Finite Group Databasebeta

← #46846311

Subgroups — #46846311

Order 34688.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D271C211yesmaximal, characteristic
2#117×D271C211yesmaximal, characteristic
3#31774239C211yesmaximal, characteristic, complemented
4#31774238C211yesmaximal, characteristic, complemented
5#31774235C211yesmaximal, characteristic, complemented
6#31774234C211yesmaximal, characteristic, complemented
7#31774231C211yesmaximal, characteristic, complemented
8#31774229C211yesmaximal, characteristic, complemented
9#31774226C211yesmaximal, characteristic, complemented
10#31773938C211yesmaximal, characteristic, complemented
11#31773936C211yesmaximal, characteristic, complemented
12#31773904C211yesmaximal, characteristic, complemented
13#31773904C211yesmaximal, characteristic, complemented
14#31773859C211yesmaximal, characteristic
15#485×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18C2×#16864637C2211yescharacteristic
19C2×#16864637C2211yescharacteristic
20#22576140C2211yescharacteristic
21#22576139C2211yescharacteristic
22#22576138—21no—
23#22576137—21no—
24#22576136C2211yescharacteristic
25#22576119C2211yescharacteristic
26#22576118C2211yescharacteristic
27#22576118C2211yescharacteristic
28#22576117—21no—
29#22576116—21no—
30C4⋊C4×D271—21no—
31C4⋊C4×D271C2211yescharacteristic
32C4⋊C4×D271C2211yescharacteristic
33C4⋊C4×D271C2211yescharacteristic
34C22⋊C4×D271—21no—
35C22⋊C4×D271—21no—
36C42×D271—21no—
37#22576094C2211yescharacteristic, complemented
38#22576093C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41#22576091C2211yescharacteristic, complemented
42#22576091C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576084C2211yescharacteristic, complemented
46#22576084C2211yescharacteristic, complemented
47#22576083C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52C22⋊Q8×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic
59C23×D271D411yescharacteristic
60C2×Dic542—21no—
61C2×Dic542C2311yescharacteristic
62Q8×D271—22no—
63Q8×D271—22no—
64C22×Dic271D411yescharacteristic
65C2×C4×D271—22no—
66C2×C4×D271—22no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271D411yescharacteristic
72C2×C4×D271D411yescharacteristic
73C2×C4×D271C2311yescharacteristic
74C2×C4×D271C2311yescharacteristic
75#16864637—22no—
76#16864637—22no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C4⋊Dic271—21no—
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271—21no—
97C2×Q8×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C2×C8×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C4⋊C4×C271—21no—
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C22×D271—22no—
109C22×D271—22no—
110C22×D271—41no—
111C22×D271C2×D411yescharacteristic
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—42no—
116C2×Dic271—22no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C4×D271—22no—
124C4×D271—21no—
125C4×D271—21no—
126C4×D271—22no—
127C4×D271—42no—
128C4×D271—42no—
129C4×D271—42no—
130C4×D271—42no—
131C4×D271#4112yes—
132C4×D271#4112yes—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271C2×D411yescharacteristic
136Q8×C271—21no—
137Q8×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271—21no—
148C22×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271C22×D411yescharacteristic
152C2×D271—21no—
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—44no—
157C2×D271#12312yes—
158C2×D271#13712yes—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271—41no—
164C4×C271C2×#4111yescharacteristic
165C4×C271C2×#4111yescharacteristic
166Dic271—21no—
167Dic271—41no—
168Dic271—41no—
169Dic271—41no—
170Dic271—41no—
171Dic271#12312yes—
172C2×C271—41no—
173C2×C271C2×#12311yescharacteristic
174C2×C271C2×#13711yescharacteristic
175C2×C271#57511yescharacteristic
176D271—22no—
177D271—22no—
178C271#309511yescharacteristic, complemented, Sylow
179#3095—2711nomaximal, Sylow
180C2×C22⋊Q8—2711no—
181C2×#117—2711no—
182#485—2711no—
183#485—2711no—
184#485—2711no—
185#485D27111yescharacteristic, complemented
186#484—2711no—
187#484—2711no—
188#483—2711no—
189#483—2711no—
190#428—2711no—
191#428—2711no—
192#419—2711no—
193#419—2711no—
