Finite Group Databasebeta

← #46847945

Subgroups — #46847945

Order 34688.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576190C211yesmaximal, characteristic, complemented
2C2×#22576180C211yesmaximal, characteristic, complemented
3#31774556C211yesmaximal, characteristic, complemented
4#31774284C211yesmaximal, characteristic, complemented
5#31774148C211yesmaximal, characteristic, complemented
6#31774124C211yesmaximal, characteristic, complemented
7#31774119C211yesmaximal, characteristic, complemented
8#31774094C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774042C211yesmaximal, characteristic, complemented
11#31774040C211yesmaximal, characteristic, complemented
12#31774035C211yesmaximal, characteristic, complemented
13#31774002C211yesmaximal, characteristic, complemented
14#31773969C211yesmaximal, characteristic, complemented
15#550×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18#22576190—24no—
19#22576180—24no—
20C2×#16864644C2211yescharacteristic
21C2×#16864644C2211yescharacteristic
22C2×#16864642C2211yescharacteristic
23C2×#16864642C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864638C2211yescharacteristic
30C22×C271⋊C8C2211yescharacteristic
31#22576148C2211yescharacteristic
32#22576146—21no—
33#22576139C2211yescharacteristic
34#22576138—21no—
35#22576134C2211yescharacteristic
36#22576132—21no—
37C4×Dic542C2211yescharacteristic
38(C22⋊D271)⋊C4—21no—
39#22576091C2211yescharacteristic
40C4⋊(C271⋊C8)C2211yescharacteristic
41#22576084C2211yescharacteristic
42#22576083C2211yescharacteristic
43#22576083C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576080C2211yescharacteristic
47C22⋊(C271⋊C8)C2211yescharacteristic
48C22⋊(C271⋊C8)C2211yescharacteristic
49#22576071C2211yescharacteristic
50C2×#41×C271C2211yescharacteristic
51C2×#41×C271C2211yescharacteristic
52#127×C271—21no—
53#127×C271C2211yescharacteristic
54#121×C271—21no—
55#121×C271—21no—
56#121×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic
58C4×Q8×C271C2211yescharacteristic
59C4×D4×C271—21no—
60C4×D4×C271C2211yescharacteristic
61#16864654—22no—
62#16864654—41no—
63C2×Dic542—22no—
64C2×Dic542—21no—
65C2×Dic542—41no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C22×Dic271—21no—
72C2×C4×D271—21no—
73#16864644—22no—
74#16864644D414yescomplemented
75#16864642—22no—
76#16864642—22no—
77#16864642—22no—
78#16864642—41no—
79#16864641—22no—
80#16864639—22no—
81#16864639—41no—
82#16864638—22no—
83C2×C271⋊C8—22no—
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95C4×Dic271—21no—
96#41×C271—22no—
97#41×C271—22no—
98#41×C271—41no—
99#41×C271—41no—
100C2×Q8×C271C2311yescharacteristic
101C2×D4×C271—21no—
102C2×D4×C271—21no—
103C2×D4×C271—21no—
104C2×D4×C271—21no—
105C2×D4×C271—21no—
106C2×D4×C271—21no—
107C2×D4×C271D411yescharacteristic
108C2×D4×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271—21no—
111C22×C4×C271—21no—
112C22×C4×C271C2311yescharacteristic
113C22×C4×C271C2311yescharacteristic
114C4⋊C4×C271—21no—
115C4⋊C4×C271D411yescharacteristic
116C4⋊C4×C271C2311yescharacteristic
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271D411yescharacteristic
120C22⋊C4×C271D411yescharacteristic
121C42×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—42no—
128Dic542C2×D412yes—
129D1084—21no—
130D1084—21no—
131C22⋊D271—42no—
132C22⋊D271—42no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—42no—
137C4×D271—42no—
138C271⋊C8—21no—
139C271⋊C8—21no—
140C271⋊C8—21no—
141C271⋊C8C2×D412yes—
142C23×C271—21no—
143C23×C271—21no—
144C23×C271C2×D411yescharacteristic
145C23×C271C2×D411yescharacteristic
146Q8×C271—21no—
147Q8×C271C2×D412yes—
148D4×C271—21no—
149D4×C271—21no—
150D4×C271—21no—
151D4×C271—21no—
152D4×C271—41no—
153D4×C271—42no—
154D4×C271—42no—
155D4×C271—42no—
156D4×C271—81no—
157D4×C271—81no—
158D4×C271—81no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—42no—
166C2×C4×C271—42no—
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2×D411yescharacteristic
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2411yescharacteristic
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—21no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271C22×D411yescharacteristic
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C4×C271—21no—
189C4×C271—21no—
190C4×C271—22no—
191C4×C271—21no—
192C4×C271—21no—
193C4×C271—41no—
194C4×C271C22×D411yescharacteristic
195C4×C271H(5,𝔽2)11yescharacteristic
196Dic271—22no—
