Finite Group Databasebeta

← #30641491

Subgroups — #30641491

Order 16256.

596 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861180C211yesmaximal, characteristic
2#22549776C211yesmaximal, characteristic, complemented
3#22549776C211yesmaximal, characteristic, complemented
4#22549770C211yesmaximal, characteristic, complemented
5#22549764C211yesmaximal, characteristic, complemented
6#22549761C211yesmaximal, characteristic, complemented
7#22549759C211yesmaximal, characteristic, complemented
8#22549756C211yesmaximal, characteristic, complemented
9#22549755C211yesmaximal, characteristic, complemented
10#22549753C211yesmaximal, characteristic, complemented
11#22549738C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
14#22549723C211yesmaximal, characteristic, complemented
15#386×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C23×Dic127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20#16861180C418yescomplemented
21C2×C4⋊Dic127C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C2×C4×Dic127C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C22×D4×C127C2211yescharacteristic
46C23×C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C23×D127C2311yescharacteristic
53C2×C22⋊D127—41no—
54C2×C22⋊D127—41no—
55C2×C22⋊D127—41no—
56C2×C22⋊D127—41no—
57C2×C22⋊D127C2×C414yescomplemented
58C2×C22⋊D127C2×C414yescomplemented
59C22×Dic127—22no—
60C22×Dic127—41no—
61C22×Dic127—41no—
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C2×C4×D127—41no—
70C2×C4×D127—41no—
71C2×C4×D127C2×C414yescomplemented
72C4⋊Dic127C2×C414yescomplemented
73Dic127⋊C4—41no—
74Dic127⋊C4—41no—
75C22⋊Dic127—22no—
76C22⋊Dic127—41no—
77C22⋊Dic127—41no—
78C22⋊Dic127C2×C414yescomplemented
79C22⋊Dic127C2×C414yescomplemented
80(C2×D127)⋊C4—22no—
81(C2×D127)⋊C4—22no—
82(C2×D127)⋊C4—41no—
83(C2×D127)⋊C4—41no—
84C4×Dic127—41no—
85C4×Dic127—41no—
86C4×Dic127—41no—
87C4×Dic127—41no—
88C24×C127C2311yescharacteristic
89C24×C127C2311yescharacteristic
90C2×D4×C127—41no—
91C2×D4×C127—41no—
92C2×D4×C127C2×C414yes—
93C22×C4×C127—22no—
94C22×C4×C127—41no—
95C22×C4×C127—41no—
96C22×C4×C127C2311yescharacteristic
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C22×C4×C127C2311yescharacteristic
100C22×C4×C127C2311yescharacteristic
101C22⋊C4×C127—22no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C22×D127—21no—
105C22×D127—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127—21no—
109C22×D127—21no—
110C22×D127C22×C412yes—
111C22⋊D127—44no—
112C22⋊D127—44no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—44no—
131C2×Dic127C22×C412yes—
132C2×Dic127C22×C412yes—
133C2×Dic127C22×C412yes—
134C2×Dic127#4112yes—
135C4×D127—44no—
136C23×C127—21no—
137C23×C127—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127—21no—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127—21no—
145C23×C127—21no—
146C23×C127—21no—
147C23×C127—21no—
148C23×C127C22×C412yes—
149C23×C127C22×C412yes—
150C23×C127C2411yescharacteristic
151D4×C127—44no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—44no—
166C2×C4×C127C22×C412yes—
167C2×C4×C127#4112yes—
168C22×C127—22no—
169C22×C127—22no—
170C22×C127—22no—
171C22×C127—22no—
172C22×C127—22no—
173C22×C127—22no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127C23×C411yescharacteristic
183C22×C127C2×#4111yescharacteristic
184C22×C127C2×#4111yescharacteristic
185C22×C127H(5,𝔽2)11yescharacteristic
186C22×C127H(5,𝔽2)11yescharacteristic
187C22×C127H(5,𝔽2)11yescharacteristic
188C22×C127H(5,𝔽2)11yescharacteristic
189C2×D127—22no—
190C2×D127—22no—
191C2×D127—22no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—41no—
196C4×C127—22no—
197C4×C127—22no—
198C4×C127—41no—
199C4×C127—41no—
200C4×C127—41no—
201Dic127—22no—
202Dic127—22no—
203Dic127—22no—
204Dic127—22no—
205Dic127—41no—
206Dic127—41no—
207Dic127—41no—
208C2×C127—42no—
209C2×C127—42no—
210C2×C127C4×#4111yescharacteristic
211C2×C127#51811yescharacteristic
212C2×C127#51811yescharacteristic
213C2×C127#53811yescharacteristic
214C2×C127#53811yescharacteristic
215C2×C127#54011yescharacteristic
216C2×C127#54011yescharacteristic
217D127—42no—
218#2333—1271nomaximal, Sylow
219C127#233311yescharacteristic, complemented, Sylow
220C2×#121—1271no—
221#390—1271no—
222#390—1271no—
223#388—1271no—
224#388—1271no—
225#387—1271no—
226#387—1271no—
227#386—1271no—
228#386—1271no—
229#386—1271no—
230#386D12711yescharacteristic, complemented
231#381—1271no—
232C4×C22⋊C4—1271no—
