Finite Group Databasebeta

← #46845189

Subgroups — #46845189

Order 34688.

506 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
1C2×#22576184C211yesmaximal, characteristic
2#31774376C211yesmaximal, characteristic, complemented
3#31774369C211yesmaximal, characteristic, complemented
4#31774009C211yesmaximal, characteristic, complemented
5#31774007C211yesmaximal, characteristic, complemented
6#31773997C211yesmaximal, characteristic, complemented
7#31773996C211yesmaximal, characteristic, complemented
8#31773988C211yesmaximal, characteristic, complemented
9C4×#16864643C211yesmaximal, characteristic, complemented
10#31773977C211yesmaximal, characteristic, complemented
11#31773938C211yesmaximal, characteristic, complemented
12#31773890C211yesmaximal, characteristic, complemented
13#102×D271C211yesmaximal, characteristic, complemented
14#31773851C211yesmaximal, characteristic, complemented
15C4×D8×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18#22576184C418yescomplemented
19C2×#16864643C2211yescharacteristic
20C2×#16864642C2211yescharacteristic
21C2×#16864639C2211yescharacteristic
22C2×#16864637C2211yescharacteristic
23C2×C8×D271C2211yescharacteristic
24C4×Dic542C2211yescharacteristic, complemented
25#22576125C2211yescharacteristic, complemented
26C4×D1084C2211yescharacteristic
27D4×Dic271C2211yescharacteristic
28D4×Dic271C2211yescharacteristic, complemented
29#22576122C2211yescharacteristic
30C4×C22⋊D271—21no—
31C4×C22⋊D271—21no—
32(C22⋊D271)⋊C4—21no—
33(C22⋊D271)⋊C4—21no—
34#22576118C2211yescharacteristic
35#22576117C2211yescharacteristic
36#22576116—21no—
37C2×Dic271⋊C4—21no—
38C4⋊C4×D271C2211yescharacteristic
39C2×(C2×D271)⋊C4—21no—
40C22⋊C4×D271—21no—
41#22576093C2211yescharacteristic, complemented
42#22576092C2211yescharacteristic, complemented
43#22576091C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576083C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic
50#22576071C2211yescharacteristic, complemented
51#22576070C2211yescharacteristic, complemented
52C8×Dic271C2211yescharacteristic, complemented
53C2×D8×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic, complemented
56#107×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic, complemented
59C4×C8×C271C2211yescharacteristic, complemented
60C23×D271—21no—
61#16864653—22no—
62#16864653C2×C414yescomplemented
63C2×Dic542C2311yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67D4×D271—22no—
68D4×D271C2×C414yescomplemented
69C22×Dic271—21no—
70C22×Dic271—21no—
71C22×Dic271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76#16864643C2×C414yescomplemented
77#16864642C2×C414yescomplemented
78#16864639C2×C414yescomplemented
79#16864637C2×C414yescomplemented
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
82C8×D271—22no—
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4D411yescharacteristic
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96(C2×D271)⋊C4D411yescharacteristic
97C4×Dic271—21no—
98C4×Dic271—21no—
99C4×Dic271—21no—
100C4×Dic271C2311yescharacteristic
101C2×D4×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271—21no—
105D8×C271C2×C414yescomplemented
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22⋊C4×C271—21no—
112C42×C271C2311yescharacteristic
113C22×D271—21no—
114C22×D271—24no—
