Finite Group Databasebeta

← #30650458

Subgroups — #30650458

Order 16256.

494 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×#16861229C211yesmaximal, characteristic, complemented
2C2×#16861228C211yesmaximal, characteristic, complemented
3#22550544C211yesmaximal, characteristic, complemented
4#22550525C211yesmaximal, characteristic, complemented
5#22550247C211yesmaximal, characteristic, complemented
6#22550108C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550085C211yesmaximal, characteristic, complemented
9#22550069C211yesmaximal, characteristic, complemented
10#22550040C211yesmaximal, characteristic, complemented
11#22550035C211yesmaximal, characteristic, complemented
12#22550022C211yesmaximal, characteristic, complemented
13#22549967C211yesmaximal, characteristic, complemented
14(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523247C2211yescharacteristic
19#16861229—24no—
20#16861228—24no—
21C2×Dic508C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523237C2211yescharacteristic
24C2×#14523236C2211yescharacteristic
25C2×#14523236C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523233C2211yescharacteristic
28C22×C127⋊C8C2211yescharacteristic
29C2×C8×D127C2211yescharacteristic
30#16861207C2211yescharacteristic
31#16861205—21no—
32#16861205C2211yescharacteristic
33#16861192—21no—
34#16861190—21no—
35#16861188C2211yescharacteristic
36#16861185C2211yescharacteristic
37#16861184C2211yescharacteristic
38#16861182—21no—
39#16861181—21no—
40#16861181—21no—
41#16861174C2211yescharacteristic
42D4×Dic127C2211yescharacteristic
43C4×C22⋊D127C2211yescharacteristic
44(C22⋊D127)⋊C4—21no—
45C2×Dic127⋊C4—21no—
46#16861140C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861132C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861130C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53C8×Dic127C2211yescharacteristic
54C2×#41×C127C2211yescharacteristic
55C2×SD16×C127C2211yescharacteristic
56C2×D8×C127C2211yescharacteristic
57#121×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#102×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61#14523248—22no—
62#14523248—41no—
63#14523247—22no—
64#14523247—22no—
65#14523247—41no—
66#14523247—41no—
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69C2×D508—21no—
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127D411yescharacteristic
77C22×Dic127—21no—
78C22×Dic127—21no—
79C22×Dic127—21no—
80C2×C4×D127—21no—
81C2×C4×D127C2311yescharacteristic
82Dic508—22no—
83#14523238—22no—
84#14523237—41no—
85#14523236—22no—
86#14523236D414yescomplemented
87#14523235—22no—
88#14523233—22no—
89C2×C127⋊C8—22no—
90C2×C127⋊C8C2311yescharacteristic
91C2×C127⋊C8C2311yescharacteristic
92C8×D127—22no—
93C4⋊Dic127C2311yescharacteristic
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4D411yescharacteristic
100C22⋊Dic127—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127D411yescharacteristic
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4D411yescharacteristic
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108#41×C127—22no—
109#41×C127—41no—
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116C22×C4×C127C2311yescharacteristic
117SD16×C127—41no—
118D8×C127—22no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127—21no—
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254C2×D412yes—
129D508—21no—
130D508—21no—
131D508—81no—
132C22⋊D127—22no—
133C22⋊D127—42no—
134C22⋊D127—42no—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C22⋊D127—81no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—42no—
142C2×Dic127—42no—
143C2×Dic127—42no—
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C2×Dic127C2×D411yescharacteristic
147C4×D127—21no—
148C4×D127—21no—
149C4×D127—42no—
150C127⋊C8—21no—
151C127⋊C8C2×D412yes—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127C2×D411yescharacteristic
155Q8×C127—21no—
156Q8×C127—21no—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127—42no—
161D4×C127—81no—
162D4×C127C2×D412yes—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—42no—
168C2×C4×C127C2×D411yescharacteristic
169C2×C4×C127C2411yescharacteristic
170C8×C127—21no—
171C8×C127—21no—
172C22×C127—21no—
173C22×C127—22no—
174C22×C127—21no—
175C22×C127—21no—
176C22×C127—41no—
177C22×C127—41no—
178C22×C127—41no—
179C22×C127—41no—
180C22×C127—41no—
181C22×C127C22×D411yescharacteristic
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—41no—
186C2×D127—41no—
187C2×D127—41no—
188C4×C127—21no—
189C4×C127—41no—
190C4×C127—41no—
191C4×C127C22×D411yescharacteristic
192C4×C127H(5,𝔽2)11yescharacteristic
193Dic127—21no—
194Dic127—22no—
195Dic127—21no—
196Dic127—41no—
197Dic127—41no—
