Finite Group Databasebeta

← #46846317

Subgroups — #46846317

Order 34688.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774463C211yesmaximal, characteristic
2#31774377C211yesmaximal, characteristic, complemented
3#31774223C211yesmaximal, characteristic, complemented
4#31774221C211yesmaximal, characteristic, complemented
5#31774220C211yesmaximal, characteristic, complemented
6#31774216C211yesmaximal, characteristic, complemented
7#31774214C211yesmaximal, characteristic, complemented
8#31774212C211yesmaximal, characteristic, complemented
9#31774208C211yesmaximal, characteristic, complemented
10#31773938C211yesmaximal, characteristic, complemented
11#31773932C211yesmaximal, characteristic, complemented
12#31773900C211yesmaximal, characteristic, complemented
13#102×D271C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#482×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864637C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576156—21no—
21#22576146—21no—
22#22576135C2211yescharacteristic
23#22576134C2211yescharacteristic
24#22576131C2211yescharacteristic
25#22576130—21no—
26#22576128—21no—
27C4×Dic542—21no—
28C4×D1084—21no—
29(C22⋊D271)⋊C4—21no—
30(C22⋊D271)⋊C4—21no—
31#22576118C2211yescharacteristic
32#22576118C2211yescharacteristic
33#22576118C2211yescharacteristic, complemented
34C2×C4⋊Dic271C2211yescharacteristic, complemented
35C4⋊C4×D271C2211yescharacteristic
36C4⋊C4×D271C2211yescharacteristic, complemented
37#22576093C2211yescharacteristic, complemented
38#22576093C2211yescharacteristic, complemented
39#22576093C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41#22576092C2211yescharacteristic, complemented
42#22576091C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52#121×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic, complemented
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864654—22no—
61#16864654—22no—
62C2×Dic542D411yescharacteristic
63C2×D1084—21no—
64C2×D1084D411yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271D411yescharacteristic
70C2×C22⋊D271D411yescharacteristic
71D4×D271—22no—
72D4×D271—22no—
73C22×Dic271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77C2×C4×D271C2311yescharacteristic
78#16864637—22no—
79C2×C271⋊C8C2311yescharacteristic
80C2×C271⋊C8C2311yescharacteristic
81C8×D271—22no—
82C4⋊Dic271—21no—
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88C22⋊Dic271—21no—
89(C2×D271)⋊C4—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×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C22×D271—21no—
109C22×D271—42no—
110C22×D271—81no—
111C22×D271—81no—
112C22×D271C2×D411yescharacteristic
113C22×D271C2×D411yescharacteristic
114Dic542—22no—
115D1084—22no—
116D1084—21no—
117D1084—21no—
118D1084—81no—
119C22⋊D271—22no—
120C22⋊D271—22no—
121C22⋊D271—42no—
122C22⋊D271—81no—
123C22⋊D271—81no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—81no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—81no—
133C4×D271—81no—
134C4×D271#4112yes—
135C4×D271#4112yes—
136C271⋊C8—21no—
137C271⋊C8—21no—
138C23×C271—21no—
139C23×C271C2×D411yescharacteristic
140D4×C271—21no—
141D4×C271—21no—
142D4×C271—81no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—84no—
168C2×D271#12312yes—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271—41no—
173C4×C271C2×#4111yescharacteristic
174C4×C271C2×#4111yescharacteristic
175Dic271—21no—
176Dic271—41no—
177Dic271—41no—
178Dic271—41no—
179Dic271#12312yes—
180C2×C271—41no—
181C2×C271—81no—
182C2×C271C2×#12311yescharacteristic
183C2×C271#57611yescharacteristic
184C2×C271#57711yescharacteristic
185D271—22no—
186D271—41no—
187D271—81no—
188C271#309611yescharacteristic, complemented, Sylow
189#3096—2711nomaximal, Sylow
190#535—2711no—
191#520—2711no—
192#482—2711no—
193#482—2711no—
194#482—2711no—
195#482D27111yescharacteristic, complemented
196#481—2711no—
197#481—2711no—
198#480—2711no—
199#480—2711no—
200#428—2711no—
201#427—2711no—
202#418—2711no—
203C2×#102—2711no—
204#408—2711no—
205C2×#41—2711no—
206C22×D4—2711no—
207C2×M16—2711no—
208C22×C8—2711no—
209#127—5421no—
210#124—5421no—
211#121—2711no—
212#121—2711no—
213#121—2711no—
