Finite Group Databasebeta

← #30648255

Subgroups — #30648255

Order 16256.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861241C211yesmaximal, characteristic
2C2×Q16×D127C211yesmaximal, characteristic, complemented
3#22550451C211yesmaximal, characteristic
4#22550352C211yesmaximal, characteristic, complemented
5#22550101C211yesmaximal, characteristic, complemented
6#22550100C211yesmaximal, characteristic, complemented
7#22550099C211yesmaximal, characteristic, complemented
8#22550098C211yesmaximal, characteristic, complemented
9#22550085C211yesmaximal, characteristic, complemented
10#22550084C211yesmaximal, characteristic, complemented
11#22550083C211yesmaximal, characteristic, complemented
12#22549901C211yesmaximal, characteristic, complemented
13#22549883C211yesmaximal, characteristic, complemented
14#22549871C211yesmaximal, characteristic
15#462×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19#16861241—24no—
20C2×Dic508C2211yescharacteristic
21C2×#14523238C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24Q16×D127—24no—
25C2×#14523237C2211yescharacteristic
26C2×#14523235C2211yescharacteristic
27C2×#14523231C2211yescharacteristic
28C2×C8×D127C2211yescharacteristic
29#16861207C2211yescharacteristic
30#16861198C2211yescharacteristic
31#16861197C2211yescharacteristic
32#16861194—21no—
33#16861194—21no—
34#16861187—21no—
35#16861187—21no—
36Q8×Dic127C2211yescharacteristic
37C4×Dic254—21no—
38C4×Dic254C2211yescharacteristic
39#16861174—21no—
40#16861174C2211yescharacteristic
41#16861167—21no—
42#16861167C2211yescharacteristic
43#16861166—21no—
44#16861166C2211yescharacteristic
45C4⋊(C127⋊C8)C2211yescharacteristic
46Dic127⋊C8C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861133C2211yescharacteristic
50#16861133C2211yescharacteristic
51#16861132C2211yescharacteristic
52#16861131C2211yescharacteristic
53#16861129C2211yescharacteristic
54C2×Q16×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic
56C4⋊Q8×C127C2211yescharacteristic
57C4×Q8×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#103×C127C2211yescharacteristic
61#14523249—22no—
62#14523249—22no—
63C2×Dic254—21no—
64C2×Dic254—21no—
65C2×Dic254C2311yescharacteristic
66C2×Dic254C2311yescharacteristic
67Q8×D127—22no—
68Q8×D127—22no—
69Q8×D127—22no—
70Q8×D127D414yes—
71C2×D508—21no—
72C2×D508C2311yescharacteristic
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127—21no—
76C2×C4×D127C2311yescharacteristic
77Dic508—22no—
78#14523238—22no—
79#14523238—22no—
80#14523238—22no—
81#14523237—22no—
82#14523235—22no—
83#14523231—22no—
84C2×C127⋊C8C2311yescharacteristic
85C2×C127⋊C8C2311yescharacteristic
86C8×D127—22no—
87C4⋊Dic127—21no—
88C4⋊Dic127C2311yescharacteristic
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93Dic127⋊C4D411yescharacteristic
94Dic127⋊C4D411yescharacteristic
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4D411yescharacteristic
101(C2×D127)⋊C4D411yescharacteristic
102C4×Dic127—21no—
103C4×Dic127—21no—
104C4×Dic127—21no—
105C4×Dic127—21no—
106C4×Dic127C2311yescharacteristic
107C2×Q8×C127C2311yescharacteristic
108C2×Q8×C127C2311yescharacteristic
109C2×Q8×C127C2311yescharacteristic
110Q16×C127—22no—
111Q16×C127—22no—
112C2×C8×C127C2311yescharacteristic
113C2×C8×C127C2311yescharacteristic
114C4⋊C4×C127—21no—
115C4⋊C4×C127—21no—
116C4⋊C4×C127—21no—
117C4⋊C4×C127C2311yescharacteristic
118C42×C127—21no—
119C42×C127C2311yescharacteristic
120C22×D127—21no—
121C22×D127C2×D411yescharacteristic
122Dic254—21no—
123Dic254—24no—
124Dic254—21no—
125Dic254—21no—
126Dic254—42no—
127Dic254C2×D412yes—
128D508—21no—
129D508—21no—
130D508—42no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127C2×D411yescharacteristic
135C2×Dic127C2×D411yescharacteristic
136C2×Dic127C2×D411yescharacteristic
137C4×D127—24no—
138C4×D127—42no—
139C4×D127—42no—
140C4×D127C2×D412yes—
141C4×D127#4112yes—
142C127⋊C8—21no—
143C127⋊C8—21no—
144Q8×C127—21no—
145Q8×C127—21no—
146Q8×C127—21no—
147Q8×C127—21no—
148Q8×C127—21no—
149Q8×C127C2×D412yes—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2×D411yescharacteristic
156C2×C4×C127C2411yescharacteristic
157C8×C127—21no—
158C8×C127—21no—
159C22×C127C22×D411yescharacteristic
160C2×D127—21no—
161C2×D127—21no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127#12112yes—
166C4×C127—22no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127C22×D411yescharacteristic
173C4×C127C2×#4111yescharacteristic
