Finite Group Databasebeta

← #30650309

Subgroups — #30650309

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861232C211yesmaximal, characteristic, complemented
2#22550545C211yesmaximal, characteristic, complemented
3#22550537C211yesmaximal, characteristic, complemented
4#22550262C211yesmaximal, characteristic, complemented
5#22550129C211yesmaximal, characteristic, complemented
6#22550102C211yesmaximal, characteristic, complemented
7#22550100C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550037C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550020C211yesmaximal, characteristic, complemented
12#22550015C211yesmaximal, characteristic, complemented
13#22549983C211yesmaximal, characteristic, complemented
14#22549945C211yesmaximal, characteristic, complemented
15C2×#137×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×#14523249C2211yescharacteristic
18#16861232—24no—
19C2×#14523235C2211yescharacteristic
20C2×#14523235C2211yescharacteristic
21C2×#14523235C2211yescharacteristic
22C2×D1016C2211yescharacteristic
23C2×#14523233C2211yescharacteristic
24C2×C8×D127C2211yescharacteristic
25#16861205—21no—
26#16861205C2211yescharacteristic
27#16861198—21no—
28#16861198C2211yescharacteristic
29#16861185—21no—
30#16861185C2211yescharacteristic
31#16861183—21no—
32#16861183—21no—
33#16861183C2211yescharacteristic
34#16861183C2211yescharacteristic
35#16861180—21no—
36#16861180C2211yescharacteristic
37Q8×Dic127C2211yescharacteristic
38Q8×Dic127C2211yescharacteristic
39C4×C22⋊D127—21no—
40C4×C22⋊D127—21no—
41C4×C22⋊D127C2211yescharacteristic
42#16861141C2211yescharacteristic
43Dic127⋊C8C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861133C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861130C2211yescharacteristic
50C22⋊(C127⋊C8)C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52#16861119C2211yescharacteristic
53C2×#41×C127C2211yescharacteristic
54C22×Q8×C127C2211yescharacteristic
55#137×C127—24no—
56C2×Q16×C127C2211yescharacteristic
57C2×Q16×C127C2211yescharacteristic
58C2×SD16×C127C2211yescharacteristic
59C2×SD16×C127C2211yescharacteristic
60C2×M16×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—22no—
63#14523249—41no—
64#14523249—41no—
65C2×D508—21no—
66C2×D508—21no—
67C2×D508C2311yescharacteristic
68C2×D508C2311yescharacteristic
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127D411yescharacteristic
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127C2311yescharacteristic
78#14523235—22no—
79#14523235—22no—
80#14523235—41no—
81D1016—22no—
82#14523233—41no—
83C2×C127⋊C8C2311yescharacteristic
84C2×C127⋊C8C2311yescharacteristic
85C8×D127—22no—
86C4⋊Dic127—21no—
87C4⋊Dic127—21no—
88C4⋊Dic127C2311yescharacteristic
89C4⋊Dic127C2311yescharacteristic
90Dic127⋊C4—21no—
91Dic127⋊C4D411yescharacteristic
92C22⋊Dic127—21no—
93C22⋊Dic127D411yescharacteristic
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4D411yescharacteristic
100C4×Dic127—21no—
101C4×Dic127—21no—
102C4×Dic127C2311yescharacteristic
103#41×C127—22no—
104#41×C127—41no—
105C2×Q8×C127—21no—
106C2×Q8×C127—22no—
107C2×Q8×C127—41no—
108C2×Q8×C127C2311yescharacteristic
109C2×Q8×C127C2311yescharacteristic
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—22no—
117Q16×C127D414yescomplemented
118SD16×C127—22no—
119SD16×C127—22no—
120M16×C127—22no—
121C2×C8×C127C2311yescharacteristic
122C2×C8×C127C2311yescharacteristic
123C22×D127—21no—
124C22×D127—21no—
125C22×D127C2×D411yescharacteristic
126D508—21no—
127D508—21no—
128D508—21no—
129D508—21no—
130D508—42no—
131D508—42no—
132C22⋊D127—41no—
133C22⋊D127—42no—
134C22⋊D127—42no—
135C22⋊D127—81no—
136C22⋊D127—81no—
137C22⋊D127—81no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C2×Dic127C2×D411yescharacteristic
143C4×D127—21no—
144C4×D127—21no—
145C4×D127—42no—
146C4×D127—42no—
147C127⋊C8—21no—
148C127⋊C8—21no—
149C23×C127—21no—
150C23×C127C2×D411yescharacteristic
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127—21no—
155Q8×C127—42no—
156Q8×C127C2×D412yes—
157Q8×C127C2×D412yes—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—42no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—42no—
167C2×C4×C127—42no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127C2×D412yes—
172C22×C127—21no—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—41no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127C22×D411yescharacteristic
179C2×D127—21no—
180C2×D127—21no—
181C2×D127—21no—
182C2×D127—41no—
183C2×D127—41no—
184C2×D127—41no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C4×C127—21no—
189C4×C127—22no—
190C4×C127—22no—
191C4×C127—41no—
192C4×C127C22×D411yescharacteristic
193C4×C127H(5,𝔽2)11yescharacteristic
194Dic127—21no—
