Finite Group Databasebeta

← #46845191

Subgroups — #46845191

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×#22576187C211yesmaximal, characteristic
2#31774379C211yesmaximal, characteristic, complemented
3#31774369C211yesmaximal, characteristic, complemented
4#31774008C211yesmaximal, characteristic, complemented
5#31774007C211yesmaximal, characteristic, complemented
6#31773998C211yesmaximal, characteristic, complemented
7#31773995C211yesmaximal, characteristic, complemented
8#31773982C211yesmaximal, characteristic, complemented
9C4×D2168C211yesmaximal, characteristic, complemented
10#31773977C211yesmaximal, characteristic, complemented
11#31773936C211yesmaximal, characteristic, complemented
12#31773892C211yesmaximal, characteristic, complemented
13#102×D271C211yesmaximal, characteristic, complemented
14#31773851C211yesmaximal, characteristic, complemented
15C4×SD16×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18#22576187C418yescomplemented
19C2×#16864641C2211yescharacteristic
20C2×D2168C2211yescharacteristic
21C2×#16864639C2211yescharacteristic
22C2×#16864637C2211yescharacteristic
23C2×C8×D271C2211yescharacteristic
24Q8×Dic271C2211yescharacteristic, complemented
25C4×D1084—21no—
26C4×D1084C2211yescharacteristic
27C4×D1084C2211yescharacteristic, complemented
28D4×Dic271C2211yescharacteristic
29#22576122—21no—
30#22576122C2211yescharacteristic
31#22576122C2211yescharacteristic, complemented
32C4×C22⋊D271—21no—
33(C22⋊D271)⋊C4—21no—
34#22576118C2211yescharacteristic
35#22576117—21no—
36#22576117C2211yescharacteristic
37C4⋊C4×D271—21no—
38C4⋊C4×D271C2211yescharacteristic
39C2×(C2×D271)⋊C4—21no—
40C22⋊C4×D271—21no—
41#22576094C2211yescharacteristic, complemented
42#22576093C2211yescharacteristic, complemented
43#22576091C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic, complemented
50#22576071C2211yescharacteristic, complemented
51#22576070C2211yescharacteristic, complemented
52C8×Dic271C2211yescharacteristic, complemented
53C2×SD16×C271C2211yescharacteristic
54C4×Q8×C271C2211yescharacteristic, complemented
55C4×D4×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic
59C4×C8×C271C2211yescharacteristic, complemented
60C23×D271—21no—
61#16864655—22no—
62#16864655C2×C414yescomplemented
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C22⋊D271—21no—
67D4×D271—22no—
68D4×D271C2×C414yescomplemented
69C22×Dic271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77#16864641C2×C414yescomplemented
78D2168C2×C414yescomplemented
79#16864639C2×C414yescomplemented
80#16864637C2×C414yescomplemented
81C2×C271⋊C8C2311yescharacteristic
82C2×C271⋊C8C2311yescharacteristic
83C8×D271—22no—
84C4⋊Dic271—21no—
85C4⋊Dic271C2311yescharacteristic
86C4⋊Dic271C2311yescharacteristic
87Dic271⋊C4—21no—
88Dic271⋊C4D411yescharacteristic
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97(C2×D271)⋊C4D411yescharacteristic
98C4×Dic271—21no—
99C4×Dic271—21no—
100C4×Dic271C2311yescharacteristic
101C2×Q8×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104SD16×C271C2×C414yescomplemented
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271—21no—
108C4⋊C4×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C42×C271—21no—
112C42×C271C2311yescharacteristic
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—24no—
116C22×D271—81no—
117C22×D271—81no—
118C22×D271C2×D411yescharacteristic
119D1084—24no—
120D1084—21no—
121D1084—21no—
122D1084C22×C412yes—
123D1084C22×C412yes—
124C22⋊D271—24no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—81no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C2×Dic271C2×D411yescharacteristic
131C4×D271—24no—
132C4×D271—81no—
133C4×D271—81no—
134C4×D271—81no—
135C4×D271—81no—
136C4×D271—81no—
137C4×D271C22×C412yes—
138C4×D271#4112yes—
139C271⋊C8—21no—
140C271⋊C8C22×C412yes—
141C23×C271—21no—
142Q8×C271—21no—
143Q8×C271C22×C412yes—
144D4×C271—21no—
145D4×C271C22×C412yes—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
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—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—22no—
162C2×D271—22no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—84no—
168C2×D271C4×D412yes—
169C4×C271—22no—
170C4×C271—21no—
171C4×C271—21no—
172C4×C271—41no—
173C4×C271—41no—
174C4×C271C23×C411yescharacteristic
175C4×C271C2×#4111yescharacteristic
176Dic271—21no—
177Dic271—21no—
178Dic271—41no—
179Dic271—41no—
180Dic271C4×D412yes—
181C2×C271—42no—
182C2×C271C2×C4×D411yescharacteristic
183C2×C271#57611yescharacteristic
184C2×C271#57711yescharacteristic
185D271—22no—
186D271—42no—