197Dic271—41no—
198Dic271—41no—
199C2×C271—41no—
200C2×C271—41no—
201C2×C271—81no—
202C2×C271—81no—
203C2×C271D4211yescharacteristic
204C2×C271#57511yescharacteristic
205C2×C271#57711yescharacteristic
206D271—81no—
207C271#327111yescharacteristic, complemented, Sylow
208#3271—2711nomaximal, Sylow
209C2×#137—2711no—
210C2×#135—2711no—
211#550D27111yescharacteristic, complemented
212#548—2711no—
213#496—2711no—
214#469—2711no—
215#465—2711no—
216#464—2711no—
217#460—2711no—
218#452—2711no—
219#449—2711no—
220#449—2711no—
221#448—2711no—
222#441—2711no—
223#435—2711no—
224C2×#41—2711no—
225C2×#41C2×D27111yescharacteristic
226C2×#41C2×D27111yescharacteristic
227C22×Q8—2711no—
228#137—5424no—
229#135—5424no—
230C2×Q16—2711no—
231C2×Q16—2711no—
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16—2711no—
237C2×D8—2711no—
238C2×D8—2711no—
239C2×M16—2711no—
240C22×C8—2711no—
241#127—21no—
242#127C2×D27111yescharacteristic
243#124—2711no—
244#124—5421no—
245C22⋊Q8—2711no—
246C22⋊Q8—5421no—
247#121—21no—
248#121—21no—
249#121—2711no—
250#121—5421no—
251#121C2×D27111yescharacteristic
252#121C2×D27111yescharacteristic
253C4×Q8—2711no—
254C4×Q8C2×D27111yescharacteristic
255C4×D4—21no—
256C4×D4—5421no—
257C4×D4C2×D27111yescharacteristic
258#106—2711no—
259C4⋊C8—2711no—
260#103—2711no—
261#103—2711no—
262#103—2711no—
263#102—2711no—
264#102—2711no—
265#102—2711no—
266C22⋊C8—2711no—
267C22⋊C8—2711no—
268#97—2711no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—5422no—
274#41—10841no—
275C2×Q8—2711no—
276C2×Q8—2711no—
277C2×Q8—5422no—
278C2×Q8—5421no—
279C2×Q8—10841no—
280C2×Q8C22×D27111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2711no—
288C2×D4—5421no—
289C2×D4—5421no—
290C2×D4C22⋊D27111yescharacteristic
291C2×D4C22×D27111yescharacteristic
292C2×D4C22×D27111yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4C22×D27111yescharacteristic
298C22×C4C22×D27111yescharacteristic
299Q16—2714no—
300Q16—5422no—
301SD16—5422no—
302SD16—5422no—
303SD16—5422no—
304SD16—5422no—
305SD16—10841no—
306D8—5422no—
307D8—10841no—
308M16—5422no—
309C2×C8—2711no—
310C2×C8—2711no—
311C2×C8—2711no—
312C2×C8—2711no—
313C2×C8—5422no—
314C4⋊C4—21no—
315C4⋊C4—2711no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C4⋊C4C22⋊D27111yescharacteristic
319C4⋊C4C22×D27111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4C22⋊D27111yescharacteristic
327C22⋊C4C22⋊D27111yescharacteristic
328C42—21no—
329C42—5421no—
330C42C22×D27111yescharacteristic
331C23—21no—
332C23—21no—
333C23—5421no—
334C23C2×C22⋊D27111yescharacteristic
335C23C2×C22⋊D27111yescharacteristic
336Q8—21no—
337Q8—2712no—
338Q8—5421no—
339Q8—5421no—
340Q8—5421no—
341Q8—10842no—
342Q8D4×D27112yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—5421no—
355D4—5421no—
356D4—10842no—
357D4—10842no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—5421no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—10842no—
370C2×C4—10842no—
371C2×C4C2×C22⋊D27111yescharacteristic
372C2×C4C2×C22⋊D27111yescharacteristic
373C2×C4C2×C22⋊D27111yescharacteristic
374C2×C4C2×C22⋊D27111yescharacteristic
375C2×C4C23×D27111yescharacteristic
376C8—2712no—
377C8—5421no—
378C8—5421no—
379C8—5421no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22C22×C22⋊D27111yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—41no—
402C4—5422no—
403C4—10841no—
404C4—10841no—
405C4C2×D4×D27111yescharacteristic
406C4#2257620511yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—81no—
411C2—21681no—
412C2D4×C22⋊D27111yescharacteristic
413C2#3177469511yescharacteristic
414C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#550×C271C2
socleC22×C271C22×D4
hypercenter#550D271
hypocenterC271#3271
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22576190C2semidirect
2. C2×#22576180C2semidirect
3. #31774556C2semidirect
4. #31774284C2semidirect
5. #31774148C2semidirect
6. #31774124C2semidirect
7. #31774119C2semidirect
8. #31774094C2semidirect
9. #31774059C2semidirect
10. #31774042C2semidirect
11. #31774040C2semidirect
12. #31774035C2semidirect
13. #31774002C2semidirect
14. #31773969C2semidirect
15. #550×C271C2semidirect
74. #16864644D4semidirect
207. C271#3271semidirect
211. #550D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761902
C2×#225761802
#317745562
#317742842
#317741482
#317741242
#317741192
#317740942
#317740592
#317740422
#317740402
#317740352
#317740022
#317739692
#550×C2712
#3271271