Finite Group Databasebeta

← #30650321

Subgroups — #30650321

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic, complemented
2C2×#16861230C211yesmaximal, characteristic, complemented
3#22550548C211yesmaximal, characteristic, complemented
4#22550534C211yesmaximal, characteristic, complemented
5#22550266C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550077C211yesmaximal, characteristic, complemented
9#22550037C211yesmaximal, characteristic, complemented
10#22550022C211yesmaximal, characteristic, complemented
11#22550020C211yesmaximal, characteristic, complemented
12#22550016C211yesmaximal, characteristic, complemented
13#22549981C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#464×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18C2×#14523247C2211yescharacteristic
19C2×Q8×D127C2211yescharacteristic
20#16861241—24no—
21#16861230—24no—
22C2×#14523238C2211yescharacteristic
23C2×#14523237C2211yescharacteristic
24C2×#14523237C2211yescharacteristic
25C2×#14523235C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C2×#14523233C2211yescharacteristic
29C2×#14523231C2211yescharacteristic
30C2×C8×D127C2211yescharacteristic
31#16861206—21no—
32#16861206C2211yescharacteristic
33#16861195—21no—
34#16861186—21no—
35#16861185C2211yescharacteristic
36#16861184—21no—
37#16861184—21no—
38#16861184C2211yescharacteristic
39#16861184C2211yescharacteristic
40#16861182—21no—
41#16861180C2211yescharacteristic
42C4×Dic254C2211yescharacteristic
43C4×D508—21no—
44C4×D508C2211yescharacteristic
45(C22⋊D127)⋊C4—21no—
46#16861140C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53#16861120C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×Q8×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—22no—
63#14523249—41no—
64#14523249—41no—
65#14523247—22no—
66#14523247—41no—
67C2×Dic254—21no—
68C2×Dic254C2311yescharacteristic
69Q8×D127—22no—
70Q8×D127—41no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—21no—
74C2×D508—21no—
75C2×D508—21no—
76C2×D508—21no—
77C2×D508D411yescharacteristic
78C2×D508C2311yescharacteristic
79C2×D508C2311yescharacteristic
80C2×C22⋊D127—21no—
81C2×C22⋊D127—21no—
82C22×Dic127—21no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127—21no—
86C2×C4×D127C2311yescharacteristic
87C2×C4×D127C2311yescharacteristic
88#14523238D414yescomplemented
89#14523237—22no—
90#14523237—22no—
91#14523235—22no—
92#14523235—41no—
93#14523233—22no—
94#14523233—41no—
95#14523231—22no—
96C2×C127⋊C8C2311yescharacteristic
97C2×C127⋊C8C2311yescharacteristic
98C8×D127—22no—
99C4⋊Dic127D411yescharacteristic
100C4⋊Dic127C2311yescharacteristic
101Dic127⋊C4—21no—
102Dic127⋊C4—21no—
103C22⋊Dic127—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108(C2×D127)⋊C4D411yescharacteristic
109C4×Dic127—21no—
110C2×Q8×C127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113Q16×C127—22no—
114SD16×C127—22no—
115C2×C8×C127C2311yescharacteristic
116C2×C8×C127C2311yescharacteristic
117C4⋊C4×C127—21no—
118C4⋊C4×C127C2311yescharacteristic
119C22⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C42×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127C2×D411yescharacteristic
126C22×D127C2×D411yescharacteristic
127Dic254—21no—
128Dic254—42no—
129Dic254C2×D412yes—
130D508—21no—
131D508—21no—
132D508—21no—
133D508—21no—
134D508—41no—
135D508—42no—
136D508—42no—
137D508—42no—
138D508—81no—
139D508—81no—
140D508—81no—
141C22⋊D127—42no—
142C22⋊D127—42no—
143C2×Dic127—21no—
144C2×Dic127—21no—
145C2×Dic127—42no—
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—21no—
151C4×D127—21no—
152C4×D127—42no—
153C4×D127—42no—
154C4×D127—42no—
155C127⋊C8—21no—
156C127⋊C8C2×D412yes—
157C23×C127—21no—
158Q8×C127—21no—
159Q8×C127—21no—
160Q8×C127—21no—
161Q8×C127C2×D412yes—
162D4×C127—21no—
163D4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127C2×D411yescharacteristic
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127—21no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127C22×D411yescharacteristic
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—21no—
180C2×D127—21no—
181C2×D127—21no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C4×C127—22no—
189C4×C127—21no—
190C4×C127—21no—
191C4×C127—41no—
192C4×C127—41no—
193C4×C127C22×D411yescharacteristic
194C4×C127H(5,𝔽2)11yescharacteristic
195Dic127—21no—
196Dic127—21no—
197Dic127—22no—
198Dic127—41no—
199C2×C127—81no—
200C2×C127D4211yescharacteristic
201C2×C127#57511yescharacteristic
202C2×C127#57711yescharacteristic
203D127—41no—
204D127—41no—
