Finite Group Databasebeta

← #46848233

Subgroups — #46848233

Order 34688.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774605C211yesmaximal, characteristic
2Q8×C22⋊D271C211yesmaximal, characteristic
3#31774246C211yesmaximal, characteristic, complemented
4#31774233C211yesmaximal, characteristic, complemented
5#31774231C211yesmaximal, characteristic, complemented
6#31774190C211yesmaximal, characteristic, complemented
7#31774136C211yesmaximal, characteristic, complemented
8#31774135C211yesmaximal, characteristic, complemented
9#31774105C211yesmaximal, characteristic, complemented
10#31774051C211yesmaximal, characteristic, complemented
11Q16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic
13C2×#22576084C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#451×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17C2×#16864644C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576159C2211yescharacteristic
21#22576158—21no—
22#22576158C2211yescharacteristic
23#22576150—21no—
24#22576140—21no—
25#22576139—21no—
26#22576139C2211yescharacteristic
27#22576137—21no—
28#22576136—21no—
29#22576136C2211yescharacteristic
30#22576136C2211yescharacteristic
31Q8×Dic271C2211yescharacteristic
32Q8×Dic271C2211yescharacteristic
33C4×C22⋊D271—21no—
34C4×C22⋊D271C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36C2×C4⋊Dic271C2211yescharacteristic
37C4⋊C4×D271C2211yescharacteristic
38#22576094C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576093C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576084C2211yescharacteristic
48#22576084C2211yescharacteristic
49#22576083C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C22×Q8×C271C2211yescharacteristic
53C2×Q16×C271C2211yescharacteristic
54C2×Q16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542—21no—
61C2×Dic542C2311yescharacteristic
62Q8×D271—22no—
63Q8×D271—41no—
64C2×C22⋊D271D411yescharacteristic
65C22×Dic271—21no—
66C2×C4×D271—21no—
67C2×C4×D271—21no—
68C2×C4×D271C2311yescharacteristic
69#16864644—41no—
70C2×C271⋊C8—41no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C8×D271—41no—
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4D411yescharacteristic
88C22⋊Dic271—21no—
89C22⋊Dic271D411yescharacteristic
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4D411yescharacteristic
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96C2×Q8×C271—21no—
97C2×Q8×C271—22no—
98C2×Q8×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×Q8×C271C2311yescharacteristic
101C2×Q8×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104Q16×C271—22no—
105Q16×C271—41no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271C2×D411yescharacteristic
112Dic542—21no—
113Dic542—21no—
114Dic542—42no—
115Dic542—81no—
116C22⋊D271Q1614yescomplemented
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—81no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—21no—
126C4×D271—21no—
127C4×D271—42no—
128C4×D271—81no—
129C271⋊C8—21no—
130C271⋊C8—21no—
131C23×C271C2×D411yescharacteristic
132Q8×C271—21no—
133Q8×C271—21no—
134Q8×C271—21no—
135Q8×C271—21no—
136Q8×C271—21no—
137Q8×C271—42no—
138Q8×C271#4112yes—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—42no—
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C8×C271—21no—
149C8×C271—21no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271C2×Q1612yes—
153C22×C271C22×D411yescharacteristic
154C2×D271—21no—
155C2×D271—21no—
156C2×D271C2×Q1612yes—
157C4×C271—21no—
158C4×C271—22no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271C2×#4111yescharacteristic
163C4×C271H(5,𝔽2)11yescharacteristic
164Dic271—21no—
165Dic271—21no—
166Dic271—41no—
167Dic271—41no—
168Dic271—41no—
169Dic271—41no—
170Dic271C2×Q1612yes—
171C2×C271—22no—
172C2×C271#54611yescharacteristic
173C2×C271C22×Q1611yescharacteristic
174C2×C271#57711yescharacteristic
175D271—22no—