115C22×D271—81no—
116C22×D271—81no—
117C22×D271C2×D411yescharacteristic
118Dic542—21no—
119Dic542C22×C412yes—
120D1084—21no—
121D1084C22×C412yes—
122C22⋊D271—24no—
123C22⋊D271—24no—
124C2×Dic271—21no—
125C2×Dic271—24no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—81no—
129C2×Dic271—81no—
130C2×Dic271C2×D411yescharacteristic
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—81no—
134C4×D271—81no—
135C4×D271—81no—
136C4×D271C22×C412yes—
137C4×D271#4112yes—
138C271⋊C8—21no—
139C271⋊C8C22×C412yes—
140C23×C271—21no—
141C23×C271—21no—
142D4×C271—21no—
143D4×C271—21no—
144D4×C271C22×C412yes—
145D4×C271C22×C412yes—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—81no—
149C2×C4×C271—81no—
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271C22×C412yes—
155C22×C271—22no—
156C22×C271—22no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—22no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—84no—
168C2×D271C4×D412yes—
169C4×C271—21no—
170C4×C271—21no—
171C4×C271—41no—
172C4×C271—41no—
173C4×C271C23×C411yescharacteristic
174C4×C271C2×#4111yescharacteristic
175Dic271—22no—
176Dic271—21no—
177Dic271—21no—
178Dic271—41no—
179Dic271—41no—
180Dic271C4×D412yes—
181C2×C271—42no—
182C2×C271—42no—
183C2×C271C2×C4×D411yescharacteristic
184C2×C271#57611yescharacteristic
185C2×C271#57711yescharacteristic
186D271—22no—
187D271—42no—
188C271#294611yescharacteristic, complemented, Sylow
189#2946—2711nomaximal, Sylow
190C2×AGL(1,ℤ/8)—2711no—
191#520—2711no—
192#518—2711no—
193#442—2711no—
194#442—2711no—
195#440—2711no—
196#440—2711no—
197C4×SD16—2711no—
198C4×SD16—2711no—
199C4×D8—2711no—
200C4×D8D27111yescharacteristic, complemented
201#428—2711no—
202#416—2711no—
203C2×#102—2711no—
204#405—2711no—
205C2×#41—2711no—
206C22×D4—2711no—
207AGL(1,ℤ/8)—2718no—
208C2×SD16—2711no—
209C2×SD16—2711no—
210C2×D8—2711no—
211C2×D8C2×D27111yescharacteristic
212C2×M16—2711no—
213C22×C8—2711no—
214C4×Q8—2711no—
215C4×Q8—2711no—
216C4×D4—2711no—
217C4×D4—2711no—
218C4×D4—2711no—
219C4×D4—2711no—
220C4×D4—5421no—
221C4×D4—5421no—
222C4×D4—5421no—
223C4×D4—5421no—
224C4×D4C2×D27111yescharacteristic
225C4×D4C2×D27111yescharacteristic, complemented
226#117—2711no—
227#117—2711no—
228#117—5421no—
229C2×C4⋊C4—2711no—
230C2×C4⋊C4—5421no—
231C2×C22⋊C4—5421no—
232C2×C22⋊C4—5421no—
233#107—2711no—
234#107C2×D27111yescharacteristic, complemented
235#106—2711no—
236#106—2711no—
237#103—2711no—
238#103—2711no—
239#102—2711no—
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102C2×D27111yescharacteristic
244#102C2×D27111yescharacteristic, complemented
245#97—2711no—
246#97—2711no—
247C4×C8—2711no—
248C4×C8C2×D27111yescharacteristic, complemented
249C24—5421no—
250#41—2714no—
251#41—5422no—
252C2×Q8—2711no—
253C2×D4—2714no—
254C2×D4—2711no—
255C2×D4—5422no—
256C2×D4—5421no—
257C2×D4—5421no—
258C2×D4C22×D27111yescharacteristic
259C2×D4C22×D27111yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—2711no—
263C22×C4—5421no—
264C22×C4—5421no—
265C22×C4—5421no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269SD16—2714no—
270SD16—2714no—
271D8—2714no—
272D8C4×D27114yescomplemented
273M16—2714no—
274C2×C8—2711no—
275C2×C8—2711no—
276C2×C8—5422no—
277C2×C8C22×D27111yescharacteristic
278C2×C8C22×D27111yescharacteristic