214#121C2×D27111yescharacteristic
215#120—5421no—
216#120—5421no—
217C4×Q8—5421no—
218C4×D4—5421no—
219C4×D4—5421no—
220C4×D4—5421no—
221#117—2711no—
222#117—2711no—
223#117—2711no—
224#117C2×D27111yescharacteristic, complemented
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—2711no—
227C2×C4⋊C4—2711no—
228#107—2711no—
229#107—2711no—
230#107—2711no—
231#107C2×D27111yescharacteristic, complemented
232#106—2711no—
233#106—2711no—
234#106—2711no—
235#106C2×D27111yescharacteristic, complemented
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102—2711no—
242#102C2×D27111yescharacteristic
243#102C2×D27111yescharacteristic
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246C22⋊C8—2711no—
247C22⋊C8C2×D27111yescharacteristic, complemented
248C24—5421no—
249#41—5422no—
250#41—5422no—
251C2×Q8—2711no—
252C2×D4—21no—
253C2×D4—2711no—
254C2×D4—2711no—
255C2×D4—2711no—
256C2×D4—2711no—
257C2×D4—5421no—
258C2×D4—5422no—
259C2×D4—5422no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4C22×D27111yescharacteristic
264C22×C4—2711no—
265C22×C4—2711no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4C22×D27111yescharacteristic
270M16—5422no—
271C2×C8—2711no—
272C2×C8—2711no—
273C2×C8—5422no—
274C2×C8C22×D27111yescharacteristic
275C2×C8C22×D27111yescharacteristic
276C4⋊C4—2711no—
277C4⋊C4—2711no—
278C4⋊C4—2711no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4C22×D27111yescharacteristic
283C4⋊C4C22×D27111yescharacteristic
284C4⋊C4C22×D27111yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C42—21no—
294C42—5421no—
295C42—5421no—
296C42—5421no—
297C23—21no—
298C23—2711no—
299C23—2711no—
300C23—5421no—
301C23—10842no—
302C23—21681no—
303C23—21681no—
304C23D4×D27111yescharacteristic
305Q8—5422no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—5422no—
310D4—5422no—
311D4—5422no—
312D4—5421no—
313D4—5421no—
314D4—10842no—
315D4—21681no—
316D4—21681no—
317D4—21681no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—2711no—
324C2×C4—2712no—
325C2×C4—2712no—
326C2×C4—2711no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—21681no—
333C2×C4—21681no—
334C2×C4—21681no—
335C2×C4D4×D27111yescharacteristic
336C2×C4C23×D27111yescharacteristic
337C8—21no—
338C8—21no—
339C8—5421no—
340C8—5421no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—2712no—
348C22—5421no—
349C22—5421no—
350C22—5421no—
351C22—5421no—
352C22—5421no—
353C22—10841no—
354C22—10841no—
355C22—10841no—
356C22—21684no—
357C22C2×D4×D27111yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—2712no—
363C4—5421no—
364C4—10841no—
365C4—10841no—
366C4—10841no—
367C4#41×D27111yescharacteristic
368C4#41×D27111yescharacteristic
369C2—41no—
370C2—81no—
371C2—5422no—
372C2—10841no—
373C2—21681no—
374C2#123×D27111yescharacteristic
375C2#3177470111yescharacteristic
376C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#482×C271C2
socleC22×C271C22×D4
prime radical#31774463C2
hypercenter#482D271
hypocenterC271#3096
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #31774377C2semidirect
3. #31774223C2semidirect
4. #31774221C2semidirect
5. #31774220C2semidirect
6. #31774216C2semidirect
7. #31774214C2semidirect
8. #31774212C2semidirect
9. #31774208C2semidirect
10. #31773938C2semidirect
11. #31773932C2semidirect
12. #31773900C2semidirect
13. #102×D271C2semidirect
14. #31773862C2semidirect
15. #482×C271C2semidirect
33. #22576118C22semidirect
34. C2×C4⋊Dic271C22semidirect
36. C4⋊C4×D271C22semidirect
37. #22576093C22semidirect
38. #22576093C22semidirect
39. #22576093C22semidirect
40. #22576092C22semidirect
41. #22576092C22semidirect
42. #22576091C22semidirect
49. C22⋊(C271⋊C8)C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. (C2×D271)⋊C8C22semidirect
53. #117×C271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
58. C22⋊C8×C271C22semidirect
188. C271#3096semidirect
195. #482D271semidirect
224. #117C2×D271semidirect
231. #107C2×D271semidirect
235. #106C2×D271semidirect
247. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744632
#317743772
#317742232
#317742212
#317742202
#317742162
#317742142
#317742122
#317742082
#317739382
#317739322
#317739002
#102×D2712
#317738622
#482×C2712
#3096271