Finite Group Databasebeta

← #30650601

Subgroups — #30650601

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550584C211yesmaximal, characteristic
2#22550539C211yesmaximal, characteristic
3#22550225C211yesmaximal, characteristic, complemented
4#22550213C211yesmaximal, characteristic, complemented
5#22550209C211yesmaximal, characteristic, complemented
6#22550170C211yesmaximal, characteristic, complemented
7#22550116C211yesmaximal, characteristic, complemented
8#22550084C211yesmaximal, characteristic, complemented
9#22550031C211yesmaximal, characteristic, complemented
10#22550030C211yesmaximal, characteristic, complemented
11#22549968C211yesmaximal, characteristic, complemented
12C8×C22⋊D127C211yesmaximal, characteristic
13C2×#16861134C211yesmaximal, characteristic, complemented
14#103×D127C211yesmaximal, characteristic, complemented
15#467×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×#14523238C2211yescharacteristic
19C2×#14523238C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861208—21no—
22#16861208C2211yescharacteristic
23#16861207C2211yescharacteristic
24#16861199—21no—
25#16861189—21no—
26#16861189C2211yescharacteristic
27#16861187—21no—
28#16861186—21no—
29#16861186—21no—
30#16861185C2211yescharacteristic
31Q8×Dic127C2211yescharacteristic
32#16861174C2211yescharacteristic
33C4×C22⋊D127C2211yescharacteristic
34(C22⋊D127)⋊C4—21no—
35(C22⋊D127)⋊C4—21no—
36C2×C4⋊Dic127C2211yescharacteristic
37C4⋊C4×D127C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861142C2211yescharacteristic
40Dic127⋊C8C2211yescharacteristic
41#16861134C2211yescharacteristic
42#16861134C2211yescharacteristic
43#16861134C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861132C2211yescharacteristic
48#16861132C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C2×Q16×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C22⋊Q8×C127C2211yescharacteristic
55C22⋊Q8×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254—21no—
61C2×Dic254—22no—
62C2×Dic254—21no—
63C2×Dic254—41no—
64C2×Dic254C2311yescharacteristic
65C2×Dic254C2311yescharacteristic
66Q8×D127—22no—
67Q8×D127—41no—
68C2×C22⋊D127D411yescharacteristic
69C22×Dic127—21no—
70C22×Dic127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74#14523238—22no—
75#14523238—41no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C8×D127—41no—
79C4⋊Dic127—21no—
80C4⋊Dic127—21no—
81C4⋊Dic127—21no—
82C4⋊Dic127C2311yescharacteristic
83C4⋊Dic127C2311yescharacteristic
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4D411yescharacteristic
91C22⋊Dic127D411yescharacteristic
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127—21no—
97C4×Dic127C2311yescharacteristic
98C2×Q8×C127C2311yescharacteristic
99C2×Q8×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101Q16×C127—41no—
102C2×C8×C127—41no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127—21no—
106C4⋊C4×C127—21no—
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
109C22⋊C4×C127—21no—
110C22⋊C4×C127—21no—
111C22×D127C2×D411yescharacteristic
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—42no—
116Dic254—42no—
117Dic254—81no—
118Dic254#4112yes—
119C22⋊D127Q1614yescomplemented
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—42no—
125C2×Dic127—81no—
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—21no—
130C4×D127—21no—
131C4×D127—42no—
132C4×D127—81no—
133C127⋊C8—21no—
134C127⋊C8—21no—
135C23×C127C2×D411yescharacteristic
136Q8×C127—21no—
137Q8×C127—21no—
138Q8×C127—21no—
139Q8×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—21no—
149C8×C127—21no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127C2×Q1612yes—
153C22×C127C22×D411yescharacteristic
154C2×D127—21no—
155C2×D127—21no—
156C2×D127C2×Q1612yes—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127C2×#4111yescharacteristic
163C4×C127H(5,𝔽2)11yescharacteristic
164Dic127—21no—
165Dic127—22no—
166Dic127—21no—
167Dic127—41no—
168Dic127—41no—
169Dic127—41no—
170Dic127C2×Q1612yes—
171C2×C127—22no—
172C2×C127#54611yescharacteristic
173C2×C127C22×Q1611yescharacteristic
