Finite Group Databasebeta

← #30641524

Subgroups — #30641524

Order 16256.

532 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22549779C211yesmaximal, characteristic
2#22549775C211yesmaximal, characteristic, complemented
3#22549769C211yesmaximal, characteristic, complemented
4#22549769C211yesmaximal, characteristic, complemented
5#22549769C211yesmaximal, characteristic, complemented
6#22549769C211yesmaximal, characteristic, complemented
7#22549762C211yesmaximal, characteristic, complemented
8#22549762C211yesmaximal, characteristic, complemented
9#22549756C211yesmaximal, characteristic
10#22549751C211yesmaximal, characteristic, complemented
11#22549736C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic127C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
15C2×#123×C127C211yesmaximal, characteristic
16C23×Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C4⋊Dic127C2211yescharacteristic
22C2×C22⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×C4×Dic127C2211yescharacteristic
30C2×C4×Dic127C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C127⋊#95C2211yescharacteristic
35C127⋊#95C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C22×D4×C127C2211yescharacteristic
45C23×C4×C127C2211yescharacteristic
46#123×C127C418yescomplemented
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C4⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C22⋊C4×C127C2211yescharacteristic
52C22×Dic127—22no—
53C22×Dic127—41no—
54C22×Dic127—41no—
55C22×Dic127C2311yescharacteristic
56C22×Dic127C2311yescharacteristic
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C4⋊Dic127—22no—
64C4⋊Dic127—41no—
65C4⋊Dic127—41no—
66C4⋊Dic127—41no—
67C4⋊Dic127—41no—
68C22⋊Dic127—22no—
69C22⋊Dic127—22no—
70C22⋊Dic127—22no—
71C22⋊Dic127—41no—
72C22⋊Dic127—41no—
73C22⋊Dic127—41no—
74C22⋊Dic127—41no—
75C4×Dic127—22no—
76C4×Dic127—41no—
77C4×Dic127—41no—
78C4×Dic127—41no—
79C4×Dic127—41no—
80C24×C127C2311yescharacteristic
81C24×C127C2311yescharacteristic
82C2×D4×C127—41no—
83C2×D4×C127—41no—
84C2×D4×C127C2×C414yes—
85C22×C4×C127—41no—
86C22×C4×C127—41no—
87C22×C4×C127C2×C414yescomplemented
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C4⋊C4×C127C2×C414yescomplemented
94C4⋊C4×C127C2×C414yescomplemented
95C22⋊C4×C127C2×C414yes—
96C22⋊C4×C127C2×C414yes—
97C22⋊C4×C127C2×C414yescomplemented
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—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—44no—
121C2×Dic127#4112yes—
122C2×Dic127#4112yes—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127C22×C412yes—
136C23×C127C22×C412yes—
137C23×C127C2411yescharacteristic
138D4×C127—44no—
139C2×C4×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×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—44no—
150C2×C4×C127C22×C412yes—
151C2×C4×C127C22×C412yes—
152C2×C4×C127C22×C412yes—
153C2×C4×C127C22×C412yes—
154C2×C4×C127C22×C412yes—
155C22×C127—22no—
156C22×C127—22no—
157C22×C127—22no—
158C22×C127—22no—
159C22×C127—22no—
160C22×C127—22no—
161C22×C127—41no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127C23×C411yescharacteristic
170C22×C127C2×#4111yescharacteristic
171C22×C127C2×#4111yescharacteristic
172C22×C127H(5,𝔽2)11yescharacteristic
173C22×C127H(5,𝔽2)11yescharacteristic
174C22×C127H(5,𝔽2)11yescharacteristic
175C22×C1272−1+411yescharacteristic
176C4×C127—22no—
177C4×C127—22no—
178C4×C127—22no—
179C4×C127—22no—
180C4×C127—22no—
181Dic127—22no—
182Dic127—22no—
183Dic127—41no—
184Dic127—41no—
185Dic127—41no—
186Dic127—41no—
187Dic127—41no—
188Dic127—41no—
189C2×C127—42no—
190C2×C127—42no—
191C2×C127C4×#4111yescharacteristic
192C2×C127#51811yescharacteristic
193C2×C127#52011yescharacteristic
194C2×C127#53811yescharacteristic
195C2×C127#53911yescharacteristic
196C2×C127#54011yescharacteristic
197C2×C127#54211yescharacteristic
198#2335—1271nomaximal, Sylow
199C127#233511yescharacteristic, complemented, Sylow
200C2×#123D12711yescharacteristic
201#390—1271no—
202#389—1271no—
203#388—1271no—
204#388—1271no—
205#388—1271no—
206#388—1271no—
207#387—1271no—
208#387—1271no—
209#386—1271no—
210#385—1271no—
211#380—1271no—