195Dic127—21no—
196Dic127—21no—
197Dic127—41no—
198Dic127—41no—
199C2×C127—41no—
200C2×C127—81no—
201C2×C127D4211yescharacteristic
202C2×C127#57511yescharacteristic
203C2×C127#57711yescharacteristic
204D127—41no—
205D127—81no—
206D127—81no—
207#3271—1271nomaximal, Sylow
208C127#327111yescharacteristic, complemented, Sylow
209C2×#137D12711yescharacteristic, complemented
210C2×#135—1271no—
211#550—1271no—
212#548—1271no—
213#496—1271no—
214#469—1271no—
215#465—1271no—
216#464—1271no—
217#460—1271no—
218#452—1271no—
219#449—1271no—
220#449—1271no—
221#448—1271no—
222#441—1271no—
223#435—1271no—
224C2×#41—1271no—
225C2×#41—1271no—
226C2×#41C2×D12711yescharacteristic
227C22×Q8C2×D12711yescharacteristic
228#137—24no—
229#135—2544no—
230C2×Q16C2×D12711yescharacteristic
231C2×Q16C2×D12711yescharacteristic
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16C2×D12711yescharacteristic
236C2×SD16C2×D12711yescharacteristic
237C2×D8—1271no—
238C2×D8—1271no—
239C2×M16C2×D12711yescharacteristic
240C22×C8—1271no—
241#127—1271no—
242#127—2541no—
243#124—1271no—
244#124—2541no—
245C22⋊Q8—1271no—
246C22⋊Q8—2541no—
247#121—1271no—
248#121—1271no—
249#121—1271no—
250#121—2541no—
251#121—2541no—
252#121—2541no—
253C4×Q8—1271no—
254C4×Q8—1271no—
255C4×D4—1271no—
256C4×D4—2541no—
257C4×D4—2541no—
258#106—1271no—
259C4⋊C8—1271no—
260#103—1271no—
261#103—1271no—
262#103—1271no—
263#102—1271no—
264#102—1271no—
265#102—1271no—
266C22⋊C8—1271no—
267C22⋊C8—1271no—
268#97—1271no—
269#41—22no—
270#41—41no—
271#41—5081no—
272#41—5081no—
273#41—2542no—
274#41—2542no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D12711yescharacteristic
279C2×Q8C22×D12711yescharacteristic
280C2×Q8C22×D12711yescharacteristic
281C2×D4—21no—
282C2×D4—1271no—
283C2×D4—1271no—
284C2×D4—1271no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4—2541no—
291C2×D4—2541no—
292C2×D4C22×D12711yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—1271no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4C22×D12711yescharacteristic
299Q16—22no—
300Q16C22⋊D12714yescomplemented
301SD16—22no—
302SD16—22no—
303SD16—5081no—
304SD16—2542no—
305SD16—2542no—
306D8—5081no—
307D8—2542no—
308M16—22no—
309C2×C8—1271no—
310C2×C8—1271no—
311C2×C8—2542no—
312C2×C8C22×D12711yescharacteristic
313C2×C8C22×D12711yescharacteristic
314C4⋊C4—1271no—
315C4⋊C4—1271no—
316C4⋊C4—1271no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C22⋊C4—1271no—
321C22⋊C4—1271no—
322C22⋊C4—2541no—
323C22⋊C4—2541no—
324C22⋊C4—2541no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C42—1271no—
329C42—2541no—
330C42—2541no—
331C23—21no—
332C23—1271no—
333C23—2541no—
334C23—2541no—
335C23D4×D12711yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8C2×C22⋊D12712yes—
342Q8C2×C22⋊D12712yes—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—10161no—
347D4—10161no—
348D4—10161no—
349D4—5082no—
350D4—5082no—
351D4—5082no—
352D4—5081no—
353D4—5082no—
354D4—2541no—
355D4—2541no—
356D4—2541no—
357D4—2541no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—1271no—
366C2×C4—1271no—
367C2×C4—1271no—
368C2×C4—5082no—
369C2×C4—5082no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4D4×D12711yescharacteristic
375C2×C4C23×D12711yescharacteristic
376C8—21no—
377C8—2541no—
378C8—2541no—
379C8C2×C22⋊D12712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—5081no—
387C22—5081no—
388C22—5081no—
389C22—5081no—
390C22—5081no—
391C22—5081no—
392C22—2541no—
393C22—2541no—
394C22—2541no—
395C22C2×D4×D12711yescharacteristic
396C4—21no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—5081no—
401C4—5081no—
402C4—2541no—
403C4—2541no—
404C4—2541no—
405C4C22×C22⋊D12711yescharacteristic
406C4#1686125411yescharacteristic
407C2—41no—
408C2—81no—
409C2—10161no—
410C2—10161no—
411C2—5081no—
412C2D4×C22⋊D12711yescharacteristic
413C2#2255067511yescharacteristic
414C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
hypercenterC2×#137D127
hypocenterC127#3271
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16861232C2semidirect
2. #22550545C2semidirect
3. #22550537C2semidirect
4. #22550262C2semidirect
5. #22550129C2semidirect
6. #22550102C2semidirect
7. #22550100C2semidirect
8. #22550074C2semidirect
9. #22550037C2semidirect
10. #22550021C2semidirect
11. #22550020C2semidirect
12. #22550015C2semidirect
13. #22549983C2semidirect
14. #22549945C2semidirect
15. C2×#137×C127C2semidirect
117. Q16×C127D4semidirect
208. C127#3271semidirect
209. C2×#137D127semidirect
300. Q16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612322
#225505452
#225505372
#225502622
#225501292
#225501022
#225501002
#225500742
#225500372
#225500212
#225500202
#225500152
#225499832
#225499452
C2×#137×C1272
#3271127