Finite Group Databasebeta

← #30641897

Subgroups — #30641897

Order 16256.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861186C211yesmaximal, characteristic
2#22549782C211yesmaximal, characteristic, complemented
3#22549780C211yesmaximal, characteristic, complemented
4#22549773C211yesmaximal, characteristic, complemented
5#22549772C211yesmaximal, characteristic, complemented
6#22549770C211yesmaximal, characteristic
7#22549770C211yesmaximal, characteristic
8#22549769C211yesmaximal, characteristic
9#22549761C211yesmaximal, characteristic, complemented
10#22549760C211yesmaximal, characteristic, complemented
11#22549758C211yesmaximal, characteristic, complemented
12#22549757C211yesmaximal, characteristic, complemented
13#22549737C211yesmaximal, characteristic
14#22549734C211yesmaximal, characteristic
15#388×C127C211yesmaximal, characteristic
16C22×Dic254C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18#16861186C418yescomplemented
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C22⋊Dic127C2211yescharacteristic
30C2×C4×Dic127C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C127⋊#95C2211yescharacteristic
34C127⋊#95C2211yescharacteristic
35C127⋊#95C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C2×C4⋊C4×C127C2211yescharacteristic
46C2×C22⋊C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C2×Dic254—41no—
53C2×Dic254—41no—
54C2×Dic254C2×C414yes—
55C22×Dic127—41no—
56C22×Dic127—41no—
57C22×Dic127C2×C414yes—
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C4⋊Dic127—41no—
67C4⋊Dic127C2×C414yes—
68Dic127⋊C4—41no—
69Dic127⋊C4—41no—
70Dic127⋊C4—41no—
71Dic127⋊C4—41no—
72Dic127⋊C4C2×C414yes—
73Dic127⋊C4C2×C414yes—
74C22⋊Dic127—41no—
75C22⋊Dic127—41no—
76C22⋊Dic127C2×C414yes—
77C4×Dic127—41no—
78C4×Dic127—41no—
79C4×Dic127—41no—
80C24×C127C2311yescharacteristic
81C22×C4×C127C2311yescharacteristic
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C4⋊C4×C127—41no—
88C22⋊C4×C127—41no—
89C22⋊C4×C127—41no—
90C22⋊C4×C127C2×C414yes—
91C42×C127—41no—
92Dic254—44no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—21no—
96C2×Dic127—21no—
97C2×Dic127—21no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—44no—
114C2×Dic127C22×C412yes—
115C2×Dic127C22×C412yes—
116C2×Dic127C22×C412yes—
117C2×Dic127C22×C412yes—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127C22×C412yes—
125C23×C127C2411yescharacteristic
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127C22×C412yes—
143C2×C4×C127C22×C412yes—
144C22×C127—22no—
145C22×C127—22no—
146C22×C127—22no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127C23×C411yescharacteristic
152C22×C127H(5,𝔽2)11yescharacteristic
153C22×C127H(5,𝔽2)11yescharacteristic
154C22×C127H(5,𝔽2)11yescharacteristic
155C22×C1272−1+411yescharacteristic
156C22×C1272−1+411yescharacteristic
157C22×C1272−1+411yescharacteristic
158C4×C127—22no—
159C4×C127—22no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—41no—
164Dic127—22no—
165Dic127—22no—
166Dic127—22no—
167Dic127—22no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171Dic127—41no—
172C2×C127—42no—
173C2×C127#51811yescharacteristic
174C2×C127#51911yescharacteristic
175C2×C127#52011yescharacteristic
176C2×C127#56211yescharacteristic
177C2×C127#56211yescharacteristic
178C2×C127#56311yescharacteristic
179C2×C127#56311yescharacteristic
180#2378—1271nomaximal, Sylow
181C127#237811yescharacteristic, complemented, Sylow
182C2×C22⋊Q8—1271no—
183#391—1271no—