194#407—2711no—
195C22×Q8—2711no—
196C23×C4—2711no—
197C2×M16—2711no—
198C2×M16—2711no—
199C22⋊Q8—2711no—
200C22⋊Q8—2711no—
201C22⋊Q8—2711no—
202C22⋊Q8—5421no—
203C22⋊Q8—5421no—
204C22⋊Q8C2×D27111yescharacteristic
205#117—2711no—
206#117—2711no—
207#117—2711no—
208#117—5421no—
209#117—5421no—
210#117C2×D27111yescharacteristic
211C2×C4⋊C4—2711no—
212C2×C4⋊C4—2711no—
213C2×C4⋊C4—2711no—
214C2×C4⋊C4—5421no—
215C2×C22⋊C4—5421no—
216C2×C22⋊C4—5421no—
217C2×C42—5421no—
218#107—2711no—
219#107—2711no—
220#107—2711no—
221#107C2×D27111yescharacteristic, complemented
222#106—2711no—
223#106—2711no—
224#106—2711no—
225#106C2×D27111yescharacteristic, complemented
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103—2711no—
231#103—2711no—
232#103C2×D27111yescharacteristic, complemented
233#103C2×D27111yescharacteristic, complemented
234C22⋊C8—2711no—
235C22⋊C8—2711no—
236C22⋊C8—2711no—
237C22⋊C8C2×D27111yescharacteristic
238C24—2711no—
239C2×Q8—2711no—
240C2×Q8—5421no—
241C2×Q8—5422no—
242C2×Q8—5422no—
243C2×Q8C22×D27111yescharacteristic
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—2711no—
248C22×C4—2711no—
249C22×C4—5422no—
250C22×C4—5422no—
251C22×C4—5421no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4C22×D27111yescharacteristic
256M16—5422no—
257M16—5422no—
258C2×C8—2711no—
259C2×C8—2711no—
260C2×C8C22×D27111yescharacteristic
261C2×C8C22×D27111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—2711no—
264C4⋊C4—2711no—
265C4⋊C4—2711no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4C22×D27111yescharacteristic
273C4⋊C4C22×D27111yescharacteristic
274C4⋊C4C22×D27111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C42—21no—
284C42—5421no—
285C42—5421no—
286C42—5421no—
287C23—2711no—
288C23—2711no—
289C23—5422no—
290C23—5422no—
291C23—10841no—
292C23D4×D27111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—5421no—
296Q8—5421no—
297Q8—10842no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—2711no—
305C2×C4—2712no—
306C2×C4—2712no—
307C2×C4—2711no—
308C2×C4—5422no—
309C2×C4—5422no—
310C2×C4—5422no—
311C2×C4—5421no—
312C2×C4—5421no—
313C2×C4—5421no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—10842no—
318C2×C4—10842no—
319C2×C4—10842no—
320C2×C4—10842no—
321C2×C4D4×D27111yescharacteristic
322C2×C4C23×D27111yescharacteristic
323C8—21no—
324C8—21no—
325C8—5421no—
326C8—5421no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—2712no—
331C22—2712no—
332C22—5421no—
333C22—5421no—
334C22—5421no—
335C22—5421no—
336C22—10844no—
337C22C2×D4×D27111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—2712no—
344C4—5421no—
345C4—10841no—
346C4—10841no—
347C4—10841no—
348C4—10841no—
349C4#41×D27111yescharacteristic
350C4#41×D27111yescharacteristic
351C2—41no—
352C2—5422no—
353C2—5422no—
354C2#123×D27111yescharacteristic
355C2#137×D27111yescharacteristic
356C2#3177469411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#485×C271C2
socleC22×C271C22×D4
prime radicalC23×D271D4
hypercenter#485D271
hypocenterC271#3095
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #31774239C2semidirect
4. #31774238C2semidirect
5. #31774235C2semidirect
6. #31774234C2semidirect
7. #31774231C2semidirect
8. #31774229C2semidirect
9. #31774226C2semidirect
10. #31773938C2semidirect
11. #31773936C2semidirect
12. #31773904C2semidirect
13. #31773904C2semidirect
15. #485×C271C2semidirect
37. #22576094C22semidirect
38. #22576093C22semidirect
39. #22576093C22semidirect
40. #22576092C22semidirect
41. #22576091C22semidirect
42. #22576091C22semidirect
43. #22576085C22semidirect
44. #22576085C22semidirect
45. #22576084C22semidirect
46. #22576084C22semidirect
47. #22576083C22semidirect
48. #22576083C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
56. #103×C271C22semidirect
57. #103×C271C22semidirect
178. C271#3095semidirect
185. #485D271semidirect
221. #107C2×D271semidirect
225. #106C2×D271semidirect
232. #103C2×D271semidirect
233. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D2712
#117×D2712
#317742392
#317742382
#317742352
#317742342
#317742312
#317742292
#317742262
#317739382
#317739362
#317739042
#317738592
#485×C2712
#3095271