205D127—81no—
206D127—81no—
207#3271—1271nomaximal, Sylow
208C127#327111yescharacteristic, complemented, Sylow
209C2×#137—1271no—
210C2×#135—1271no—
211#550—1271no—
212#548—1271no—
213#496—1271no—
214#469—1271no—
215#465—1271no—
216#464D12711yescharacteristic, complemented
217#460—1271no—
218#452—1271no—
219#449—1271no—
220#449—1271no—
221#448—1271no—
222#441—1271no—
223#435—1271no—
224C2×#41—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C22×Q8—1271no—
228#137—2544no—
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16C2×D12711yescharacteristic
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16C2×D12711yescharacteristic
237C2×D8—1271no—
238C2×D8—1271no—
239C2×M16—1271no—
240C22×C8—1271no—
241#127—1271no—
242#127—2541no—
243#124—2541no—
244#124C2×D12711yescharacteristic
245C22⋊Q8—1271no—
246C22⋊Q8—2541no—
247#121—1271no—
248#121—1271no—
249#121—1271no—
250#121—2541no—
251#121—2541no—
252#121—2541no—
253C4×Q8—1271no—
254C4×Q8C2×D12711yescharacteristic
255C4×D4—1271no—
256C4×D4—2541no—
257C4×D4—2541no—
258#106—1271no—
259C4⋊C8C2×D12711yescharacteristic
260#103—1271no—
261#103—1271no—
262#103C2×D12711yescharacteristic
263#102—1271no—
264#102—1271no—
265#102C2×D12711yescharacteristic
266C22⋊C8—1271no—
267C22⋊C8—1271no—
268#97—1271no—
269#41—5081no—
270#41—5081no—
271#41—5081no—
272#41—2542no—
273#41—2542no—
274#41—2542no—
275C2×Q8—1271no—
276C2×Q8—5081no—
277C2×Q8—2541no—
278C2×Q8—2542no—
279C2×Q8C22×D12711yescharacteristic
280C2×Q8C22×D12711yescharacteristic
281C2×D4—1271no—
282C2×D4—1271no—
283C2×D4—1271no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4C22×D12711yescharacteristic
293C22×C4—1271no—
294C22×C4—1271no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4—2541no—
299Q16—22no—
300Q16—1274no—
301SD16—22no—
302SD16—5081no—
303SD16—2542no—
304SD16—2542no—
305SD16—2542no—
306D8—5081no—
307D8—2542no—
308M16—2542no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8—2542no—
312C2×C8C22×D12711yescharacteristic
313C2×C8C22×D12711yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—1271no—
316C4⋊C4—1271no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4C22×D12711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1271no—
323C22⋊C4—1271no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C42—21no—
329C42—2541no—
330C42C22×D12711yescharacteristic
331C23—21no—
332C23—1271no—
333C23—1271no—
334C23—2541no—
335C23—2541no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—1272no—
340Q8—5082no—
341Q8—2541no—
342Q8D4×D12712yes—
343D4—21no—
344D4—21no—
345D4—10161no—
346D4—10161no—
347D4—10161no—
348D4—5082no—
349D4—5082no—
350D4—5082no—
351D4—5081no—
352D4—5082no—
353D4—5082no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357D4—2541no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—1271no—
362C2×C4—1271no—
363C2×C4—5082no—
364C2×C4—5082no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—2541no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4D4×D12711yescharacteristic
374C2×C4D4×D12711yescharacteristic
375C2×C4C23×D12711yescharacteristic
376C8—21no—
377C8—21no—
378C8—1272no—
379C8—2541no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—5081no—
384C22—5081no—
385C22—5081no—
386C22—5081no—
387C22—5081no—
388C22—5081no—
389C22—2541no—
390C22—2541no—
391C22—2541no—
392C22—2541no—
393C22—2541no—
394C22—2541no—
395C22C2×D4×D12711yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—5081no—
402C4—2541no—
403C4—2541no—
404C4—2542no—
405C4C2×D4×D12711yescharacteristic
406C4#1686125511yescharacteristic
407C2—81no—
408C2—10161no—
409C2—10161no—
410C2—5081no—
411C2—5081no—
412C2#2255051011yescharacteristic
413C2#2255067511yescharacteristic
414C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#464×C127C2
socleC22×C127C22×D4
hypercenter#464D127
hypocenterC127#3271
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16861241C2semidirect
2. C2×#16861230C2semidirect
3. #22550548C2semidirect
4. #22550534C2semidirect
5. #22550266C2semidirect
6. #22550128C2semidirect
7. #22550108C2semidirect
8. #22550077C2semidirect
9. #22550037C2semidirect
10. #22550022C2semidirect
11. #22550020C2semidirect
12. #22550016C2semidirect
13. #22549981C2semidirect
14. #22549946C2semidirect
15. #464×C127C2semidirect
88. #14523238D4semidirect
208. C127#3271semidirect
216. #464D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×#168612302
#225505482
#225505342
#225502662
#225501282
#225501082
#225500772
#225500372
#225500222
#225500202
#225500162
#225499812
#225499462
#464×C1272
#3271127