233C4×C22⋊C4—1271no—
234C4×C22⋊C4—1271no—
235C22×D4—1271no—
236C22×D4—1271no—
237C22×D4C2×D12711yescharacteristic
238C23×C4—1271no—
239C23×C4—1271no—
240C23×C4C2×D12711yescharacteristic
241#121—1278no—
242C2×C4⋊C4—1271no—
243C2×C4⋊C4—1271no—
244C2×C4⋊C4—1271no—
245C2×C22⋊C4—1271no—
246C2×C22⋊C4—1271no—
247C2×C22⋊C4—1271no—
248C2×C22⋊C4—1271no—
249C2×C22⋊C4—1271no—
250C2×C22⋊C4—1271no—
251C2×C22⋊C4—1271no—
252C2×C22⋊C4—1271no—
253C2×C22⋊C4—1271no—
254C2×C22⋊C4C2×D12711yescharacteristic
255C2×C22⋊C4C2×D12711yescharacteristic
256C2×C22⋊C4C2×D12711yescharacteristic
257C2×C42—1271no—
258C2×C42—1271no—
259C2×C42—1271no—
260C2×C42—1271no—
261#95—1271no—
262#95—1271no—
263#95—1271no—
264#95—1271no—
265#95—1271no—
266#95—1271no—
267#95—1271no—
268#95—1271no—
269#95C2×D12711yescharacteristic
270#95C2×D12711yescharacteristic
271C24—1271no—
272C24C22×D12711yescharacteristic
273C24C22×D12711yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—1274no—
277C2×D4—1274no—
278C2×D4—5081no—
279C2×D4—5081no—
280C2×D4—5081no—
281C2×D4—5081no—
282C2×D4C4×D12714yes—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—1271no—
287C22×C4—1271no—
288C22×C4—1271no—
289C22×C4—1271no—
290C22×C4—1274no—
291C22×C4—1271no—
292C22×C4—1271no—
293C22×C4—1271no—
294C22×C4—5081no—
295C22×C4—5081no—
296C22×C4—5081no—
297C22×C4—5081no—
298C22×C4—2542no—
299C22×C4C22×D12711yescharacteristic
300C22×C4C22×D12711yescharacteristic
301C22×C4C22×D12711yescharacteristic
302C22×C4C22×D12711yescharacteristic
303C22×C4C22×D12711yescharacteristic
304C4⋊C4—1274no—
305C4⋊C4—5081no—
306C4⋊C4—5081no—
307C22⋊C4—22no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—1274no—
311C22⋊C4—1274no—
312C22⋊C4—5081no—
313C22⋊C4—5081no—
314C22⋊C4—5081no—
315C22⋊C4—5081no—
316C22⋊C4—2542no—
317C22⋊C4—2542no—
318C22⋊C4—2542no—
319C42—5081no—
320C42—5081no—
321C42—5081no—
322C42—5081no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—1272no—
336C23—2541no—
337C23—2541no—
338C23—2541no—
339C23—2541no—
340C23—2541no—
341C23—2541no—
342C23C2×C4×D12712yes—
343C23C2×C4×D12712yes—
344C23C23×D12711yescharacteristic
345D4—44no—
346D4—5084no—
347D4—5084no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—1272no—
363C2×C4—1272no—
364C2×C4—1272no—
365C2×C4—1272no—
366C2×C4—5084no—
367C2×C4—5084no—
368C2×C4—2541no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4—2541no—
375C2×C4—2541no—
376C2×C4—2541no—
377C2×C4—2541no—
378C2×C4—2541no—
379C2×C4—2541no—
380C2×C4—2541no—
381C2×C4—2541no—
382C2×C4—2541no—
383C2×C4—2541no—
384C2×C4—2541no—
385C2×C4C2×C4×D12712yes—
386C2×C4#1452324712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—5081no—
402C22—5081no—
403C22—5081no—
404C22—5081no—
405C22—2542no—
406C22—2542no—
407C22—2542no—
408C22C22×C4×D12711yescharacteristic
409C22#41×D12711yescharacteristic
410C22C2×#1452324711yescharacteristic
411C22#1686125411yescharacteristic
412C22#1686125411yescharacteristic
413C22#1686125411yescharacteristic
414C22#1686125511yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—5081no—
421C4—5081no—
422C4—5081no—
423C4—2542no—
424C4—2542no—
425C4—2542no—
426C4—2542no—
427C2—42no—
428C2—42no—
429C2—5082no—
430C2C4×#1452324711yescharacteristic
431C2#2255034811yescharacteristic
432C2#2255034911yescharacteristic
433C2#2255046111yescharacteristic
434C2#2255046211yescharacteristic
435C2#2255047611yescharacteristic
436C2#2255047911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#386×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861180C2
hypercenter#386D127
hypocenterC127#2333
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #22549776C2semidirect
3. #22549776C2semidirect
4. #22549770C2semidirect
5. #22549764C2semidirect
6. #22549761C2semidirect
7. #22549759C2semidirect
8. #22549756C2semidirect
9. #22549755C2semidirect
10. #22549753C2semidirect
11. #22549738C2semidirect
12. C22⋊C4×Dic127C2semidirect
13. C4×C22⋊Dic127C2semidirect
14. #22549723C2semidirect
15. #386×C127C2semidirect
20. #16861180C4semidirect
57. C2×C22⋊D127C2×C4semidirect
58. C2×C22⋊D127C2×C4semidirect
71. C2×C4×D127C2×C4semidirect
72. C4⋊Dic127C2×C4semidirect
78. C22⋊Dic127C2×C4semidirect
79. C22⋊Dic127C2×C4semidirect
219. C127#2333semidirect
230. #386D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611802
#225497762
#225497702
#225497642
#225497612
#225497592
#225497562
#225497552
#225497532
#225497382
C22⋊C4×Dic1272
C4×C22⋊Dic1272
#225497232
#386×C1272
#2333127