Finite Group Databasebeta

← #30650475

Subgroups — #30650475

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523238C211yesmaximal, characteristic, complemented
2C2×Q16×D127C211yesmaximal, characteristic, complemented
3Q8×C22⋊D127C211yesmaximal, characteristic
4#22550539C211yesmaximal, characteristic
5#22550254C211yesmaximal, characteristic, complemented
6#22550116C211yesmaximal, characteristic, complemented
7#22550115C211yesmaximal, characteristic, complemented
8#22550085C211yesmaximal, characteristic, complemented
9#22550084C211yesmaximal, characteristic, complemented
10#22550034C211yesmaximal, characteristic, complemented
11#22550033C211yesmaximal, characteristic, complemented
12#22550030C211yesmaximal, characteristic, complemented
13#22549968C211yesmaximal, characteristic, complemented
14(C22⋊D127)⋊C8C211yesmaximal, characteristic
15#451×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×Q8×D127C2211yescharacteristic
19C2×Dic508C2211yescharacteristic
20C2×Dic508C2211yescharacteristic
21C2×#14523238—24no—
22C2×#14523238C2211yescharacteristic
23C2×#14523238C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25C2×#14523238C2211yescharacteristic
26C2×#14523238C2211yescharacteristic
27Q16×D127—24no—
28C22×C127⋊C8C2211yescharacteristic
29C2×C8×D127C2211yescharacteristic
30#16861208—21no—
31#16861207—21no—
32#16861207C2211yescharacteristic
33#16861207C2211yescharacteristic
34#16861189C2211yescharacteristic
35#16861188—21no—
36#16861188C2211yescharacteristic
37#16861187—21no—
38#16861186—21no—
39#16861185—21no—
40#16861185C2211yescharacteristic
41Q8×Dic127C2211yescharacteristic
42#16861174C2211yescharacteristic
43C4×C22⋊D127—21no—
44C4×C22⋊D127C2211yescharacteristic
45(C22⋊D127)⋊C4—21no—
46#16861142C2211yescharacteristic
47Dic127⋊C8C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861134C2211yescharacteristic
50#16861133C2211yescharacteristic
51#16861132C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53C8×Dic127C2211yescharacteristic
54C22×Q8×C127C2211yescharacteristic
55C2×Q16×C127C2211yescharacteristic
56C2×Q16×C127C2211yescharacteristic
57C22⋊Q8×C127C2211yescharacteristic
58#103×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60C22⋊C8×C127C2211yescharacteristic
61C2×Dic254—21no—
62C2×Dic254—21no—
63C2×Dic254—22no—
64C2×Dic254—21no—
65C2×Dic254—41no—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2311yescharacteristic
68C2×Dic254C2311yescharacteristic
69Q8×D127—22no—
70Q8×D127—22no—
71Q8×D127—41no—
72Q8×D127—41no—
73C2×C22⋊D127D411yescharacteristic
74C22×Dic127—21no—
75C2×C4×D127—21no—
76C2×C4×D127—21no—
77C2×C4×D127C2311yescharacteristic
78Dic508—22no—
79Dic508—41no—
80#14523238—22no—
81#14523238—22no—
82#14523238—22no—
83#14523238—22no—
84#14523238D414yescomplemented
85C2×C127⋊C8—22no—
86C2×C127⋊C8C2311yescharacteristic
87C2×C127⋊C8C2311yescharacteristic
88C8×D127—22no—
89C4⋊Dic127—21no—
90C4⋊Dic127—21no—
91C4⋊Dic127C2311yescharacteristic
92Dic127⋊C4—21no—
93Dic127⋊C4—21no—
94Dic127⋊C4—21no—
95Dic127⋊C4—21no—
96Dic127⋊C4—21no—
97Dic127⋊C4—21no—
98Dic127⋊C4—21no—
99Dic127⋊C4D411yescharacteristic
100C22⋊Dic127—21no—
101C22⋊Dic127D411yescharacteristic
102(C2×D127)⋊C4—21no—