184#391—1271no—
185#389—1271no—
186#389—1271no—
187#388—1271no—
188#388—1271no—
189#388—1271no—
190#388D12711yescharacteristic
191#387—1271no—
192#387—1271no—
193#387—1271no—
194#387—1271no—
195#381—1271no—
196#380—1271no—
197C22×Q8—1271no—
198C23×C4—1271no—
199C22⋊Q8—1278no—
200C2×C4⋊C4—1271no—
201C2×C4⋊C4—1271no—
202C2×C4⋊C4—1271no—
203C2×C4⋊C4—1271no—
204C2×C4⋊C4—1271no—
205C2×C4⋊C4—1271no—
206C2×C4⋊C4—1271no—
207C2×C4⋊C4—1271no—
208C2×C4⋊C4C2×D12711yescharacteristic
209C2×C22⋊C4—1271no—
210C2×C22⋊C4—1271no—
211C2×C22⋊C4—1271no—
212C2×C22⋊C4C2×D12711yescharacteristic
213C2×C22⋊C4C2×D12711yescharacteristic
214C2×C22⋊C4C2×D12711yescharacteristic
215C2×C42—1271no—
216C2×C42—1271no—
217C2×C42—1271no—
218C2×C42C2×D12711yescharacteristic
219#95—1271no—
220#95—1271no—
221#95—1271no—
222#95—1271no—
223#95—1271no—
224#95—1271no—
225#95—1271no—
226#95—1271no—
227#95—1271no—
228#95—1271no—
229#95—1271no—
230#95—1271no—
231#95C2×D12711yescharacteristic
232#95C2×D12711yescharacteristic
233C24C22×D12711yescharacteristic
234C2×Q8—1274no—
235C2×Q8—5081no—
236C2×Q8—5081no—
237C22×C4—1274no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—1271no—
241C22×C4—1271no—
242C22×C4—1271no—
243C22×C4—1271no—
244C22×C4—1271no—
245C22×C4—1271no—
246C22×C4—5081no—
247C22×C4—5081no—
248C22×C4C22×D12711yescharacteristic
249C22×C4C22×D12711yescharacteristic
250C22×C4C22×D12711yescharacteristic
251C22×C4C22×D12711yescharacteristic
252C22×C4C22×D12711yescharacteristic
253C22×C4C22×D12711yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—1274no—
256C4⋊C4—1274no—
257C4⋊C4—1274no—
258C4⋊C4—5081no—
259C4⋊C4—5081no—
260C4⋊C4—5081no—
261C4⋊C4—5081no—
262C4⋊C4—5081no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—1274no—
266C22⋊C4—5081no—
267C22⋊C4—5081no—
268C22⋊C4C4×D12714yes—
269C42—41no—
270C42—5081no—
271C42—5081no—
272C42—5081no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×C4×D12712yes—
280C23C23×D12711yescharacteristic
281Q8—5084no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—1272no—
299C2×C4—1272no—
300C2×C4—1272no—
301C2×C4—1272no—
302C2×C4—5084no—
303C2×C4—2541no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—2541no—
323C2×C4C2×C4×D12712yes—
324C2×C4C2×C4×D12712yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D12711yescharacteristic
333C22#1686125411yescharacteristic
334C22#1686125411yescharacteristic
335C22#1686125511yescharacteristic
336C22#1686125611yescharacteristic
337C22#1686125611yescharacteristic
338C22#1686125611yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4—5081no—
349C4—2542no—
350C4—2542no—
351C4—2542no—
352C4—2542no—
353C2—42no—
354C2#2255034811yescharacteristic
355C2#2255035311yescharacteristic
356C2#2255035511yescharacteristic
357C2#2255062011yescharacteristic
358C2#2255062111yescharacteristic
359C2#2255062811yescharacteristic
360C2#2255062911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenter#388D127
hypocenterC127#2378
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #22549782C2semidirect
3. #22549780C2semidirect
4. #22549773C2semidirect
5. #22549772C2semidirect
9. #22549761C2semidirect
10. #22549760C2semidirect
11. #22549758C2semidirect
12. #22549757C2semidirect
18. #16861186C4semidirect
181. C127#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#168611862
#225497822
#225497802
#225497732
#225497722
#225497702
#225497692
#225497612
#225497602
#225497582
#225497572
#225497372
#225497342
#388×C1272
#2378127