Finite Group Databasebeta

← #30642069

Subgroups — #30642069

Order 16256.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861185C211yesmaximal, characteristic, complemented
2C2×#16861180C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
4#22549790C212yesmaximal, complemented
5#22549784C212yesmaximal, complemented
6#22549783C211yesmaximal, characteristic, complemented
7#22549779C211yesmaximal, characteristic, complemented
8#22549755C212yesmaximal, complemented
9#22549749C212yesmaximal, complemented
10#22549736C211yesmaximal, characteristic, complemented
11C22×#41×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C22×C22⋊D127C2212yes—
14C22×C4×D127C2211yescharacteristic
15#16861185—24no—
16#16861180—24no—
17C4×C22⋊D127—24no—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×Dic127⋊C4C2211yescharacteristic
22C2×C22⋊Dic127C2212yes—
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2212yes—
25C2×C22⋊Dic127C2212yes—
26C2×C22⋊Dic127C2212yes—
27C2×(C2×D127)⋊C4C2212yes—
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×C4×Dic127C2211yescharacteristic
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C127⋊#95C2212yes—
33C2×#41×C127—24no—
34C2×#41×C127—24no—
35C2×#41×C127—24no—
36C22×Q8×C127C2211yescharacteristic
37C22×D4×C127C2212yes—
38C22×D4×C127C2211yescharacteristic
39C23×C4×C127C2212yes—
40C23×C4×C127C2211yescharacteristic
41C23×D127C2311yescharacteristic
42C2×C22⋊D127—22no—
43C2×C22⋊D127—24no—
44C2×C22⋊D127—22no—
45C2×C22⋊D127—22no—
46C2×C22⋊D127—24no—
47C2×C22⋊D127—42no—
48C22×Dic127C2311yescharacteristic
49C22×Dic127C2312yes—
50C22×Dic127C2311yescharacteristic
51C22×Dic127C2312yes—
52C22×Dic127C2311yescharacteristic
53C2×C4×D127—22no—
54C2×C4×D127—22no—
55C2×C4×D127—22no—
56C4⋊Dic127—22no—
57C4⋊Dic127—22no—
58Dic127⋊C4—24no—
59Dic127⋊C4—22no—
60C22⋊Dic127—24no—
61C22⋊Dic127—22no—
62C22⋊Dic127—24no—
63C22⋊Dic127—42no—
64C22⋊Dic127—42no—
65(C2×D127)⋊C4—24no—
66(C2×D127)⋊C4—22no—
67C4×Dic127—22no—
68C24×C127C2312yes—
69C24×C127C2311yescharacteristic
70#41×C127—232no—
71C2×Q8×C127—22no—
72C2×Q8×C127—22no—
73C2×Q8×C127D414yescomplemented
74C2×D4×C127—24no—
75C2×D4×C127—24no—
76C2×D4×C127—24no—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127D414yescomplemented
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127D414yescomplemented
86C22×C4×C127C2311yescharacteristic
87C22×C4×C127C2312yes—
88C22×C4×C127C2311yescharacteristic
89C22×D127—21no—
90C22×D127—21no—
91C22×D127—21no—
92C22×D127—21no—
93C22×D127—21no—
94C22×D127—21no—
95C22×D127#4112yescomplemented
96C22⋊D127—28no—
97C22⋊D127—48no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—22no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—22no—
105C2×Dic127—21no—
106C2×Dic127—22no—
107C2×Dic127—22no—
108C2×Dic127—21no—
109C2×Dic127—22no—
110C2×Dic127—22no—
111C2×Dic127—21no—
112C2×Dic127#4112yescomplemented
113C4×D127—44no—
114C23×C127—22no—
115C23×C127—21no—
116C23×C127—22no—
117C23×C127—21no—
