Finite Group Databasebeta

← #46847870

Subgroups — #46847870

Order 34688.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576190C211yesmaximal, characteristic, complemented
2C22×#16864642C211yesmaximal, characteristic, complemented
3D4×Dic542C211yesmaximal, characteristic, complemented
4#31774292C211yesmaximal, characteristic, complemented
5#31774141C211yesmaximal, characteristic, complemented
6#31774136C211yesmaximal, characteristic, complemented
7#31774116C211yesmaximal, characteristic, complemented
8#31774102C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774052C211yesmaximal, characteristic, complemented
11#31774048C211yesmaximal, characteristic, complemented
12#31774035C211yesmaximal, characteristic, complemented
13#31773997C211yesmaximal, characteristic, complemented
14#31773969C211yesmaximal, characteristic, complemented
15#546×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic
17C22×Dic542C2211yescharacteristic, complemented
18#22576190—24no—
19C2×#16864644C2211yescharacteristic, complemented
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864642—24no—
23C2×#16864642C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864642C2211yescharacteristic
27C2×#16864642C2211yescharacteristic
28C2×#16864642C2211yescharacteristic
29C2×#16864638C2211yescharacteristic, complemented
30C22×C271⋊C8C2211yescharacteristic
31#22576160C2211yescharacteristic, complemented
32#22576159—21no—
33#22576139C2211yescharacteristic
34#22576139C2211yescharacteristic, complemented
35#22576137—21no—
36#22576137—21no—
37C4×Dic542C2211yescharacteristic, complemented
38D4×Dic271—21no—
39#22576093C2211yescharacteristic, complemented
40C4⋊(C271⋊C8)C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576083C2211yescharacteristic, complemented
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47C22⋊(C271⋊C8)C2211yescharacteristic
48C22⋊(C271⋊C8)C2211yescharacteristic, complemented
49#22576071C2211yescharacteristic, complemented
50C2×#41×C271C2211yescharacteristic
51C22×D4×C271C2211yescharacteristic
52#124×C271C2211yescharacteristic
53#123×C271—21no—
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271—21no—
56#121×C271C2211yescharacteristic
57#120×C271—21no—
58C4×D4×C271C2211yescharacteristic
59C4×D4×C271C2211yescharacteristic
60C2×C22⋊C4×C271—21no—
61C2×Dic542—22no—
62C2×Dic542—22no—
63C2×Dic542—21no—
64C2×Dic542—21no—
65C2×Dic542—41no—
66C2×Dic542—41no—
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70C22×Dic271—21no—
71C22×Dic271—21no—
72#16864644—22no—
73#16864644—22no—
74#16864644—41no—
75#16864642—22no—
76#16864642—22no—
77#16864642—22no—
78#16864642—22no—
79#16864642—41no—
80#16864642D414yescomplemented
81#16864638—22no—
82C2×C271⋊C8—22no—
83C2×C271⋊C8C2311yescharacteristic
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C4⋊Dic271—21no—
88C4⋊Dic271—21no—
89C4⋊Dic271C2311yescharacteristic
90Dic271⋊C4—21no—
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95C4×Dic271—21no—
96C24×C271—21no—
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—22no—
101C2×D4×C271—21no—
102C2×D4×C271—21no—
103C2×D4×C271—21no—
104C2×D4×C271—41no—
105C2×D4×C271D411yescharacteristic
106C2×D4×C271C2311yescharacteristic
107C2×D4×C271C2311yescharacteristic
108C22×C4×C271—21no—
109C22×C4×C271—21no—
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C4⋊C4×C271—21no—
113C4⋊C4×C271D411yescharacteristic
114C4⋊C4×C271C2311yescharacteristic
115C22⋊C4×C271—21no—
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
122C42×C271C2311yescharacteristic
123Dic542—21no—
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—42no—
129Dic542—42no—
130Dic542C2×D412yes—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—42no—
136C2×Dic271—42no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139C271⋊C8—21no—
140C271⋊C8C2×D412yes—
141C23×C271—21no—
142C23×C271—21no—
143C23×C271—42no—
144C23×C271—42no—
145C23×C271—81no—
146C23×C271C2×D411yescharacteristic
147C23×C271C2×D411yescharacteristic
148Q8×C271—21no—
149Q8×C271—21no—
150D4×C271—22no—
151D4×C271—21no—
152D4×C271—21no—
153D4×C271—21no—
154D4×C271—42no—
155D4×C271—42no—
156D4×C271—81no—
157D4×C271C2×D412yes—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—42no—
165C2×C4×C271—81no—
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2×D411yescharacteristic
170C2×C4×C271C2411yescharacteristic
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—22no—
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—84no—
183C22×C271#13712yescomplemented
184C22×C271C22×D411yescharacteristic
185C4×C271—21no—
186C4×C271—21no—
187C4×C271—21no—
188C4×C271—21no—
189C4×C271—41no—
190C4×C271—41no—
191C4×C271C22×D411yescharacteristic
192C4×C271H(5,𝔽2)11yescharacteristic
193Dic271—22no—
194Dic271—41no—
195Dic271—41no—
196Dic271—41no—
197C2×C271—22no—
198C2×C271—41no—
