Finite Group Databasebeta

← #30650240

Subgroups — #30650240

Order 16256.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic, complemented
2Q8×C22⋊D127C211yesmaximal, characteristic, complemented
3#22550520C211yesmaximal, characteristic
4#22550267C211yesmaximal, characteristic, complemented
5#22550119C211yesmaximal, characteristic, complemented
6#22550117C211yesmaximal, characteristic, complemented
7#22550096C211yesmaximal, characteristic, complemented
8#22550079C211yesmaximal, characteristic, complemented
9#22550035C211yesmaximal, characteristic, complemented
10#22550032C211yesmaximal, characteristic, complemented
11#22550027C211yesmaximal, characteristic, complemented
12#22550013C211yesmaximal, characteristic, complemented
13#22549977C211yesmaximal, characteristic, complemented
14#22549947C211yesmaximal, characteristic, complemented
15C22×SD16×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic, complemented
18#16861238—24no—
19C2×Dic508C2211yescharacteristic, complemented
20C2×#14523238C2211yescharacteristic, complemented
21C2×#14523238C2211yescharacteristic, complemented
22C2×#14523236C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523231C2211yescharacteristic, complemented
25#16861207—21no—
26#16861207C2211yescharacteristic, complemented
27#16861196C2211yescharacteristic
28#16861190—21no—
29#16861188—21no—
30#16861188C2211yescharacteristic
31#16861188C2211yescharacteristic
32#16861185—21no—
33#16861185C2211yescharacteristic, complemented
34#16861181—21no—
35#16861180C2211yescharacteristic
36#16861178—21no—
37Q8×Dic127C2211yescharacteristic, complemented
38D4×Dic127C2211yescharacteristic
39C4×C22⋊D127—21no—
40C4×C22⋊D127C2211yescharacteristic
41C2×C22⋊Dic127—21no—
42#16861143C2211yescharacteristic, complemented
43Dic127⋊C8C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861134C2211yescharacteristic, complemented
46#16861133C2211yescharacteristic, complemented
47#16861133C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50C22⋊(C127⋊C8)C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52#16861119C2211yescharacteristic, complemented
53C22×Q8×C127C2211yescharacteristic
54C22×D4×C127C2211yescharacteristic
55C2×SD16×C127—24no—
56C2×SD16×C127C2211yescharacteristic
57C2×SD16×C127C2211yescharacteristic
58C2×SD16×C127C2211yescharacteristic
59C2×SD16×C127C2211yescharacteristic
60C22×C8×C127C2211yescharacteristic
61#14523247—22no—
62#14523247—41no—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65C2×Dic254C2311yescharacteristic
66Q8×D127—22no—
67Q8×D127—41no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127D411yescharacteristic
71C22×Dic127—21no—
72C22×Dic127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2311yescharacteristic
75Dic508—22no—
76#14523238—22no—
77#14523238—41no—
78#14523236—22no—
79#14523236—41no—
80#14523231—22no—
81C2×C127⋊C8C2311yescharacteristic
82C2×C127⋊C8C2311yescharacteristic
83C4⋊Dic127—21no—
84C4⋊Dic127C2311yescharacteristic
85C4⋊Dic127C2311yescharacteristic
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91Dic127⋊C4D411yescharacteristic
92C22⋊Dic127—21no—
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96C22⋊Dic127—21no—
97C22⋊Dic127—21no—
98C22⋊Dic127D411yescharacteristic
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4D411yescharacteristic
101C4×Dic127—21no—
102C4×Dic127C2311yescharacteristic
103C24×C127—21no—
104C2×Q8×C127—21no—
105C2×Q8×C127—22no—