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4D411yescharacteristic
105C4×Dic127—21no—
106C4×Dic127—21no—
107C4×Dic127C2311yescharacteristic
108C2×Q8×C127—21no—
109C2×Q8×C127—22no—
110C2×Q8×C127—41no—
111C2×Q8×C127C2311yescharacteristic
112C2×Q8×C127C2311yescharacteristic
113C2×Q8×C127C2311yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127C2311yescharacteristic
116Q16×C127—22no—
117Q16×C127—41no—
118C2×C8×C127C2311yescharacteristic
119C2×C8×C127C2311yescharacteristic
120C4⋊C4×C127—21no—
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22×D127C2×D411yescharacteristic
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—42no—
130Dic254—42no—
131Dic254—42no—
132Dic254C2×D412yes—
133C22⋊D127Q1614yescomplemented
134C2×Dic127—21no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—42no—
139C2×Dic127C2×D411yescharacteristic
140C2×Dic127C2×D411yescharacteristic
141C2×Dic127C2×D411yescharacteristic
142C4×D127—21no—
143C4×D127—21no—
144C4×D127—42no—
145C4×D127—42no—
146C127⋊C8—21no—
147C127⋊C8C2×D412yes—
148C23×C127C2×D411yescharacteristic
149Q8×C127—21no—
150Q8×C127—21no—
151Q8×C127—21no—
152Q8×C127—21no—
153Q8×C127—21no—
154Q8×C127—42no—
155Q8×C127C2×D412yes—
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—42no—
163C2×C4×C127C2×D411yescharacteristic
164C2×C4×C127C2411yescharacteristic
165C8×C127—21no—
166C8×C127—21no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127C2×Q1612yescomplemented
170C22×C127C22×D411yescharacteristic
171C2×D127—21no—
172C2×D127—21no—
173C2×D127C2×Q1612yescomplemented
174C4×C127—21no—
175C4×C127—22no—
176C4×C127—41no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127C22×D411yescharacteristic
180C4×C127H(5,𝔽2)11yescharacteristic
181Dic127—21no—
182Dic127—22no—
183Dic127—21no—
184Dic127—41no—
185Dic127—41no—
186Dic127—41no—
187Dic127C2×Q1612yescomplemented
188C2×C127—22no—
189C2×C127D4211yescharacteristic
190C2×C127C22×Q1611yescharacteristic
191C2×C127#57811yescharacteristic
192D127—22no—
193D4×Q16—1271nomaximal, Sylow
194C127D4×Q1611yescharacteristic, complemented, Sylow
195C22×Q16—1271no—
196C22×Q16—1271no—
197D4×Q8—1271no—
198D4×Q8—1271no—
199#494—1271no—
200#467—1271no—
201#467—1271no—
202#462—1271no—
203#462—1271no—
204#451—1271no—
205#451—1271no—
206#451—1271no—
207#451D12711yescharacteristic, complemented
208C4×Q16—1271no—
209C8×D4—1271no—
210C22×Q8—1271no—
211C22×Q8—1271no—
212C22×Q8—1271no—
213C22×Q8C2×D12711yescharacteristic
214C2×Q16—1271no—
215C2×Q16—1271no—
216C2×Q16—1271no—
217C2×Q16—1271no—
218C2×Q16—1271no—
219C2×Q16—1271no—
220C2×Q16—1271no—
221C2×Q16—2544no—
222C2×Q16—2544no—
223C2×Q16C2×D12711yescharacteristic
224C2×Q16C2×D12711yescharacteristic
225C22×C8—1271no—
226C22×C8—1271no—
227C4⋊Q8—1271no—
228C4⋊Q8—1271no—
229C4⋊Q8—2541no—
230C4⋊Q8—2541no—
231C22⋊Q8—1271no—
232C22⋊Q8—1271no—
233C22⋊Q8—1271no—
234C22⋊Q8—2541no—
235C22⋊Q8—2541no—
236C22⋊Q8—2541no—
237C22⋊Q8—2541no—
238C22⋊Q8C2×D12711yescharacteristic
239C4×Q8—1271no—
240C4×Q8—1271no—
241C4×D4—1271no—
242C4×D4—2541no—
243C4×D4—2541no—
244#107—1271no—
245C4⋊C8—1271no—
246#103—1271no—
247#103—1271no—
248#103—1271no—
249#103—1271no—
250#103C2×D12711yescharacteristic