199C2×C271—42no—
200C2×C271—81no—
201C2×C271D4211yescharacteristic
202C2×C271C2×#13711yescharacteristic
203C2×C271#57711yescharacteristic
204C271#326611yescharacteristic, complemented, Sylow
205#3266—2711nomaximal, Sylow
206C2×#137—2711no—
207C22×SD16—2711no—
208D4×Q8—2711no—
209#546D27111yescharacteristic
210#497—2711no—
211#468—2711no—
212#467—2711no—
213#464—2711no—
214#461—2711no—
215#452—2711no—
216#451—2711no—
217#450—2711no—
218#448—2711no—
219#440—2711no—
220#435—2711no—
221C2×#41C2×D27111yescharacteristic
222C22×Q8—2711no—
223C22×Q8—2711no—
224C22×D4C2×D27111yescharacteristic
225#137—5424no—
226C2×Q16—2711no—
227C2×Q16—2711no—
228C2×Q16—2711no—
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—5424no—
236C2×M16—2711no—
237C22×C8—2711no—
238C4⋊Q8—2711no—
239C4⋊Q8—5421no—
240#124C2×D27111yescharacteristic
241#123—21no—
242C22⋊Q8—2711no—
243C22⋊Q8—2711no—
244C22⋊Q8—5421no—
245C22⋊Q8—5421no—
246C22⋊Q8C2×D27111yescharacteristic
247#121—21no—
248#121C2×D27111yescharacteristic
249#120—21no—
250C4×Q8—2711no—
251C4×D4—5421no—
252C4×D4C2×D27111yescharacteristic
253C4×D4C2×D27111yescharacteristic
254C2×C22⋊C4—21no—
255#107—2711no—
256C4⋊C8—2711no—
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#103—2711no—
261#102—2711no—
262#102—2711no—
263C22⋊C8—2711no—
264C22⋊C8—2711no—
265#97—2711no—
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—2711no—
270C2×Q8—2711no—
271C2×Q8—2711no—
272C2×Q8—5422no—
273C2×Q8—5422no—
274C2×Q8—5421no—
275C2×Q8—5421no—
276C2×Q8—10841no—
277C2×Q8—10841no—
278C2×Q8C22×D27111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D27111yescharacteristic
285C2×D4C22×D27111yescharacteristic
286C2×D4C22×D27111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4C22×D27111yescharacteristic
292C22×C4C22×D27111yescharacteristic
293Q16—5422no—
294Q16—5422no—
295Q16—10841no—
296SD16—2714no—
297SD16—5422no—
298SD16—5422no—
299SD16—5422no—
300SD16—5422no—
301SD16—10841no—
302M16—5422no—
303C2×C8—2711no—
304C2×C8—2711no—
305C2×C8—2711no—
306C2×C8—2711no—
307C2×C8—5422no—
308C4⋊C4—21no—
309C4⋊C4—2711no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4—5421no—
313C4⋊C4—5421no—
314C4⋊C4—5421no—
315C4⋊C4C22⋊D27111yescharacteristic
316C4⋊C4C22×D27111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4C22⋊D27111yescharacteristic
325C22⋊C4C22⋊D27111yescharacteristic
326C42—5421no—
327C42C22×D27111yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D27111yescharacteristic
334C23C2×C22⋊D27111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—2712no—
338Q8—5421no—
339Q8—5421no—
340Q8—5421no—
341Q8—5421no—
342Q8—5421no—
343Q8—10842no—
344Q8—10842no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D27112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—10842no—
366C2×C4—10842no—
367C2×C4C2×C22⋊D27111yescharacteristic
368C2×C4C2×C22⋊D27111yescharacteristic
369C2×C4C2×C22⋊D27111yescharacteristic
370C2×C4C2×C22⋊D27111yescharacteristic
371C2×C4C23×D27111yescharacteristic
372C8—2712no—
373C8—5421no—
374C8—5421no—
375C8—5421no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#2257619012yescomplemented
389C22C22×C22⋊D27111yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—5422no—
397C4—10841no—
398C4—10841no—
399C4—10841no—
400C4C2×D4×D27111yescharacteristic
401C4#2257620511yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D27111yescharacteristic
407C2C2×#2257619011yescharacteristic
408C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting, prime radical#546×C271C2
socleC22×C271C22×D4
hypercenter#546D271
hypocenterC271#3266
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22576190C2semidirect
2. C22×#16864642C2semidirect
3. D4×Dic542C2semidirect
4. #31774292C2semidirect
5. #31774141C2semidirect
6. #31774136C2semidirect
7. #31774116C2semidirect
8. #31774102C2semidirect
9. #31774059C2semidirect
10. #31774052C2semidirect
11. #31774048C2semidirect
12. #31774035C2semidirect
13. #31773997C2semidirect
14. #31773969C2semidirect
17. C22×Dic542C22semidirect
19. C2×#16864644C22semidirect
20. C2×#16864644C22semidirect
21. C2×#16864644C22semidirect
29. C2×#16864638C22semidirect
31. #22576160C22semidirect
34. #22576139C22semidirect
37. C4×Dic542C22semidirect
39. #22576093C22semidirect
40. C4⋊(C271⋊C8)C22semidirect
41. #22576084C22semidirect
42. #22576083C22semidirect
43. #22576083C22semidirect
44. #22576083C22semidirect
48. C22⋊(C271⋊C8)C22semidirect
49. #22576071C22semidirect
80. #16864642D4semidirect
183. C22×C271#137semidirect
204. C271#3266semidirect
388. C22#22576190semidirect

Maximal subgroups

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

subgroupindex
C2×#225761902
C22×#168646422
D4×Dic5422
#317742922
#317741412
#317741362
#317741162
#317741022
#317740592
#317740522
#317740482
#317740352
#317739972
#317739692
#546×C2712
#3266271