacteristic
113M16×C271—22no—
114C2×C8×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C2×C8×C271C2311yescharacteristic
119C4⋊C4×C271D411yescharacteristic
120C22⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—81no—
130Dic542—21no—
131Dic542—21no—
132Dic542—21no—
133Dic542C2×D412yes—
134D1084—21no—
135D1084—21no—
136D1084—21no—
137D1084—21no—
138D1084—21no—
139D1084—21no—
140D1084—21no—
141D1084—42no—
142D1084—42no—
143D1084—81no—
144D1084C2×D412yes—
145C22⋊D271—42no—
146C22⋊D271—42no—
147C22⋊D271—42no—
148C22⋊D271—81no—
149C22⋊D271—81no—
150C22⋊D271—81no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—21no—
155C4×D271—42no—
156C4×D271—42no—
157C4×D271—42no—
158C4×D271—81no—
159C23×C271C2×D411yescharacteristic
160C23×C271C2×D411yescharacteristic
161D4×C271—41no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2×D411yescharacteristic
170C2×C4×C271C2411yescharacteristic
171C8×C271—21no—
172C8×C271—21no—
173C8×C271—21no—
174C8×C271C2×D412yes—
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—21no—
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271C22×D411yescharacteristic
182C2×D271—22no—
183C2×D271—41no—
184C2×D271—41no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—41no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—84no—
195C4×C271—21no—
196C4×C271—21no—
197C4×C271—21no—
198C4×C271—21no—
199C4×C271C22×D411yescharacteristic
200C4×C271H(5,𝔽2)11yescharacteristic
201Dic271—22no—
202Dic271—41no—
203Dic271—41no—
204Dic271—41no—
205C2×C271—41no—
206C2×C271—41no—
207C2×C271D4211yescharacteristic
208C2×C271#57511yescharacteristic
209C2×C271#57611yescharacteristic
210D271—42no—
211D271—81no—
212D271—81no—
213D271—81no—
214C271#326811yescharacteristic, complemented, Sylow
215#3268—2711nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2711no—
217C2×#135—2711no—
218#550—2711no—
219#546—2711no—
220#496—2711no—
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#435D27111yescharacteristic, complemented
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×SD16—2711no—
241C2×SD16—2711no—
242C2×D8—2711no—
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8—2711no—
246C2×M16C2×D27111yescharacteristic
247C22×C8C2×D27111yescharacteristic, complemented
248#127—2711no—
249#127—5421no—
250#124—2711no—
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—5421no—
263C4×D4C2×D27111yescharacteristic
264C2×C22⋊C4—5421no—
265#107—2711no—
266C4⋊C8C2×D27111yescharacteristic, complemented
267#103—2711no—
268#103—2711no—
269#102—2711no—
270#102—2711no—
271#102—2711no—
272#102—2711no—
273C22⋊C8C2×D27111yescharacteristic
274C22⋊C8C2×D27111yescharacteristic, complemented
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×Q8—2711no—
285C2×D4—2711no—
286C2×D4—2711no—
287C2×D4—2711no—
288C2×D4—2711no—
289C2×D4—5422no—
290C2×D4—5421no—
291C2×D4—5421no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—5421no—
299C2×D4—10841no—
300C2×D4D108411yescharacteristic
301C22×C4—5421no—
302C22×C4—5421no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4C22×D27111yescharacteristic
306C22×C4C22×D27111yescharacteristic
307Q16—5422no—
308SD16—2714no—
309SD16—5422no—
310SD16—5422no—
311SD16—10841no—
312D8—5422no—
313D8—5422no—
314D8—5422no—
315D8—10841no—
316M16—22no—
317C2×C8—22no—
318C2×C8C22×D27111yescharacteristic
319C2×C8C22×D27111yescharacteristic
320C2×C8C22×D27111yescharacteristic
321C2×C8C22×D27111yescharacteristic
322C4⋊C4—2711no—
323C4⋊C4—2711no—
324C4⋊C4—5421no—
325C4⋊C4—5421no—
326C4⋊C4D108411yescharacteristic
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4—5421no—
330C22⋊C4—5421no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4D108411yescharacteristic
335C22⋊C4D108411yescharacteristic
336C42—5421no—
337C42C22×D27111yescharacteristic
338C23—5421no—
339C23—5421no—
340C23—5421no—
341C23—5421no—
342C23—10842no—
343C23—10842no—
344C23—21681no—
345C23C2×D108411yescharacteristic
346C23C2×D108411yescharacteristic
347Q8—2712no—
348Q8—5421no—
349Q8—5421no—
350Q8—5421no—
351D4—41no—
352D4—2712no—
353D4—5421no—
354D4—5421no—
355D4—5421no—
356D4—5421no—
357D4—5421no—
358D4—5421no—
359D4—5421no—
360D4—10842no—
361D4—10842no—
362D4—10842no—
363D4—10842no—
364D4—10842no—
365D4—21681no—
366D4—21681no—
367D4—21681no—
368D4—21681no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—5421no—
374C2×C4—5421no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—10842no—
378C2×C4—10842no—
379C2×C4—10842no—
380C2×C4—21681no—
381C2×C4C2×D108411yescharacteristic
382C2×C4C2×D108411yescharacteristic
383C2×C4C2×D108411yescharacteristic
384C2×C4C2×D108411yescharacteristic
385C2×C4C23×D27111yescharacteristic
386C8—21no—
387C8—21no—
388C8—21no—
389C8D4×D27112yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—5422no—
397C22—10841no—
398C22—10841no—
399C22—10841no—
400C22—10841no—
401C22—10841no—
402C22—10841no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—21684no—
409C22C22×D108411yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—5422no—
415C4—10841no—
416C4—10841no—
417C4—10841no—
418C4C2×D4×D27111yescharacteristic
419C4#2257620611yescharacteristic
420C2—41no—
421C2—41no—
422C2—10842no—
423C2—21681no—
424C2—21681no—
425C2—21681no—
426C2D4×D108411yescharacteristic
427C2#3177469711yescharacteristic
428C2#3177470211yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#435×C271C2
socleC22×C271C22×D4
hypercenter#435D271
hypocenterC271#3268
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C2×#22576182C2semidirect
3. #31774568C2semidirect
4. #31774543C2semidirect
5. #31774286C2semidirect
6. #31774143C2semidirect
7. #31774133C2semidirect
8. #31774120C2semidirect
9. #31774093C2semidirect
10. #31774061C2semidirect
11. #31774044C2semidirect
12. #31774044C2semidirect
13. #31774033C2semidirect
14. #31773999C2semidirect
15. #435×C271C2semidirect
18. C2×#16864654C22semidirect
22. C2×Dic1084C22semidirect
25. C2×#16864643C22semidirect
26. C2×#16864643C22semidirect
31. #22576157C22semidirect
38. #22576134C22semidirect
43. C4×Dic542C22semidirect
47. #22576094C22semidirect
48. #22576085C22semidirect
49. #22576085C22semidirect
52. #22576082C22semidirect
53. #22576082C22semidirect
55. C22×C8×C271C22semidirect
57. C4⋊C8×C271C22semidirect
59. C22⋊C8×C271C22semidirect
60. #97×C271C22semidirect
93. #16864643D4semidirect
214. C271#3268semidirect
230. #435D271semidirect
247. C22×C8C2×D271semidirect
266. C4⋊C8C2×D271semidirect
274. C22⋊C8C2×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×#225761882
C2×#225761822
#317745682
#317745432
#317742862
#317741432
#317741332
#317741202
#317740932
#317740612
#317740442
#317740332
#317739992
#435×C2712
#3268271