118C23×C127—22no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—22no—
122C23×C127—21no—
123C23×C127—22no—
124C23×C127—21no—
125C23×C127—22no—
126C23×C127C2×D414yes—
127C23×C127C2×D412yes—
128C23×C127C2411yescharacteristic
129Q8×C127—28no—
130D4×C127—216no—
131D4×C127—28no—
132C2×C4×C127—216no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127—28no—
138C2×C4×C127C2×D412yes—
139C2×C4×C127C2×D414yes—
140C2×C4×C127C2×D412yes—
141C2×C4×C127C2×D412yes—
142C2×C4×C127C2×D412yes—
143C22×C127—24no—
144C22×C127—22no—
145C22×C127—22no—
146C22×C127—22no—
147C22×C127—24no—
148C22×C127—24no—
149C22×C127—42no—
150C22×C127—42no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—42no—
155C22×C127—42no—
156C22×C127—41no—
157C22×C127C22×D411yescharacteristic
158C22×C127C22×D411yescharacteristic
159C22×C127C22×D411yescharacteristic
160C22×C127C2×#4111yescharacteristic
161C22×C127C2×#4111yescharacteristic
162C22×C127H(5,𝔽2)11yescharacteristic
163C22×C1272−1+411yescharacteristic
164C2×D127—22no—
165C2×D127—22no—
166C2×D127—22no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C4×C127—22no—
172C4×C127—24no—
173C4×C127#12014yes—
174Dic127—22no—
175Dic127—41no—
176Dic127—41no—
177Dic127—42no—
178Dic127—42no—
179C2×C127—42no—
180C2×C127—44no—
181C2×C127C2×#12011yescharacteristic
182C2×C127#52511yescharacteristic
183C2×C127#53711yescharacteristic
184C2×C127#54611yescharacteristic
185C2×C127#54711yescharacteristic
186C2×C127#54811yescharacteristic
187C2×C127#55011yescharacteristic
188D127—42no—
189#2407—1271nomaximal, Sylow
190C127#240711yescharacteristic, complemented, Sylow
191C22×#41D12711yescharacteristic, complemented
192C2×C22⋊Q8—1271no—
193C2×#121—1271no—
194C2×C4×D4—1271no—
195#394—1272no—
196#392—1272no—
197#391—1271no—
198#390—1271no—
199#386—1272no—
200#385—1272no—
201#380—1271no—
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—24no—
205C22×Q8C2×D12711yescharacteristic
206C22×D4—1272no—
207C22×D4—1271no—
208C22×D4C2×D12712yes—
209C22×D4C2×D12711yescharacteristic
210C23×C4—1271no—
211C23×C4C2×D12712yes—
212C23×C4C2×D12711yescharacteristic
213C22⋊Q8—2544no—
214#121—2544no—
215C4×D4—2544no—
216C2×C4⋊C4—1271no—
217C2×C4⋊C4—1271no—
218C2×C4⋊C4—1271no—
219C2×C4⋊C4—1272no—
220C2×C22⋊C4—1272no—
221C2×C22⋊C4—1272no—
222C2×C22⋊C4—1271no—
223C2×C22⋊C4—1272no—
224C2×C22⋊C4—1272no—
225C2×C22⋊C4—1271no—
226C2×C22⋊C4—1272no—
227C2×C42—1271no—
228#95—1272no—
229#95—1272no—
230#95—1272no—
231C24—1271no—
232C24C22×D12712yes—
233C24C22×D12711yescharacteristic
234#41—232no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8C22⋊D12714yescomplemented
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—5082no—
244C2×D4—2542no—
245C2×D4—2544no—
246C2×D4—2544no—
247C2×D4—2542no—
248C2×D4—2542no—
249C2×D4C22⋊D12714yescomplemented
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—1271no—
256C22×C4—1272no—
257C22×C4—1271no—
258C22×C4—1272no—
259C22×C4—1271no—
260C22×C4—2542no—
261C22×C4—2542no—
262C22×C4—2542no—