106C2×Q8×C127—41no—
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×D4×C127—22no—
110C2×D4×C127—21no—
111C2×D4×C127—41no—
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116SD16×C127—22no—
117SD16×C127—22no—
118SD16×C127—22no—
119SD16×C127D414yescomplemented
120C2×C8×C127—22no—
121C2×C8×C127C2311yescharacteristic
122C2×C8×C127C2311yescharacteristic
123C22×D127C2×D411yescharacteristic
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129C22⋊D127—22no—
130C22⋊D127—42no—
131C22⋊D127—81no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—42no—
137C2×Dic127—81no—
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C2×Dic127C2×D411yescharacteristic
141C4×D127—21no—
142C4×D127—21no—
143C4×D127—42no—
144C127⋊C8—21no—
145C127⋊C8—21no—
146C23×C127—21no—
147C23×C127—21no—
148C23×C127—42no—
149C23×C127—42no—
150C23×C127—81no—
151C23×C127C2×D411yescharacteristic
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127—21no—
155Q8×C127—42no—
156Q8×C127C2×D412yes—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—42no—
161D4×C127C2×D412yes—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—42no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2411yescharacteristic
169C8×C127—21no—
170C8×C127C2×D412yes—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—22no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—84no—
180C22×C127#13712yescomplemented
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C4×C127—21no—
186C4×C127—22no—
187C4×C127—41no—
188C4×C127C22×D411yescharacteristic
189C4×C127H(5,𝔽2)11yescharacteristic
190Dic127—21no—
191Dic127—21no—
192Dic127—21no—
193Dic127—41no—
194Dic127—41no—
195Dic127—41no—
196Dic127—41no—
197C2×C127—22no—
198C2×C127—42no—
199C2×C127—81no—
200C2×C127D4211yescharacteristic
201C2×C127C2×#13711yescharacteristic
202C2×C127#57711yescharacteristic
203D127—41no—
204#3266—1271nomaximal, Sylow
205C127#326611yescharacteristic, complemented, Sylow
206C2×#137—1271no—
207C22×SD16D12711yescharacteristic, complemented
208D4×Q8—1271no—
209#546—1271no—
210#497—1271no—
211#468—1271no—
212#467—1271no—
213#464—1271no—
214#461—1271no—
215#452—1271no—
216#451—1271no—
217#450—1271no—
218#448—1271no—
219#440—1271no—
220#435—1271no—
221C2×#41—1271no—
222C22×Q8—1271no—
223C22×Q8C2×D12711yescharacteristic
224C22×D4C2×D12711yescharacteristic
225#137—2544no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×Q16—1271no—
229C2×SD16—24no—
230C2×SD16—1271no—
231C2×SD16—1271no—
232C2×SD16C2×D12711yescharacteristic
233C2×SD16C2×D12711yescharacteristic
234C2×SD16C2×D12711yescharacteristic
235C2×SD16C2×D12711yescharacteristic
236C2×M16—1271no—
237C22×C8C2×D12711yescharacteristic
238C4⋊Q8—1271no—
239C4⋊Q8—2541no—
240#124—1271no—
241#123—2541no—
242C22⋊Q8—1271no—
243C22⋊Q8—1271no—
244C22⋊Q8—1271no—
245C22⋊Q8—2541no—
246C22⋊Q8—2541no—
247#121—1271no—
248#121—2541no—
249#120—2541no—
250C4×Q8—1271no—
251C4×D4—1271no—
252C4×D4—1271no—
253C4×D4—2541no—
254C2×C22⋊C4—2541no—
255#107—1271no—
256C4⋊C8—1271no—
257#103—1271no—
258#103—1271no—
259#103—1271no—
260#103—1271no—
261#102—1271no—
262#102—1271no—
263C22⋊C8—1271no—
264C22⋊C8—1271no—
265#97—1271no—
266C24—21no—
267#41—5081no—
268#41—2542no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1271no—
273C2×Q8—1271no—
274C2×Q8—5081no—
275C2×Q8—2541no—
276C2×Q8—2542no—
277C2×Q8C22×D12711yescharacteristic
278C2×Q8C22×D12711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—1271no—