174Dic127—22no—
175Dic127—21no—
176Dic127—21no—
177Dic127—41no—
178Dic127—41no—
179Dic127#12112yes—
180C2×C127C2×#12111yescharacteristic
181C2×C127#57811yescharacteristic
182C2×C127#57811yescharacteristic
183D127—22no—
184D127—81no—
185#3044—1271nomaximal, Sylow
186C127#304411yescharacteristic, complemented, Sylow
187C2×#137—1271no—
188C22×Q16—1271no—
189#536—1271no—
190#519—1271no—
191#464—1271no—
192#464—1271no—
193#464—1271no—
194#464—1271no—
195#462—1271no—
196#462—1271no—
197#462—1271no—
198#462D12711yescharacteristic, complemented
199#424—1271no—
200#419—1271no—
201#416—1271no—
202C2×#41—1271no—
203C22×Q8—1271no—
204C22×Q8—1271no—
205#137—2544no—
206C2×Q16—1271no—
207C2×Q16—1271no—
208C2×Q16—1271no—
209C2×Q16—1271no—
210C2×Q16—2544no—
211C2×Q16C2×D12711yescharacteristic
212C2×Q16C2×D12711yescharacteristic
213C2×SD16—1271no—
214C2×SD16—1271no—
215C2×M16—1271no—
216C22×C8—1271no—
217C4⋊Q8—1271no—
218C4⋊Q8C2×D12711yescharacteristic
219#124—1271no—
220#124—1271no—
221#123—2541no—
222#123—2541no—
223C22⋊Q8—2541no—
224C22⋊Q8—2541no—
225C4×Q8—1271no—
226C4×Q8—1271no—
227C4×Q8—1271no—
228C4×Q8—2541no—
229C4×Q8—2541no—
230C4×Q8C2×D12711yescharacteristic
231#117—1271no—
232#117—1271no—
233#117—2541no—
234#117—2541no—
235C4⋊C8—1271no—
236C4⋊C8—1271no—
237C4⋊C8—1271no—
238C4⋊C8C2×D12711yescharacteristic
239#103—1271no—
240#103—1271no—
241#103—1271no—
242#103—1271no—
243#103C2×D12711yescharacteristic
244#103C2×D12711yescharacteristic
245#102—1271no—
246#102—1271no—
247#41—2542no—
248#41—2542no—
249C2×Q8—1274no—
250C2×Q8—1271no—
251C2×Q8—1271no—
252C2×Q8—2541no—
253C2×Q8—2541no—
254C2×Q8—2542no—
255C2×Q8—2542no—
256C2×Q8—2542no—
257C2×Q8C22×D12711yescharacteristic
258C2×Q8C22×D12711yescharacteristic
259C2×Q8C22×D12711yescharacteristic
260C2×D4—1271no—
261C2×D4—2541no—
262C22×C4—1271no—
263C22×C4—2541no—
264C22×C4—2541no—
265C22×C4—2541no—
266Q16—22no—
267Q16—22no—
268Q16—2542no—
269Q16—2542no—
270Q16—2542no—
271Q16—2542no—
272SD16—2542no—
273SD16—2542no—
274M16—2542no—
275C2×C8—1271no—
276C2×C8—1271no—
277C2×C8—2542no—
278C2×C8C22×D12711yescharacteristic
279C2×C8C22×D12711yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—1271no—
284C4⋊C4—1271no—
285C4⋊C4—1271no—
286C4⋊C4—2541no—
287C4⋊C4—2541no—
288C4⋊C4—2541no—
289C4⋊C4—2541no—
290C4⋊C4—2541no—
291C4⋊C4C22×D12711yescharacteristic
292C22⋊C4—1271no—
293C22⋊C4—1271no—
294C22⋊C4—2541no—
295C22⋊C4—2541no—
296C22⋊C4—2541no—
297C22⋊C4—2541no—
298C22⋊C4—2541no—
299C42—21no—
300C42—1271no—
301C42—2541no—
302C42—2541no—
303C42—2541no—
304C42—2541no—
305C42C22×D12711yescharacteristic
306C23—1271no—
307C23—2541no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—1272no—
314Q8—5082no—
315Q8—2544no—
316Q8—2541no—
317Q8—2541no—
318Q8—2541no—
319Q8D4×D12712yes—
320D4—5082no—
321D4—2541no—
322D4—2541no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—1271no—
328C2×C4—1272no—
329C2×C4—1272no—
330C2×C4—1271no—
331C2×C4—1271no—
332C2×C4—5082no—
333C2×C4—5082no—
334C2×C4—2544no—
335C2×C4—2541no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4D4×D12711yescharacteristic
339C2×C4D4×D12711yescharacteristic
340C2×C4C23×D12711yescharacteristic
341C8—21no—
342C8—21no—
343C8—2541no—
344C8—2541no—
345C22—1272no—
346C22—5081no—
347C22—5081no—
348C22—5081no—
349C22—2541no—
350C22—2541no—
351C22C2×D4×D12711yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—1272no—
359C4—5081no—
360C4—5081no—
361C4—2542no—
362C4—2541no—
363C4—2541no—
364C4C2×D4×D12711yescharacteristic
365C4#41×D12711yescharacteristic
366C2—10161no—
367C2—2542no—
368C2#121×D12711yescharacteristic
369C2#2255068911yescharacteristic
370C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#462×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523249C22
hypercenter#462D127
hypocenterC127#3044
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D127C2semidirect
4. #22550352C2semidirect
5. #22550101C2semidirect
6. #22550100C2semidirect
7. #22550099C2semidirect
8. #22550098C2semidirect
9. #22550085C2semidirect
10. #22550084C2semidirect
11. #22550083C2semidirect
12. #22549901C2semidirect
13. #22549883C2semidirect
15. #462×C127C2semidirect
186. C127#3044semidirect
198. #462D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612412
C2×Q16×D1272
#225504512
#225503522
#225501012
#225501002
#225500992
#225500982
#225500852
#225500842
#225500832
#225499012
#225498832
#225498712
#462×C1272
#3044127