176C271#329911yescharacteristic, complemented, Sylow
177#3299—2711nomaximal, Sylow
178#554—2711no—
179D4×Q8—2711no—
180#487—2711no—
181#484—2711no—
182#484—2711no—
183#477—2711no—
184#467—2711no—
185#467—2711no—
186#462—2711no—
187#451—2711no—
188#451D27111yescharacteristic, complemented
189C4×Q16—2711no—
190C8×D4—2711no—
191C2×#103—2711no—
192C2×#103—2711no—
193C22×Q8—2711no—
194C22×Q8C2×D27111yescharacteristic
195C2×Q16—2711no—
196C2×Q16C2×D27111yescharacteristic
197C2×Q16C2×D27111yescharacteristic
198C22×C8—2711no—
199C22×C8—2711no—
200C4⋊Q8—2711no—
201C4⋊Q8—2711no—
202C4⋊Q8—5421no—
203#125—5421no—
204C22⋊Q8—2711no—
205C22⋊Q8—2711no—
206C22⋊Q8—2711no—
207C22⋊Q8—5421no—
208C22⋊Q8—5421no—
209C22⋊Q8—5421no—
210C22⋊Q8—5421no—
211C22⋊Q8C2×D27111yescharacteristic
212C4×Q8—2711no—
213C4×Q8—2711no—
214C4×D4—2711no—
215C4×D4—5421no—
216C4×D4—5421no—
217C2×C4⋊C4—2711no—
218C2×C4⋊C4—2711no—
219#107—2711no—
220#107—2711no—
221#107—2711no—
222C4⋊C8—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103—2711no—
231#103C2×D27111yescharacteristic
232#103C2×D27111yescharacteristic
233C22⋊C8—2711no—
234C22⋊C8C2×D27111yescharacteristic
235C4×C8—2711no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—2711no—
240C2×Q8—5421no—
241C2×Q8—5421no—
242C2×Q8—5422no—
243C2×Q8—10841no—
244C2×Q8C22×D27111yescharacteristic
245C2×Q8C22×D27111yescharacteristic
246C2×Q8C22×D27111yescharacteristic
247C2×D4—2711no—
248C22×C4—21no—
249C22×C4—2711no—
250C22×C4—5421no—
251C22×C4—5421no—
252C22×C4—5421no—
253C22×C4C22×D27111yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—10841no—
257C2×C8—2711no—
258C2×C8—2711no—
259C2×C8—10841no—
260C2×C8—10841no—
261C2×C8C22×D27111yescharacteristic
262C2×C8C22×D27111yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—2711no—
265C4⋊C4—2711no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4C22×D27111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—2711no—
281C22⋊C4—2711no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C42—2711no—
286C42—5421no—
287C42—5421no—
288C23—2711no—
289C23D4×D27111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—5421no—
297Q8—5421no—
298Q8—10842no—
299Q8—21681no—
300Q8#1686465312yes—
301D4—2714no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2711no—
310C2×C4—2711no—
311C2×C4—2711no—
312C2×C4—5421no—
313C2×C4—5421no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—10842no—
319C2×C4—21681no—
320C2×C4—21681no—
321C2×C4D4×D27111yescharacteristic
322C2×C4C23×D27111yescharacteristic
323C8—21no—
324C8—21no—
325C8—5421no—
326C8—5421no—
327C22—21no—
328C22—21no—
329C22—2712no—
330C22—5421no—
331C22—5421no—
332C22Q16×D27112yes—
333C22C2×D4×D27111yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—2712no—
340C4—5421no—
341C4—5421no—
342C4—10841no—
343C4—10841no—
344C4—10841no—
345C4—10841no—
346C4C2×#1686465311yescharacteristic
347C4#2257620511yescharacteristic
348C2—22no—
349C2—5422no—
350C2#3177453411yescharacteristic
351C2C2×Q16×D27111yescharacteristic
352C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#451×C271C2
socleC22×C271C22×D4
prime radicalC2×C22⋊D271D4
hypercenter#451D271
hypocenterC271#3299
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #31774246C2semidirect
4. #31774233C2semidirect
5. #31774231C2semidirect
6. #31774190C2semidirect
7. #31774136C2semidirect
8. #31774135C2semidirect
9. #31774105C2semidirect
10. #31774051C2semidirect
11. Q16×Dic271C2semidirect
13. C2×#22576084C2semidirect
14. #103×D271C2semidirect
15. #451×C271C2semidirect
116. C22⋊D271Q16semidirect
176. C271#3299semidirect
188. #451D271semidirect

Maximal subgroups

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

subgroupindex
#317746052
Q8×C22⋊D2712
#317742462
#317742332
#317742312
#317741902
#317741362
#317741352
#317741052
#317740512
Q16×Dic2712
(C22⋊D271)⋊C82
C2×#225760842
#103×D2712
#451×C2712
#3299271