212C4×C4⋊C4—1271no—
213C4×C22⋊C4—1271no—
214C4×C22⋊C4—1271no—
215C22×D4C2×D12711yescharacteristic
216C23×C4—1271no—
217C23×C4C2×D12711yescharacteristic
218#123Dic12718yescomplemented
219C2×C4⋊C4—1271no—
220C2×C4⋊C4—1271no—
221C2×C4⋊C4—1271no—
222C2×C4⋊C4—1271no—
223C2×C4⋊C4—1271no—
224C2×C4⋊C4C2×D12711yescharacteristic
225C2×C4⋊C4C2×D12711yescharacteristic
226C2×C22⋊C4—1271no—
227C2×C22⋊C4—1271no—
228C2×C22⋊C4—1271no—
229C2×C22⋊C4—1271no—
230C2×C22⋊C4—1271no—
231C2×C22⋊C4—1271no—
232C2×C22⋊C4—1271no—
233C2×C22⋊C4C2×D12711yescharacteristic
234C2×C22⋊C4C2×D12711yescharacteristic
235C2×C22⋊C4C2×D12711yescharacteristic
236C2×C42—1271no—
237C2×C42—1271no—
238C2×C42—1271no—
239C2×C42—1271no—
240C2×C42—1271no—
241#95—1271no—
242#95—1271no—
243#95—1271no—
244#95—1271no—
245#95—1271no—
246#95—1271no—
247#95—1271no—
248#95—1271no—
249#95—1271no—
250#95—1271no—
251C24C22×D12711yescharacteristic
252C24C22×D12711yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4C2×Dic12714yes—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—1271no—
259C22×C4—1271no—
260C22×C4—1271no—
261C22×C4—1271no—
262C22×C4—1271no—
263C22×C4—1271no—
264C22×C4—1271no—
265C22×C4—1271no—
266C22×C4—5081no—
267C22×C4—5081no—
268C22×C4—2542no—
269C22×C4C2×Dic12714yescomplemented
270C22×C4C22×D12711yescharacteristic
271C22×C4C22×D12711yescharacteristic
272C22×C4C22×D12711yescharacteristic
273C22×C4C22×D12711yescharacteristic
274C22×C4C22×D12711yescharacteristic
275C4⋊C4—5081no—
276C4⋊C4—5081no—
277C4⋊C4—5081no—
278C4⋊C4—5081no—
279C4⋊C4—2542no—
280C4⋊C4C2×Dic12714yescomplemented
281C4⋊C4C2×Dic12714yescomplemented
282C22⋊C4—5081no—
283C22⋊C4—5081no—
284C22⋊C4—5081no—
285C22⋊C4—5081no—
286C22⋊C4—2542no—
287C22⋊C4—2542no—
288C22⋊C4—2542no—
289C22⋊C4C2×Dic12714yes—
290C22⋊C4C2×Dic12714yes—
291C22⋊C4C2×Dic12714yescomplemented
292C42—5081no—
293C42—5081no—
294C42—5081no—
295C42—5081no—
296C42—2542no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C22×Dic12712yes—
310C23C22×Dic12712yes—
311C23C23×D12711yescharacteristic
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—1272no—
325C2×C4—1272no—
326C2×C4—5084no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4—2541no—
333C2×C4—2541no—
334C2×C4—2541no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4C22×Dic12712yes—
350C2×C4C22×Dic12712yes—
351C2×C4C22×Dic12712yes—
352C2×C4C22×Dic12712yes—
353C2×C4C22×Dic12712yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C23×Dic12711yescharacteristic
369C22#41×D12711yescharacteristic
370C22#41×D12711yescharacteristic
371C22#1686125411yescharacteristic
372C22#1686125411yescharacteristic
373C22#1686125511yescharacteristic
374C22#1686125611yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—5081no—
381C4—5081no—
382C4—5081no—
383C4—5081no—
384C4—5081no—
385C4—5081no—
386C4—2542no—
387C4—2542no—
388C2—42no—
389C2—42no—
390C2#41×Dic12711yescharacteristic
391C2#2255035111yescharacteristic
392C2#2255035911yescharacteristic
393C2#2255046211yescharacteristic
394C2#2255046911yescharacteristic
395C2#2255047811yescharacteristic
396C2#2255049411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC2×#123×C127C2
socle, cosocleC23×C127C24
prime radicalC22×D4×C127C22
hypercenterC2×#123D127
hypocenterC127#2335
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #22549775C2semidirect
3. #22549769C2semidirect
4. #22549769C2semidirect
5. #22549769C2semidirect
6. #22549769C2semidirect
7. #22549762C2semidirect
8. #22549762C2semidirect
10. #22549751C2semidirect
11. #22549736C2semidirect
12. C4⋊C4×Dic127C2semidirect
13. C22⋊C4×Dic127C2semidirect
14. C4×C22⋊Dic127C2semidirect
46. #123×C127C4semidirect
87. C22×C4×C127C2×C4semidirect
93. C4⋊C4×C127C2×C4semidirect
94. C4⋊C4×C127C2×C4semidirect
97. C22⋊C4×C127C2×C4semidirect
199. C127#2335semidirect
218. #123Dic127semidirect
269. C22×C4C2×Dic127semidirect
280. C4⋊C4C2×Dic127semidirect
281. C4⋊C4C2×Dic127semidirect
291. C22⋊C4C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225497792
#225497752
#225497692
#225497622
#225497562
#225497512
#225497362
C4⋊C4×Dic1272
C22⋊C4×Dic1272
C4×C22⋊Dic1272
C2×#123×C1272
#2335127