Finite Group Databasebeta

← #30650457

Subgroups — #30650457

Order 16256.

494 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×#16861231C211yesmaximal, characteristic, complemented
2C2×#16861230C211yesmaximal, characteristic, complemented
3#22550544C211yesmaximal, characteristic, complemented
4#22550528C211yesmaximal, characteristic, complemented
5#22550247C211yesmaximal, characteristic, complemented
6#22550106C211yesmaximal, characteristic, complemented
7#22550085C211yesmaximal, characteristic, complemented
8#22550070C211yesmaximal, characteristic, complemented
9#22550039C211yesmaximal, characteristic, complemented
10#22550036C211yesmaximal, characteristic, complemented
11#22550024C211yesmaximal, characteristic, complemented
12#22550019C211yesmaximal, characteristic, complemented
13#22549963C211yesmaximal, characteristic, complemented
14C8×C22⋊D127C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18C2×#14523248C2211yescharacteristic
19C2×#14523247C2211yescharacteristic
20#16861231—24no—
21#16861230—24no—
22C2×Dic508C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523237C2211yescharacteristic
25C2×#14523237C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523235C2211yescharacteristic
28C2×D1016C2211yescharacteristic
29C2×#14523233C2211yescharacteristic
30C2×C8×D127C2211yescharacteristic
31#16861207C2211yescharacteristic
32#16861205—21no—
33#16861205C2211yescharacteristic
34#16861194—21no—
35#16861191—21no—
36#16861189C2211yescharacteristic
37#16861184C2211yescharacteristic
38#16861182—21no—
39#16861182—21no—
40#16861181—21no—
41#16861174C2211yescharacteristic
42#16861171C2211yescharacteristic
43C4×C22⋊D127C2211yescharacteristic
44(C22⋊D127)⋊C4—21no—
45C4⋊C4×D127—21no—
46Dic127⋊C8C2211yescharacteristic
47#16861132C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861129C2211yescharacteristic
50#16861129C2211yescharacteristic
51C22⋊(C127⋊C8)C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53C8×Dic127C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic
55C22×C8×C127C2211yescharacteristic
56C22⋊Q8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—41no—
63#14523248—22no—
64#14523248—22no—
65#14523248—41no—
66#14523248—41no—
67#14523247—22no—
68#14523247—41no—
69C2×Dic254C2311yescharacteristic
70C2×Dic254C2311yescharacteristic
71C2×D508—21no—
72C2×D508—21no—
73C2×D508C2311yescharacteristic
74C2×D508C2311yescharacteristic
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127D411yescharacteristic
81C22×Dic127—21no—
82C2×C4×D127—21no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127—21no—
86C2×C4×D127C2311yescharacteristic
87Dic508—22no—
88#14523238—22no—
89#14523237—22no—
90#14523237D414yescomplemented
91#14523236—41no—
92#14523235—41no—
93D1016—22no—
94#14523233—22no—
95C2×C127⋊C8C2311yescharacteristic
96C2×C127⋊C8C2311yescharacteristic
97C8×D127—22no—
98C4⋊Dic127—21no—
99Dic127⋊C4—21no—
100Dic127⋊C4—21no—
101Dic127⋊C4—21no—
102Dic127⋊C4D411yescharacteristic
103C22⋊Dic127D411yescharacteristic
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108C4×Dic127—21no—
109C4×Dic127C2311yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114SD16×C127—22no—
115C2×C8×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127—21no—
119C4⋊C4×C127C2311yescharacteristic
120C4⋊C4×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127C2×D411yescharacteristic
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254C2×D412yes—
130D508—21no—
131D508—21no—
132D508—21no—
133D508—42no—
134D508—81no—
135D508C2×D412yes—
136C22⋊D127—22no—
137C22⋊D127—42no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—81no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—42no—
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C2×Dic127C2×D411yescharacteristic
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—42no—
151C4×D127—42no—
152C4×D127—42no—
153C4×D127—42no—
154C127⋊C8—21no—
155C127⋊C8—21no—
156C23×C127—21no—
157C23×C127C2×D411yescharacteristic
158Q8×C127—21no—
159Q8×C127—21no—
160D4×C127—21no—
161D4×C127—21no—
162D4×C127—81no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127C2×D412yes—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127C22×D411yescharacteristic
179C2×D127—21no—
180C2×D127—22no—
181C2×D127—21no—
182C2×D127—21no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C4×C127—21no—
189C4×C127—41no—
190C4×C127—41no—
191C4×C127—41no—
192C4×C127C22×D411yescharacteristic
193C4×C127H(5,𝔽2)11yescharacteristic
194Dic127—21no—
195Dic127—22no—
196Dic127—21no—