263C22×C4C22⋊D12714yescomplemented
264C22×C4C22×D12711yescharacteristic
265C22×C4C22×D12712yes—
266C22×C4C22×D12711yescharacteristic
267C4⋊C4—2542no—
268C4⋊C4—2544no—
269C4⋊C4—2542no—
270C4⋊C4—2542no—
271C22⋊C4—5082no—
272C22⋊C4—5082no—
273C22⋊C4—2544no—
274C22⋊C4—2542no—
275C22⋊C4—2544no—
276C22⋊C4—2542no—
277C22⋊C4—2544no—
278C42—2542no—
279C23—22no—
280C23—21no—
281C23—22no—
282C23—21no—
283C23—22no—
284C23—21no—
285C23—21no—
286C23—22no—
287C23—21no—
288C23—22no—
289C23—21no—
290C23—22no—
291C23—1272no—
292C23—2541no—
293C23—2541no—
294C23—2541no—
295C23—2541no—
296C23—2541no—
297C23—2541no—
298C23C2×C22⋊D12714yes—
299C23C2×C22⋊D12712yes—
300C23C23×D12711yescharacteristic
301Q8—28no—
302D4—216no—
303D4—28no—
304D4—5088no—
305D4—2548no—
306C2×C4—216no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—28no—
312C2×C4—1272no—
313C2×C4—5084no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2542no—
317C2×C4—2541no—
318C2×C4—2542no—
319C2×C4—2541no—
320C2×C4—2542no—
321C2×C4—2542no—
322C2×C4—2542no—
323C2×C4—2541no—
324C2×C4—2542no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4C2×C22⋊D12712yes—
329C2×C4C2×C22⋊D12714yes—
330C2×C4C2×C22⋊D12712yes—
331C2×C4C2×C22⋊D12712yes—
332C2×C4C2×C22⋊D12712yes—
333C22—24no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—24no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22—5081no—
348C22—5081no—
349C22—5081no—
350C22—5081no—
351C22—2542no—
352C22—2542no—
353C22—2542no—
354C22C22×C22⋊D12711yescharacteristic
355C22C22×C22⋊D12711yescharacteristic
356C22C22×C22⋊D12711yescharacteristic
357C22#41×D12711yescharacteristic
358C22#41×D12711yescharacteristic
359C22#1686125511yescharacteristic
360C22#1686125611yescharacteristic
361C4—22no—
362C4—24no—
363C4—5082no—
364C4—5081no—
365C4—5081no—
366C4—5082no—
367C4—2542no—
368C4#1686117814yes—
369C2—42no—
370C2—44no—
371C2—5082no—
372C2C2×#1686117811yescharacteristic
373C2#2255038311yescharacteristic
374C2#2255045411yescharacteristic
375C2#2255051411yescharacteristic
376C2#2255052511yescharacteristic
377C2#2255053311yescharacteristic
378C2#2255054511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC22×#41×C127C2
hypercenterC22×#41D127
hypocenterC127#2407
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16861185C2semidirect
2. C2×#16861180C2semidirect
3. C2×C4×C22⋊D127C2semidirect
4. #22549790C2semidirect
5. #22549784C2semidirect
6. #22549783C2semidirect
7. #22549779C2semidirect
8. #22549755C2semidirect
9. #22549749C2semidirect
10. #22549736C2semidirect
11. C22×#41×C127C2semidirect
73. C2×Q8×C127D4semidirect
79. C2×D4×C127D4semidirect
85. C22×C4×C127D4semidirect
95. C22×D127#41semidirect
112. C2×Dic127#41semidirect
190. C127#2407semidirect
191. C22×#41D127semidirect
237. C2×Q8C22⋊D127semidirect
249. C2×D4C22⋊D127semidirect
263. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611852
C2×#168611802
C2×C4×C22⋊D1272
#225497902
#225497842
#225497832
#225497792
#225497552
#225497492
#225497362
C22×#41×C1272
#2407127