174C2×C127#57711yescharacteristic
175D127—22no—
176#3299—1271nomaximal, Sylow
177C127#329911yescharacteristic, complemented, Sylow
178#554—1271no—
179D4×Q8—1271no—
180#487—1271no—
181#484—1271no—
182#484—1271no—
183#477—1271no—
184#467—1271no—
185#467D12711yescharacteristic, complemented
186#462—1271no—
187#451—1271no—
188#451—1271no—
189C4×Q16—1271no—
190C8×D4—1271no—
191C2×#103—1271no—
192C2×#103—1271no—
193C22×Q8—1271no—
194C22×Q8—1271no—
195C2×Q16—1271no—
196C2×Q16—1271no—
197C2×Q16C2×D12711yescharacteristic
198C22×C8—1271no—
199C22×C8C2×D12711yescharacteristic
200C4⋊Q8—1271no—
201C4⋊Q8—1271no—
202C4⋊Q8—2541no—
203#125—2541no—
204C22⋊Q8—1271no—
205C22⋊Q8—1271no—
206C22⋊Q8—2541no—
207C22⋊Q8—2541no—
208C22⋊Q8—2541no—
209C22⋊Q8—2541no—
210C22⋊Q8C2×D12711yescharacteristic
211C22⋊Q8C2×D12711yescharacteristic
212C4×Q8—1271no—
213C4×Q8—1271no—
214C4×D4—1271no—
215C4×D4—2541no—
216C4×D4—2541no—
217C2×C4⋊C4—1271no—
218C2×C4⋊C4—1271no—
219#107—1271no—
220#107—1271no—
221#107C2×D12711yescharacteristic
222C4⋊C8—1271no—
223#103—1271no—
224#103—1271no—
225#103—1271no—
226#103—1271no—
227#103—1271no—
228#103—1271no—
229#103—1271no—
230#103—1271no—
231#103C2×D12711yescharacteristic
232#103C2×D12711yescharacteristic
233C22⋊C8—1271no—
234C22⋊C8—1271no—
235C4×C8—1271no—
236C2×Q8—1271no—
237C2×Q8—1271no—
238C2×Q8—5081no—
239C2×Q8—5081no—
240C2×Q8—2541no—
241C2×Q8—2541no—
242C2×Q8—2542no—
243C2×Q8—2542no—
244C2×Q8—2541no—
245C2×Q8C22×D12711yescharacteristic
246C2×Q8C22×D12711yescharacteristic
247C2×D4—1271no—
248C22×C4—1271no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4—2541no—
252C22×C4—2541no—
253C22×C4C22×D12711yescharacteristic
254Q16—41no—
255Q16—5081no—
256Q16—2542no—
257C2×C8—41no—
258C2×C8—1271no—
259C2×C8—1271no—
260C2×C8—5081no—
261C2×C8C22×D12711yescharacteristic
262C2×C8C22×D12711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—1271no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4—2541no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4C22×D12711yescharacteristic
278C4⋊C4C22×D12711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1271no—
282C22⋊C4—1271no—
283C22⋊C4—2541no—
284C22⋊C4—2541no—
285C42—1271no—
286C42—2541no—
287C42—2541no—
288C23—1271no—
289C23D4×D12711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—1272no—
295Q8—10161no—
296Q8—5082no—
297Q8—5082no—
298Q8—2541no—
299Q8—2541no—
300Q8—2541no—
301D4—1274no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—1271no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—10161no—
312C2×C4—10161no—
313C2×C4—5082no—
314C2×C4—5082no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4D4×D12711yescharacteristic
322C2×C4C23×D12711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2541no—
326C8—2541no—
327C22—21no—
328C22—21no—
329C22—1272no—
330C22—2541no—
331C22—2541no—
332C22Q16×D12712yes—
333C22C2×D4×D12711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—1272no—
340C4—5081no—
341C4—5081no—
342C4—5081no—
343C4—2541no—
344C4—2541no—
345C4—2542no—
346C4#41×D12711yescharacteristic
347C4#1686125411yescharacteristic
348C2—22no—
349C2—2542no—
350C2#2255051511yescharacteristic
351C2C2×Q16×D12711yescharacteristic
352C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#467×C127C2
socleC22×C127C22×D4
prime radicalC2×C22⋊D127D4
hypercenter#467D127
hypocenterC127#3299
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #22550225C2semidirect
4. #22550213C2semidirect
5. #22550209C2semidirect
6. #22550170C2semidirect
7. #22550116C2semidirect
8. #22550084C2semidirect
9. #22550031C2semidirect
10. #22550030C2semidirect
11. #22549968C2semidirect
13. C2×#16861134C2semidirect
14. #103×D127C2semidirect
15. #467×C127C2semidirect
119. C22⋊D127Q16semidirect
177. C127#3299semidirect
185. #467D127semidirect

Maximal subgroups

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

subgroupindex
#225505842
#225505392
#225502252
#225502132
#225502092
#225501702
#225501162
#225500842
#225500312
#225500302
#225499682
C8×C22⋊D1272
C2×#168611342
#103×D1272
#467×C1272
#3299127