279C4⋊C4—2711no—
280C4⋊C4—2711no—
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4C22×D27111yescharacteristic
287C4⋊C4C22×D27111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—2711no—
291C22⋊C4—2711no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C42—2711no—
298C42—5421no—
299C42—5421no—
300C42—5421no—
301C42C22×D27111yescharacteristic
302C23—21no—
303C23—21no—
304C23—2711no—
305C23—5424no—
306C23—5421no—
307C23—21681no—
308C23—21681no—
309Q8—2712no—
310Q8—5421no—
311D4—21no—
312D4—21no—
313D4—2712no—
314D4—5424no—
315D4—5424no—
316D4—5421no—
317D4C2×C4×D27112yes—
318D4C2×C4×D27112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—2711no—
324C2×C4—2712no—
325C2×C4—2712no—
326C2×C4—2711no—
327C2×C4—2711no—
328C2×C4—5424no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—21681no—
333C2×C4—21681no—
334C2×C4—21681no—
335C2×C4—21681no—
336C2×C4—21681no—
337C2×C4D4×D27111yescharacteristic
338C2×C4D4×D27111yescharacteristic
339C2×C4C23×D27111yescharacteristic
340C8—21no—
341C8—2712no—
342C8—5421no—
343C8C2×C4×D27112yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—2712no—
351C22—5421no—
352C22—5421no—
353C22—5422no—
354C22—10841no—
355C22—10841no—
356C22—21684no—
357C22C2×D4×D27111yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—2712no—
363C4—5422no—
364C4—5421no—
365C4—5421no—
366C4—10841no—
367C4—10841no—
368C4C22×C4×D27111yescharacteristic
369C4#41×D27111yescharacteristic
370C2—42no—
371C2—42no—
372C2—5422no—
373C2—10842no—
374C2C4×D4×D27111yescharacteristic
375C2#3177469811yescharacteristic
376C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutatorC4×C271C23×C4
FrattiniC2×C4C23×D271
FittingC4×D8×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576184C2
cosocleC2×C4×C271C24
hypercenterC4×D8D271
hypocenterC271#2946
LCSG ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
2. #31774376C2semidirect
3. #31774369C2semidirect
4. #31774009C2semidirect
5. #31774007C2semidirect
6. #31773997C2semidirect
7. #31773996C2semidirect
8. #31773988C2semidirect
9. C4×#16864643C2semidirect
10. #31773977C2semidirect
11. #31773938C2semidirect
12. #31773890C2semidirect
13. #102×D271C2semidirect
14. #31773851C2semidirect
15. C4×D8×C271C2semidirect
18. #22576184C4semidirect
24. C4×Dic542C22semidirect
25. #22576125C22semidirect
28. D4×Dic271C22semidirect
41. #22576093C22semidirect
42. #22576092C22semidirect
43. #22576091C22semidirect
44. #22576085C22semidirect
45. #22576083C22semidirect
48. #22576081C22semidirect
50. #22576071C22semidirect
51. #22576070C22semidirect
52. C8×Dic271C22semidirect
55. C4×D4×C271C22semidirect
56. #107×C271C22semidirect
58. #102×C271C22semidirect
59. C4×C8×C271C22semidirect
62. #16864653C2×C4semidirect
68. D4×D271C2×C4semidirect
76. #16864643C2×C4semidirect
77. #16864642C2×C4semidirect
78. #16864639C2×C4semidirect
79. #16864637C2×C4semidirect
105. D8×C271C2×C4semidirect
188. C271#2946semidirect
200. C4×D8D271semidirect
225. C4×D4C2×D271semidirect
234. #107C2×D271semidirect
244. #102C2×D271semidirect
248. C4×C8C2×D271semidirect
272. D8C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
#317743762
#317743692
#317740092
#317740072
#317739972
#317739962
#317739882
C4×#168646432
#317739772
#317739382
#317738902
#102×D2712
#317738512
C4×D8×C2712
#2946271