198Dic127#13512yescomplemented
199C2×C127—41no—
200C2×C127—42no—
201C2×C127—81no—
202C2×C127D4211yescharacteristic
203C2×C127C2×#13511yescharacteristic
204C2×C127#57711yescharacteristic
205D127—41no—
206D127—81no—
207#3284—1271nomaximal, Sylow
208C127#328411yescharacteristic, complemented, Sylow
209C2×#135—1271no—
210C2×#135—1271no—
211#550—1271no—
212#547—1271no—
213#492—1271no—
214#465—1271no—
215#465—1271no—
216#462—1271no—
217#459—1271no—
218#452—1271no—
219#452—1271no—
220#449—1271no—
221#449D12711yescharacteristic, complemented
222C4×SD16—1271no—
223C8×D4—1271no—
224C2×#41—1271no—
225C2×#41—1271no—
226C2×#41—1271no—
227C2×#41C2×D12711yescharacteristic
228#135—2544no—
229#135—2544no—
230C2×Q16—1271no—
231C2×Q16—1271no—
232C2×SD16—1271no—
233C2×SD16—1271no—
234C2×SD16—1271no—
235C2×SD16—1271no—
236C2×SD16C2×D12711yescharacteristic
237C2×D8—1271no—
238C2×D8C2×D12711yescharacteristic
239C22×C8—1271no—
240C22×C8—1271no—
241C4⋊Q8—1271no—
242#127—1271no—
243#127—2541no—
244#123—2541no—
245#123—2541no—
246C22⋊Q8—1271no—
247C22⋊Q8—1271no—
248#121—1271no—
249#121—2541no—
250#121—2541no—
251#121—2541no—
252#121C2×D12711yescharacteristic
253C4×Q8—1271no—
254C4×D4—1271no—
255C4×D4—1271no—
256C4×D4—2541no—
257C2×C4⋊C4—2541no—
258#106—1271no—
259C4⋊C8—1271no—
260#103—1271no—
261#103—1271no—
262#103C2×D12711yescharacteristic
263#102—1271no—
264#102—1271no—
265#102C2×D12711yescharacteristic
266C22⋊C8—1271no—
267C22⋊C8C2×D12711yescharacteristic
268C4×C8—1271no—
269#41—22no—
270#41—41no—
271#41—5081no—
272#41—5081no—
273#41—5081no—
274#41—2542no—
275#41—2542no—
276#41—2542no—
277C2×Q8—1271no—
278C2×Q8—1271no—
279C2×Q8C22×D12711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—1271no—
283C2×D4—1271no—
284C2×D4—2541no—
285C2×D4—2541no—
286C2×D4—2541no—
287C2×D4—2541no—
288C2×D4—2541no—
289C2×D4—2541no—
290C2×D4C22×D12711yescharacteristic
291C2×D4C22×D12711yescharacteristic
292C22×C4—21no—
293C22×C4—1271no—
294C22×C4—2541no—
295C22×C4—2541no—
296C22×C4—2541no—
297C22×C4—2541no—
298C22×C4C22×D12711yescharacteristic
299Q16—2542no—
300Q16—2542no—
301SD16—41no—
302SD16—1274no—
303SD16—5081no—
304SD16—2542no—
305SD16—2542no—
306D8—22no—
307D8—2542no—
308C2×C8—1271no—
309C2×C8—1271no—
310C2×C8—2542no—
311C2×C8—2542no—
312C2×C8C22×D12711yescharacteristic
313C2×C8C22×D12711yescharacteristic
314C4⋊C4—1271no—
315C4⋊C4—1271no—
316C4⋊C4—2541no—
317C4⋊C4—2541no—
318C4⋊C4—2541no—
319C4⋊C4—2541no—
320C4⋊C4—2541no—
321C4⋊C4C22×D12711yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—1271no—
324C22⋊C4—1271no—
325C22⋊C4—2541no—
326C22⋊C4—2541no—
327C22⋊C4—2541no—
328C22⋊C4—2541no—
329C42—1271no—
330C42—2541no—
331C23—21no—
332C23—21no—
333C23—1271no—
334C23—2541no—
335C23D4×D12711yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—1272no—
339Q8—2541no—
340Q8—2541no—
341Q8—2541no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—81no—
347D4—10161no—
348D4—10161no—
349D4—5082no—
350D4—5082no—
351D4—5082no—
352D4—5082no—
353D4—2542no—
354D4—2541no—
355D4—2541no—
356D4D4×D12712yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—1271no—
363C2×C4—1271no—
364C2×C4—1271no—
365C2×C4—5082no—
366C2×C4—5082no—
367C2×C4—5082no—
368C2×C4—5082no—
369C2×C4—2541no—
370C2×C4—2541no—
371C2×C4—2541no—
372C2×C4—2541no—
373C2×C4—2541no—
374C2×C4D4×D12711yescharacteristic
375C2×C4C23×D12711yescharacteristic
376C8—21no—
377C8—21no—
378C8—1272no—
379C8—2541no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—5081no—
390C22—5081no—
391C22—5081no—
392C22—2541no—
393C22—2541no—
394C22—2541no—
395C22C2×D4×D12711yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—1272no—
400C4—5081no—
401C4—5081no—
402C4—2541no—
403C4—2541no—
404C4—2542no—
405C4C2×D4×D12711yescharacteristic
406C4#1686125411yescharacteristic
407C2—41no—
408C2—42no—
409C2—81no—
410C2—10161no—
411C2—5081no—
412C2#2255050811yescharacteristic
413C2#135×D12711yescharacteristic
414C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3284
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. C2×#16861228C2semidirect
3. #22550544C2semidirect
4. #22550525C2semidirect
5. #22550247C2semidirect
6. #22550108C2semidirect
7. #22550103C2semidirect
8. #22550085C2semidirect
9. #22550069C2semidirect
10. #22550040C2semidirect
11. #22550035C2semidirect
12. #22550022C2semidirect
13. #22549967C2semidirect
14. (C22⋊D127)⋊C8C2semidirect
15. #449×C127C2semidirect
86. #14523236D4semidirect
198. Dic127#135semidirect
208. C127#3284semidirect
221. #449D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
C2×#168612282
#225505442
#225505252
#225502472
#225501082
#225501032
#225500852
#225500692
#225500402
#225500352
#225500222
#225499672
(C22⋊D127)⋊C82
#449×C1272
#3284127