Finite Group Databasebeta

← #30650370

Subgroups — #30650370

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic, complemented
2Q8×C22⋊D127C211yesmaximal, characteristic, complemented
3#22550525C211yesmaximal, characteristic
4#22550254C211yesmaximal, characteristic, complemented
5#22550134C211yesmaximal, characteristic, complemented
6#22550131C211yesmaximal, characteristic, complemented
7#22550084C211yesmaximal, characteristic, complemented
8#22550079C211yesmaximal, characteristic, complemented
9#22550038C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550032C211yesmaximal, characteristic, complemented
12#22550030C211yesmaximal, characteristic, complemented
13SD16×Dic127C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15C2×#137×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18#16861238—24no—
19C2×Dic508C2211yescharacteristic
20C2×#14523238C2211yescharacteristic
21C2×#14523238C2211yescharacteristic
22C2×#14523238C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523231C2211yescharacteristic
25#16861207—21no—
26#16861207C2211yescharacteristic
27#16861207C2211yescharacteristic
28#16861192—21no—
29#16861190—21no—
30#16861188—21no—
31#16861188C2211yescharacteristic
32#16861188C2211yescharacteristic
33#16861185—21no—
34#16861185C2211yescharacteristic
35#16861185C2211yescharacteristic
36#16861181—21no—
37Q8×Dic127C2211yescharacteristic
38D4×Dic127C2211yescharacteristic
39C4×C22⋊D127—21no—
40C4×C22⋊D127C2211yescharacteristic
41C2×Dic127⋊C4—21no—
42#16861141C2211yescharacteristic
43Dic127⋊C8C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861133C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861133C2211yescharacteristic
49#16861130C2211yescharacteristic
50C22⋊(C127⋊C8)C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52C8×Dic127C2211yescharacteristic
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#14523247—22no—
62#14523247—41no—
63C2×Dic254—21no—
64C2×Dic254C2311yescharacteristic
65C2×Dic254C2311yescharacteristic
66Q8×D127—22no—
67Q8×D127—41no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127D411yescharacteristic
70C22×Dic127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74Dic508—22no—
75#14523238—22no—
76#14523238—41no—
77#14523238—41no—
78#14523236—22no—
79#14523231—22no—
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C4⋊Dic127—21no—
83C4⋊Dic127C2311yescharacteristic
84C4⋊Dic127C2311yescharacteristic
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4—21no—
90Dic127⋊C4—21no—
91Dic127⋊C4—21no—
92Dic127⋊C4—21no—
93Dic127⋊C4D411yescharacteristic
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96C22⋊Dic127—21no—
97C22⋊Dic127D411yescharacteristic
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4D411yescharacteristic
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×C127—22no—
118SD16×C127—22no—
119SD16×C127D414yescomplemented
120M16×C127—22no—
121C2×C8×C127C2311yescharacteristic
122C2×C8×C127C2311yescharacteristic
123C22×D127C2×D411yescharacteristic
124Dic254—21no—
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—42no—
129C22⋊D127—22no—
130C22⋊D127—42no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—21no—
135C2×Dic127—42no—
136C2×Dic127—42no—
137C2×Dic127C2×D411yescharacteristic
138C2×Dic127C2×D411yescharacteristic
