Finite Group Databasebeta

← #30641894

Subgroups — #30641894

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×#16861189C211yesmaximal, characteristic
2#22549780C211yesmaximal, characteristic, complemented
3#22549780C211yesmaximal, characteristic, complemented
4#22549774C211yesmaximal, characteristic, complemented
5#22549772C211yesmaximal, characteristic, complemented
6#22549768C211yesmaximal, characteristic
7#22549767C211yesmaximal, characteristic
8#22549766C211yesmaximal, characteristic
9#22549765C211yesmaximal, characteristic
10#22549762C211yesmaximal, characteristic, complemented
11#22549761C211yesmaximal, characteristic, complemented
12#22549759C211yesmaximal, characteristic, complemented
13#22549738C211yesmaximal, characteristic
14#22549735C211yesmaximal, characteristic
15#387×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C22×C4×D127C2211yescharacteristic
18#16861189C418yescomplemented
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
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×C4⋊C4×C127C2211yescharacteristic
47C2×C42×C127C2211yescharacteristic
48#95×C127C2211yescharacteristic
49#95×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C23×D127C2311yescharacteristic
53C2×Dic254—41no—
54C2×Dic254—41no—
55C2×Dic254C2×C414yes—
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C2×C4×D127—41no—
64C2×C4×D127—41no—
65C2×C4×D127C2×C414yes—
66C4⋊Dic127—41no—
67C4⋊Dic127C2×C414yes—
68C4⋊Dic127C2×C414yes—
69Dic127⋊C4—41no—
70Dic127⋊C4—41no—
71Dic127⋊C4—41no—
72Dic127⋊C4—41no—
73(C2×D127)⋊C4—41no—
74(C2×D127)⋊C4—41no—
75(C2×D127)⋊C4—41no—
76(C2×D127)⋊C4—41no—
77(C2×D127)⋊C4C2×C414yes—
78(C2×D127)⋊C4C2×C414yes—
79C4×Dic127—41no—
80C4×Dic127—41no—
81C4×Dic127—41no—
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C4⋊C4×C127—41no—
90C4⋊C4×C127C2×C414yes—
91C42×C127—41no—
92C22×D127—21no—
93C22×D127—21no—
94C22×D127—21no—
95C22×D127—21no—
96C22×D127—21no—
97C22×D127—21no—
98C22×D127C22×C412yes—
99Dic254—44no—
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—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127C22×C412yes—
119C2×Dic127C22×C412yes—
120C2×Dic127C22×C412yes—
121C4×D127—44no—
122C23×C127C2411yescharacteristic
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
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×C127C22×C412yes—
142C2×C4×C127C22×C412yes—
143C2×C4×C127C22×C412yes—
144C22×C127C23×C411yescharacteristic
145C22×C127H(5,𝔽2)11yescharacteristic
146C22×C127H(5,𝔽2)11yescharacteristic
147C22×C127H(5,𝔽2)11yescharacteristic
148C22×C1272−1+411yescharacteristic
149C22×C1272−1+411yescharacteristic
150C22×C1272−1+411yescharacteristic
151C2×D127—22no—
152C2×D127—22no—
153C2×D127—22no—
154C2×D127—41no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C4×C127—22no—
159C4×C127—22no—
160C4×C127—22no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127—41no—
165Dic127—22no—
166Dic127—22no—
167Dic127—22no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171Dic127—41no—
172C2×C127#51811yescharacteristic
173C2×C127#51911yescharacteristic
174C2×C127#52011yescharacteristic
175C2×C127#56211yescharacteristic
176C2×C127#56211yescharacteristic
177C2×C127#56311yescharacteristic
178C2×C127#56311yescharacteristic
179D127—42no—
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#388—1271no—
191#387—1271no—
192#387—1271no—
193#387—1271no—
194#387D12711yescharacteristic, complemented
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⋊C4C2×D12711yescharacteristic
208C2×C4⋊C4C2×D12711yescharacteristic
209C2×C22⋊C4—1271no—
210C2×C22⋊C4—1271no—
211C2×C22⋊C4—1271no—
212C2×C22⋊C4—1271no—
213C2×C22⋊C4—1271no—
214C2×C22⋊C4—1271no—
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#95C2×D12711yescharacteristic
230#95C2×D12711yescharacteristic
231#95C2×D12711yescharacteristic
232#95C2×D12711yescharacteristic
233C24—1271no—
234C2×Q8—1274no—
235C2×Q8—5081no—
236C2×Q8—5081no—
237C22×C4—1271no—
238C22×C4—1271no—
239C22×C4—1271no—
240C22×C4—1274no—
241C22×C4—1271no—
242C22×C4—1271no—
243C22×C4—1271no—
244C22×C4—1271no—
245C22×C4—5081no—
246C22×C4—5081no—
247C22×C4C22×D12711yescharacteristic
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—5081no—
258C4⋊C4—5081no—
259C4⋊C4—5081no—
260C4⋊C4—5081no—
261C4⋊C4—5081no—
262C4⋊C4C4×D12714yes—
263C22⋊C4—1274no—
264C22⋊C4—1274no—
265C22⋊C4—5081no—
266C22⋊C4—5081no—
267C22⋊C4—5081no—
268C22⋊C4—5081no—
269C42—41no—
270C42—5081no—
271C42—5081no—
272C42—5081no—
273C23—1272no—
274C23—2541no—
275C23—2541no—
276C23—2541no—
277C23—2541no—
278C23—2541no—
279C23—2541no—
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—21no—
299C2×C4—21no—
300C2×C4—1272no—
301C2×C4—1272no—
302C2×C4—1272no—
303C2×C4—5084no—
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×C4C2×C4×D12712yes—
323C2×C4C2×C4×D12712yes—
324C2×C4C2×C4×D12712yes—
325C22—5081no—
326C22—5081no—
327C22—5081no—
328C22—5081no—
329C22—2542no—
330C22—2542no—
331C22—2542no—
332C22C22×C4×D12711yescharacteristic
333C22#1686125411yescharacteristic
334C22#1686125511yescharacteristic
335C22#1686125511yescharacteristic
336C22#1686125611yescharacteristic
337C22#1686125611yescharacteristic
338C22#1686125711yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—5081no—
347C4—5081no—
348C4—5081no—
349C4—5081no—
350C4—2542no—
351C4—2542no—
352C4—2542no—
353C2—5082no—
354C2#2255035011yescharacteristic
355C2#2255035211yescharacteristic
356C2#2255035511yescharacteristic
357C2#2255062211yescharacteristic
358C2#2255062311yescharacteristic
359C2#2255062811yescharacteristic
360C2#2255063011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#387×C127C2
socle, cosocleC23×C127C24
prime radicalC23×D127C23
hypercenter#387D127
hypocenterC127#2378
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #22549780C2semidirect
3. #22549780C2semidirect
4. #22549774C2semidirect
5. #22549772C2semidirect
10. #22549762C2semidirect
11. #22549761C2semidirect
12. #22549759C2semidirect
15. #387×C127C2semidirect
18. #16861189C4semidirect
181. C127#2378semidirect
194. #387D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611892
#225497802
#225497742
#225497722
#225497682
#225497672
#225497662
#225497652
#225497622
#225497612
#225497592
#225497382
#225497352
#387×C1272
#2378127