197Dic127—41no—
198Dic127#13512yescomplemented
199C2×C127—41no—
200C2×C127—81no—
201C2×C127D4211yescharacteristic
202C2×C127C2×#13511yescharacteristic
203C2×C127#57711yescharacteristic
204D127—41no—
205D127—42no—
206D127—81no—
207#3284—1271nomaximal, Sylow
208C127#328411yescharacteristic, complemented, Sylow
209C2×#135—1271no—
210C2×#135—1271no—
211#550—1271no—
212#547—1271no—
213#492—1271no—
214#465—1271no—
215#465D12711yescharacteristic, complemented
216#462—1271no—
217#459—1271no—
218#452—1271no—
219#452—1271no—
220#449—1271no—
221#449—1271no—
222C4×SD16—1271no—
223C8×D4—1271no—
224C2×#41—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C2×#41—1271no—
228#135—2544no—
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16C2×D12711yescharacteristic
237C2×D8—1271no—
238C2×D8—1271no—
239C22×C8—1271no—
240C22×C8C2×D12711yescharacteristic
241C4⋊Q8—1271no—
242#127—1271no—
243#127—2541no—
244#123—2541no—
245#123—2541no—
246C22⋊Q8—1271no—
247C22⋊Q8C2×D12711yescharacteristic
248#121—1271no—
249#121—2541no—
250#121—2541no—
251#121—2541no—
252#121C2×D12711yescharacteristic
253C4×Q8—1271no—
254C4×D4—1271no—
255C4×D4—1271no—
256C4×D4—2541no—
257C2×C4⋊C4—2541no—
258#106C2×D12711yescharacteristic
259C4⋊C8—1271no—
260#103—1271no—
261#103—1271no—
262#103C2×D12711yescharacteristic
263#102—1271no—
264#102—1271no—
265#102C2×D12711yescharacteristic
266C22⋊C8—1271no—
267C22⋊C8—1271no—
268C4×C8—1271no—
269#41—5081no—
270#41—5081no—
271#41—5081no—
272#41—5081no—
273#41—2542no—
274#41—2542no—
275#41—2542no—
276#41—2542no—
277C2×Q8—1271no—
278C2×Q8—1271no—
279C2×Q8C22×D12711yescharacteristic
280C2×D4—21no—
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×D4C22×D12711yescharacteristic
292C22×C4—1271no—
293C22×C4—2541no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4C22×D12711yescharacteristic
299Q16—2542no—
300Q16—2542no—
301SD16—22no—
302SD16—1274no—
303SD16—5081no—
304SD16—5081no—
305SD16—2542no—
306D8—2542no—
307D8—2542no—
308C2×C8—22no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8—2542no—
312C2×C8C22×D12711yescharacteristic
313C2×C8C22×D12711yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—1271no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4C22×D12711yescharacteristic
321C4⋊C4C22×D12711yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—1271no—
325C22⋊C4—1271no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C42—1271no—
330C42—2541no—
331C23—21no—
332C23—1271no—
333C23—2541no—
334C23—2541no—
335C23D4×D12711yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—1272no—
339Q8—2541no—
340Q8—2541no—
341Q8—2541no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—1272no—
346D4—10161no—
347D4—10161no—
348D4—5082no—
349D4—5082no—
350D4—5082no—
351D4—5082no—
352D4—5082no—
353D4—2542no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—1271no—
363C2×C4—1271no—
364C2×C4—1271no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—5082no—
369C2×C4—5082no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4D4×D12711yescharacteristic
375C2×C4C23×D12711yescharacteristic
376C8—21no—
377C8—2541no—
378C8—2541no—
379C8D4×D12712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—5081no—
387C22—5081no—
388C22—5081no—
389C22—5081no—
390C22—5081no—
391C22—2541no—
392C22—2541no—
393C22—2542no—
394C22—2541no—
395C22C2×D4×D12711yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—1272no—
401C4—5081no—
402C4—2541no—
403C4—2541no—
404C4—2542no—
405C4C2×D4×D12711yescharacteristic
406C4#1686125511yescharacteristic
407C2—41no—
408C2—81no—
409C2—10161no—
410C2—5082no—
411C2—5081no—
412C2#2255051011yescharacteristic
413C2#135×D12711yescharacteristic
414C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
hypercenter#465D127
hypocenterC127#3284
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16861231C2semidirect
2. C2×#16861230C2semidirect
3. #22550544C2semidirect
4. #22550528C2semidirect
5. #22550247C2semidirect
6. #22550106C2semidirect
7. #22550085C2semidirect
8. #22550070C2semidirect
9. #22550039C2semidirect
10. #22550036C2semidirect
11. #22550024C2semidirect
12. #22550019C2semidirect
13. #22549963C2semidirect
14. C8×C22⋊D127C2semidirect
15. #465×C127C2semidirect
90. #14523237D4semidirect
198. Dic127#135semidirect
208. C127#3284semidirect
215. #465D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612312
C2×#168612302
#225505442
#225505282
#225502472
#225501062
#225500852
#225500702
#225500392
#225500362
#225500242
#225500192
#225499632
C8×C22⋊D1272
#465×C1272
#3284127