251#103C2×D12711yescharacteristic
252C22⋊C8—1271no—
253C22⋊C8C2×D12711yescharacteristic
254C4×C8—1271no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—1271no—
259C2×Q8—1271no—
260C2×Q8—1271no—
261C2×Q8—5081no—
262C2×Q8—5081no—
263C2×Q8—5081no—
264C2×Q8—2541no—
265C2×Q8—2541no—
266C2×Q8—2542no—
267C2×Q8—2542no—
268C2×Q8—2542no—
269C2×Q8—2541no—
270C2×Q8C22×D12711yescharacteristic
271C2×Q8C22×D12711yescharacteristic
272C2×Q8C22×D12711yescharacteristic
273C2×D4—1271no—
274C22×C4—21no—
275C22×C4—1271no—
276C22×C4—2541no—
277C22×C4—2541no—
278C22×C4—2541no—
279C22×C4C22×D12711yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—1274no—
283Q16—5081no—
284Q16—2542no—
285Q16—2542no—
286Q16—2542no—
287Q16—2542no—
288Q16—2542no—
289C2×C8—1271no—
290C2×C8—1271no—
291C2×C8—2542no—
292C2×C8—2542no—
293C2×C8C22×D12711yescharacteristic
294C2×C8C22×D12711yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—1271no—
297C4⋊C4—1271no—
298C4⋊C4—2541no—
299C4⋊C4—2541no—
300C4⋊C4—2541no—
301C4⋊C4—2541no—
302C4⋊C4—2541no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4C22×D12711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—1271no—
310C22⋊C4—1271no—
311C22⋊C4—2541no—
312C22⋊C4—2541no—
313C22⋊C4—2541no—
314C42—1271no—
315C42—2541no—
316C42—2541no—
317C23—1271no—
318C23D4×D12711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—1272no—
326Q8—5082no—
327Q8—5082no—
328Q8—5082no—
329Q8—2541no—
330Q8—2541no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8D4×D12712yes—
335D4—1274no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1271no—
344C2×C4—1271no—
345C2×C4—1271no—
346C2×C4—5082no—
347C2×C4—5082no—
348C2×C4—5082no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4D4×D12711yescharacteristic
356C2×C4C23×D12711yescharacteristic
357C8—21no—
358C8—21no—
359C8—1272no—
360C8—2541no—
361C22—21no—
362C22—21no—
363C22—1272no—
364C22—2541no—
365C22—2541no—
366C22Q16×D12712yescomplemented
367C22C2×D4×D12711yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—1272no—
374C4—5081no—
375C4—5081no—
376C4—5081no—
377C4—2541no—
378C4—2541no—
379C4—2542no—
380C4C2×D4×D12711yescharacteristic
381C4#1686125411yescharacteristic
382C2—22no—
383C2—2542no—
384C2#2255050811yescharacteristic
385C2C2×Q16×D12711yescharacteristic
386C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#451×C127C2
socleC22×C127C22×D4
prime radicalC2×C22⋊D127D4
hypercenter#451D127
hypocenterC127D4×Q16
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
2. C2×Q16×D127C2semidirect
5. #22550254C2semidirect
6. #22550116C2semidirect
7. #22550115C2semidirect
8. #22550085C2semidirect
9. #22550084C2semidirect
10. #22550034C2semidirect
11. #22550033C2semidirect
12. #22550030C2semidirect
13. #22549968C2semidirect
15. #451×C127C2semidirect
84. #14523238D4semidirect
133. C22⋊D127Q16semidirect
169. C22×C127C2×Q16semidirect
173. C2×D127C2×Q16semidirect
187. Dic127C2×Q16semidirect
194. C127D4×Q16semidirect
207. #451D127semidirect
366. C22Q16×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C2×Q16×D1272
Q8×C22⋊D1272
#225505392
#225502542
#225501162
#225501152
#225500852
#225500842
#225500342
#225500332
#225500302
#225499682
(C22⋊D127)⋊C82
#451×C1272
D4×Q16127