283C2×D4—2541no—
284C2×D4—2541no—
285C2×D4C22×D12711yescharacteristic
286C2×D4C22×D12711yescharacteristic
287C22×C4—21no—
288C22×C4—1271no—
289C22×C4—2541no—
290C22×C4—2541no—
291C22×C4—2541no—
292C22×C4C22×D12711yescharacteristic
293Q16—5081no—
294Q16—2542no—
295Q16—2542no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—5081no—
300SD16—2542no—
301SD16C22⋊D12714yescomplemented
302M16—2542no—
303C2×C8—22no—
304C2×C8—1271no—
305C2×C8—1271no—
306C2×C8C22×D12711yescharacteristic
307C2×C8C22×D12711yescharacteristic
308C4⋊C4—1271no—
309C4⋊C4—1271no—
310C4⋊C4—1271no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4—2541no—
314C4⋊C4—2541no—
315C4⋊C4—2541no—
316C4⋊C4—2541no—
317C22⋊C4—1271no—
318C22⋊C4—1271no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C42—1271no—
327C42—2541no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1271no—
334C23D4×D12711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—5082no—
340Q8—2541no—
341Q8—2541no—
342Q8—2541no—
343Q8—2541no—
344Q8C2×C22⋊D12712yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—10161no—
350D4—5082no—
351D4—2542no—
352D4C2×C22⋊D12712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—1271no—
359C2×C4—1271no—
360C2×C4—1271no—
361C2×C4—10161no—
362C2×C4—5082no—
363C2×C4—5082no—
364C2×C4—2541no—
365C2×C4—2541no—
366C2×C4—2541no—
367C2×C4—2541no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4D4×D12711yescharacteristic
371C2×C4C23×D12711yescharacteristic
372C8—21no—
373C8—2541no—
374C8—2541no—
375C8C2×C22⋊D12712yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2541no—
386C22—2541no—
387C22—2541no—
388C22#1686123812yescomplemented
389C22C2×D4×D12711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—5081no—
394C4—5081no—
395C4—5081no—
396C4—5081no—
397C4—2541no—
398C4—2541no—
399C4—2541no—
400C4C22×C22⋊D12711yescharacteristic
401C4#1686125411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—5081no—
406C2D4×C22⋊D12711yescharacteristic
407C2C2×#1686123811yescharacteristic
408C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×SD16×C127C2
socleC22×C127C22×D4
prime radical#22550520C2
hypercenterC22×SD16D127
hypocenterC127#3266
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. Q8×C22⋊D127C2semidirect
4. #22550267C2semidirect
5. #22550119C2semidirect
6. #22550117C2semidirect
7. #22550096C2semidirect
8. #22550079C2semidirect
9. #22550035C2semidirect
10. #22550032C2semidirect
11. #22550027C2semidirect
12. #22550013C2semidirect
13. #22549977C2semidirect
14. #22549947C2semidirect
15. C22×SD16×C127C2semidirect
17. C2×Q8×D127C22semidirect
19. C2×Dic508C22semidirect
20. C2×#14523238C22semidirect
21. C2×#14523238C22semidirect
24. C2×#14523231C22semidirect
26. #16861207C22semidirect
33. #16861185C22semidirect
37. Q8×Dic127C22semidirect
42. #16861143C22semidirect
43. Dic127⋊C8C22semidirect
44. #16861134C22semidirect
45. #16861134C22semidirect
46. #16861133C22semidirect
47. #16861133C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. #16861119C22semidirect
119. SD16×C127D4semidirect
180. C22×C127#137semidirect
205. C127#3266semidirect
207. C22×SD16D127semidirect
301. SD16C22⋊D127semidirect
388. C22#16861238semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
Q8×C22⋊D1272
#225505202
#225502672
#225501192
#225501172
#225500962
#225500792
#225500352
#225500322
#225500272
#225500132
#225499772
#225499472
C22×SD16×C1272
#3266127