Finite Group Databasebeta

← #30641883

Subgroups — #30641883

Order 16256.

576 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×#16861182C211yesmaximal, characteristic
2#22549778C211yesmaximal, characteristic, complemented
3#22549776C211yesmaximal, characteristic, complemented
4#22549768C211yesmaximal, characteristic, complemented
5#22549763C211yesmaximal, characteristic, complemented
6#22549760C211yesmaximal, characteristic, complemented
7#22549754C211yesmaximal, characteristic, complemented
8#22549753C211yesmaximal, characteristic, complemented
9#22549753C211yesmaximal, characteristic, complemented
10#22549752C211yesmaximal, characteristic, complemented
11#22549751C211yesmaximal, characteristic, complemented
12#22549747C211yesmaximal, characteristic, complemented
13#22549738C211yesmaximal, characteristic, complemented
14#22549733C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C22×D508C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C22⋊D127C2211yescharacteristic
19C23×Dic127C2211yescharacteristic
20C22×C4×D127C2211yescharacteristic
21C22×C4×D127C2211yescharacteristic
22#16861182C418yescomplemented
23C2×C4⋊Dic127C2211yescharacteristic
24C2×C4⋊Dic127C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
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
52C23×D127C2311yescharacteristic
53C23×D127C2311yescharacteristic
54C2×D508—41no—
55C2×D508—41no—
56C2×D508C2×C414yescomplemented
57C2×C22⋊D127—41no—
58C2×C22⋊D127—41no—
59C2×C22⋊D127—41no—
60C2×C22⋊D127—41no—
61C2×C22⋊D127C2×C414yes—
62C2×C22⋊D127C2×C414yescomplemented
63C22×Dic127—41no—
64C22×Dic127—41no—
65C22×Dic127—41no—
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C22×Dic127C2311yescharacteristic
70C22×Dic127C2311yescharacteristic
71C22×Dic127C2311yescharacteristic
72C2×C4×D127—41no—
73C2×C4×D127—41no—
74C2×C4×D127—41no—
75C2×C4×D127—41no—
76C2×C4×D127—41no—
77C2×C4×D127C2×C414yescomplemented
78C4⋊Dic127—41no—
79C4⋊Dic127—41no—
80Dic127⋊C4—41no—
81Dic127⋊C4C2×C414yescomplemented
82C22⋊Dic127—41no—
83(C2×D127)⋊C4—41no—
84(C2×D127)⋊C4—41no—
85(C2×D127)⋊C4—41no—
86(C2×D127)⋊C4—41no—
87(C2×D127)⋊C4—41no—
88(C2×D127)⋊C4—41no—
89(C2×D127)⋊C4—41no—
90(C2×D127)⋊C4C2×C414yescomplemented
91C4×Dic127—41no—
92C24×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C22×C4×C127C2311yescharacteristic
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C4⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127C2×C414yescomplemented
103C42×C127—41no—
104C22×D127—21no—
105C22×D127—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—21no—
111C22×D127—21no—
112C22×D127—21no—
113C22×D127—21no—
114C22×D127—21no—
115C22×D127—21no—
116C22×D127C22×C412yes—
117C22×D127C22×C412yes—
118D508—44no—
119C22⋊D127—44no—
120C22⋊D127—44no—
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—21no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—44no—
138C2×Dic127C22×C412yes—
139C2×Dic127C22×C412yes—
140C4×D127—44no—
141C4×D127—44no—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127—21no—
145C23×C127—21no—
146C23×C127—21no—
147C23×C127—21no—
148C23×C127C22×C412yes—
149C23×C127C2411yescharacteristic
150C2×C4×C127—21no—
151C2×C4×C127—21no—
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—21no—
166C2×C4×C127C22×C412yes—
167C2×C4×C127C22×C412yes—
168C22×C127—22no—
169C22×C127—22no—
170C22×C127—22no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127C23×C411yescharacteristic
176C22×C127H(5,𝔽2)11yescharacteristic
177C22×C127H(5,𝔽2)11yescharacteristic
178C22×C127H(5,𝔽2)11yescharacteristic
179C22×C127H(5,𝔽2)11yescharacteristic
180C22×C127H(5,𝔽2)11yescharacteristic
181C22×C1272−1+411yescharacteristic
182C2×D127—22no—
183C2×D127—22no—
184C2×D127—22no—
185C2×D127—22no—
186C2×D127—22no—
187C2×D127—22no—
188C2×D127—41no—
189C2×D127—41no—
190C2×D127—41no—
191C2×D127—41no—
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—
201C4×C127—41no—
202Dic127—22no—
203Dic127—22no—
204Dic127—41no—
205Dic127—41no—
206Dic127—41no—
207Dic127—41no—
208C2×C127—42no—
209C2×C127#51811yescharacteristic
210C2×C127#51811yescharacteristic
211C2×C127#52011yescharacteristic
212C2×C127#56011yescharacteristic
213C2×C127#56011yescharacteristic
214C2×C127#56211yescharacteristic
215C2×C127#56211yescharacteristic
216D127—42no—
217D127—42no—
218#2377—1271nomaximal, Sylow
219C127#237711yescharacteristic, complemented, Sylow