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×SD16D27111yescharacteristic, complemented
199C4×D8—2711no—
200C4×D8—2711no—
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×SD16C2×D27111yescharacteristic
210C2×D8—2711no—
211C2×D8—2711no—
212C2×M16—2711no—
213C22×C8—2711no—
214C4×Q8—2711no—
215C4×Q8C2×D27111yescharacteristic, complemented
216C4×D4—2711no—
217C4×D4—2711no—
218C4×D4—2711no—
219C4×D4—2711no—
220C4×D4—2711no—
221C4×D4—5421no—
222C4×D4—5421no—
223C4×D4—5421no—
224C4×D4—5421no—
225C4×D4C2×D27111yescharacteristic
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#107—2711no—
235#106—2711no—
236#106C2×D27111yescharacteristic, complemented
237#103—2711no—
238#103C2×D27111yescharacteristic, complemented
239#102—2711no—
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102C2×D27111yescharacteristic
245#97—2711no—
246#97—2711no—
247C4×C8—2711no—
248C4×C8C2×D27111yescharacteristic, complemented
249C24—5421no—
250#41—2714no—
251#41—5422no—
252C2×Q8C22×D27111yescharacteristic
253C2×D4—2714no—
254C2×D4—2711no—
255C2×D4—2711no—
256C2×D4—5421no—
257C2×D4—5422no—
258C2×D4—5421no—
259C2×D4C22×D27111yescharacteristic
260C22×C4—21no—
261C22×C4—2711no—
262C22×C4—5421no—
263C22×C4—5421no—
264C22×C4—5421no—
265C22×C4—5421no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269SD16—2714no—
270SD16C4×D27114yescomplemented
271D8—2714no—
272D8—2714no—
273M16—2714no—
274C2×C8—2711no—
275C2×C8—2711no—
276C2×C8—5422no—
277C2×C8C22×D27111yescharacteristic
278C2×C8C22×D27111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—2711no—
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4C22×D27111yescharacteristic
287C4⋊C4C22×D27111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—2711no—
290C22⋊C4—2711no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C42—21no—
298C42—2711no—
299C42—5421no—
300C42—5421no—
301C42C22×D27111yescharacteristic
302C23—21no—
303C23—2711no—
304C23—5424no—
305C23—5421no—
306C23—5421no—
307C23—21681no—
308C23—21681no—
309Q8—21no—
310Q8C2×C4×D27112yes—
311D4—21no—
312D4—2712no—
313D4—2712no—
314D4—5424no—
315D4—5424no—
316D4—5421no—
317D4—5421no—
318D4C2×C4×D27112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
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—21681no—
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—41no—
346C22—41no—
347C22—2712no—
348C22—5421no—
349C22—5421no—
350C22—5422no—
351C22—5422no—
352C22—10841no—
353C22—10841no—
354C22—10841no—
355C22—10841no—
356C22—21684no—
357C22C2×D4×D27111yescharacteristic
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—2712no—
364C4—5421no—
365C4—5421no—
366C4—10841no—
367C4—10841no—
368C4C22×C4×D27111yescharacteristic
369C4#41×D27111yescharacteristic
370C2—42no—
371C2—5422no—
372C2—10842no—
373C2—10842no—
374C2C4×D4×D27111yescharacteristic
375C2#3177469911yescharacteristic
376C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutatorC4×C271C23×C4
FrattiniC2×C4C23×D271
FittingC4×SD16×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576187C2
cosocleC2×C4×C271C24
hypercenterC4×SD16D271
hypocenterC271#2946
LCSG ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #31774379C2semidirect
3. #31774369C2semidirect
4. #31774008C2semidirect
5. #31774007C2semidirect
6. #31773998C2semidirect
7. #31773995C2semidirect
8. #31773982C2semidirect
9. C4×D2168C2semidirect
10. #31773977C2semidirect
11. #31773936C2semidirect
12. #31773892C2semidirect
13. #102×D271C2semidirect
14. #31773851C2semidirect
15. C4×SD16×C271C2semidirect
18. #22576187C4semidirect
24. Q8×Dic271C22semidirect
27. C4×D1084C22semidirect
31. #22576122C22semidirect
41. #22576094C22semidirect
42. #22576093C22semidirect
43. #22576091C22semidirect
44. #22576084C22semidirect
46. #22576082C22semidirect
49. #22576080C22semidirect
50. #22576071C22semidirect
51. #22576070C22semidirect
52. C8×Dic271C22semidirect
54. C4×Q8×C271C22semidirect
56. #106×C271C22semidirect
57. #103×C271C22semidirect
59. C4×C8×C271C22semidirect
62. #16864655C2×C4semidirect
68. D4×D271C2×C4semidirect
77. #16864641C2×C4semidirect
78. D2168C2×C4semidirect
79. #16864639C2×C4semidirect
80. #16864637C2×C4semidirect
104. SD16×C271C2×C4semidirect
188. C271#2946semidirect
198. C4×SD16D271semidirect
215. C4×Q8C2×D271semidirect
236. #106C2×D271semidirect
238. #103C2×D271semidirect
248. C4×C8C2×D271semidirect
270. SD16C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
#317743792
#317743692
#317740082
#317740072
#317739982
#317739952
#317739822
C4×D21682
#317739772
#317739362
#317738922
#102×D2712
#317738512
C4×SD16×C2712
#2946271