220C2×#121—1271no—
221#390—1271no—
222#390—1271no—
223#388—1271no—
224#388D12711yescharacteristic, complemented
225#387—1271no—
226#387—1271no—
227#386—1271no—
228#386—1271no—
229#386—1271no—
230#386—1271no—
231#385—1271no—
232#385—1271no—
233#381—1271no—
234#380—1271no—
235C22×D4—1271no—
236C22×D4—1271no—
237C22×D4—1271no—
238C23×C4—1271no—
239C23×C4—1271no—
240C23×C4—1271no—
241#121—1278no—
242C2×C4⋊C4—1271no—
243C2×C4⋊C4—1271no—
244C2×C4⋊C4—1271no—
245C2×C4⋊C4—1271no—
246C2×C4⋊C4C2×D12711yescharacteristic
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⋊C4—1271no—
255C2×C22⋊C4—1271no—
256C2×C22⋊C4C2×D12711yescharacteristic
257C2×C22⋊C4C2×D12711yescharacteristic
258C2×C22⋊C4C2×D12711yescharacteristic
259C2×C42—1271no—
260C2×C42C2×D12711yescharacteristic
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—
272C24—1271no—
273C24C22×D12711yescharacteristic
274C2×D4—1274no—
275C2×D4—1274no—
276C2×D4—1274no—
277C2×D4—5081no—
278C2×D4—5081no—
279C2×D4—5081no—
280C2×D4—5081no—
281C2×D4—5081no—
282C2×D4—5081no—
283C22×C4—1271no—
284C22×C4—1271no—
285C22×C4—1274no—
286C22×C4—1271no—
287C22×C4—1271no—
288C22×C4—1271no—
289C22×C4—1271no—
290C22×C4—5081no—
291C22×C4—5081no—
292C22×C4—5081no—
293C22×C4—5081no—
294C22×C4—5081no—
295C22×C4—5081no—
296C22×C4—5081no—
297C22×C4—5081no—
298C22×C4C22×D12711yescharacteristic
299C22×C4C22×D12711yescharacteristic
300C22×C4C22×D12711yescharacteristic
301C22×C4C22×D12711yescharacteristic
302C22×C4C22×D12711yescharacteristic
303C22×C4C22×D12711yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—1274no—
306C4⋊C4—5081no—
307C4⋊C4—5081no—
308C4⋊C4—5081no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—1274no—
312C22⋊C4—5081no—
313C22⋊C4—5081no—
314C22⋊C4—5081no—
315C22⋊C4—5081no—
316C22⋊C4—5081no—
317C22⋊C4—5081no—
318C22⋊C4—5081no—
319C22⋊C4—5081no—
320C22⋊C4C4×D12714yescomplemented
321C42—41no—
322C42—5081no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—1272no—
330C23—1272no—
331C23—2541no—
332C23—2541no—
333C23—2541no—
334C23—2541no—
335C23—2541no—
336C23—2541no—
337C23—2541no—
338C23—2541no—
339C23—2541no—
340C23—2541no—
341C23—2541no—
342C23—2541no—
343C23C2×C4×D12712yes—
344C23C23×D12711yescharacteristic
345D4—5084no—
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—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—1272no—
365C2×C4—1272no—
366C2×C4—5084no—
367C2×C4—5084no—
368C2×C4—5084no—
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×C4C2×C4×D12712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—5081no—
395C22—5081no—
396C22—5081no—
397C22—5081no—
398C22—5081no—
399C22—5081no—
400C22—5081no—
401C22—5081no—
402C22—2542no—
403C22—2542no—
404C22—2542no—
405C22—2542no—
406C22—2542no—
407C22—2542no—
408C22C22×C4×D12711yescharacteristic
409C22#1686125411yescharacteristic
410C22#1686125411yescharacteristic
411C22#1686125511yescharacteristic
412C22#1686125511yescharacteristic
413C22#1686125511yescharacteristic
414C22#1686125611yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—5081no—
422C4—5081no—
423C4—5081no—
424C4—5081no—
425C4—2542no—
426C4—2542no—
427C2—42no—
428C2—5082no—
429C2—5082no—
430C2#2255034811yescharacteristic
431C2#2255034811yescharacteristic
432C2#2255035611yescharacteristic
433C2#2255061411yescharacteristic
434C2#2255061711yescharacteristic
435C2#2255062011yescharacteristic
436C2#2255062311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861182C2
hypercenter#388D127
hypocenterC127#2377
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #22549778C2semidirect
3. #22549776C2semidirect
4. #22549768C2semidirect
5. #22549763C2semidirect
6. #22549760C2semidirect
7. #22549754C2semidirect
8. #22549753C2semidirect
9. #22549753C2semidirect
10. #22549752C2semidirect
11. #22549751C2semidirect
12. #22549747C2semidirect
13. #22549738C2semidirect
14. #22549733C2semidirect
15. #388×C127C2semidirect
22. #16861182C4semidirect
56. C2×D508C2×C4semidirect
62. C2×C22⋊D127C2×C4semidirect
77. C2×C4×D127C2×C4semidirect
81. Dic127⋊C4C2×C4semidirect
90. (C2×D127)⋊C4C2×C4semidirect
102. C22⋊C4×C127C2×C4semidirect
219. C127#2377semidirect
224. #388D127semidirect
320. C22⋊C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611822
#225497782
#225497762
#225497682
#225497632
#225497602
#225497542
#225497532
#225497522
#225497512
#225497472
#225497382
#225497332
#388×C1272
#2377127