139C2×Dic127C2×D411yescharacteristic
140C4×D127—21no—
141C4×D127—21no—
142C4×D127—42no—
143C127⋊C8—21no—
144C127⋊C8—21no—
145C23×C127—21no—
146C23×C127C2×D411yescharacteristic
147Q8×C127—21no—
148Q8×C127—21no—
149Q8×C127—21no—
150Q8×C127—21no—
151Q8×C127—21no—
152Q8×C127—42no—
153Q8×C127C2×D412yes—
154D4×C127—21no—
155D4×C127—42no—
156D4×C127C2×D412yes—
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×C127—42no—
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2411yescharacteristic
166C8×C127—21no—
167C8×C127C2×D412yes—
168C22×C127—21no—
169C22×C127—22no—
170C22×C127—21no—
171C22×C127—21no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127C22×D411yescharacteristic
175C2×D127—21no—
176C2×D127—21no—
177C2×D127—21no—
178C4×C127—21no—
179C4×C127—22no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127C22×D411yescharacteristic
183C4×C127H(5,𝔽2)11yescharacteristic
184Dic127—21no—
185Dic127—21no—
186Dic127—41no—
187Dic127—41no—
188Dic127—41no—
189Dic127—41no—
190Dic127#13712yescomplemented
191C2×C127—41no—
192C2×C127—42no—
193C2×C127D4211yescharacteristic
194C2×C127C2×#13711yescharacteristic
195C2×C127#57811yescharacteristic
196D127—41no—
197#3276—1271nomaximal, Sylow
198C127#327611yescharacteristic, complemented, Sylow
199C2×#137—1271no—
200C2×#137D12711yescharacteristic, complemented
201D4×Q8—1271no—
202#547—1271no—
203#494—1271no—
204#470—1271no—
205#470—1271no—
206#462—1271no—
207#461—1271no—
208#452—1271no—
209#452—1271no—
210#451—1271no—
211#451—1271no—
212C4×SD16—1271no—
213#436—1271no—
214C2×#41—1271no—
215C2×#41C2×D12711yescharacteristic
216C22×Q8—1271no—
217C22×Q8C2×D12711yescharacteristic
218#137—24no—
219#137—2544no—
220C2×Q16—1271no—
221C2×Q16—1271no—
222C2×Q16—1271no—
223C2×Q16—1271no—
224C2×Q16C2×D12711yescharacteristic
225C2×Q16C2×D12711yescharacteristic
226C2×SD16—1271no—
227C2×SD16C2×D12711yescharacteristic
228C2×SD16C2×D12711yescharacteristic
229C2×M16—1271no—
230C2×M16C2×D12711yescharacteristic
231C4⋊Q8—1271no—
232C4⋊Q8—1271no—
233C4⋊Q8—2541no—
234#123—2541no—
235#123—2541no—
236C22⋊Q8—1271no—
237C22⋊Q8—1271no—
238C22⋊Q8—1271no—
239C22⋊Q8—1271no—
240C22⋊Q8—2541no—
241C22⋊Q8—2541no—
242#121—2541no—
243C4×Q8—1271no—
244C4×D4—1271no—
245C4×D4—1271no—
246C4×D4—2541no—
247C2×C4⋊C4—2541no—
248#106—1271no—
249C4⋊C8—1271no—
250#103—1271no—
251#103—1271no—
252#103—1271no—
253#103—1271no—
254#103—1271no—
255#102—1271no—
256C22⋊C8—1271no—
257C22⋊C8—1271no—
258C4×C8—1271no—
259#41—22no—
260#41—41no—
261#41—5081no—
262#41—2542no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—1271no—
267C2×Q8—1271no—
268C2×Q8—5081no—
269C2×Q8—2541no—
270C2×Q8—2542no—
271C2×Q8C22×D12711yescharacteristic
272C2×Q8C22×D12711yescharacteristic
273C2×Q8C22×D12711yescharacteristic
274C2×D4—21no—
275C2×D4—1271no—
276C2×D4—2541no—
277C2×D4C22×D12711yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—1271no—
281C22×C4—2541no—
282C22×C4—2541no—
283C22×C4—2541no—
284C22×C4C22×D12711yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—5081no—
288Q16—5081no—
289Q16—2542no—
290Q16—2542no—
291SD16—22no—
292SD16—2542no—
293SD16C22⋊D12714yescomplemented
294M16—22no—
295M16—2542no—
296C2×C8—1271no—
297C2×C8—1271no—
298C2×C8C22×D12711yescharacteristic
299C2×C8C22×D12711yescharacteristic
300C4⋊C4—1271no—
301C4⋊C4—1271no—
302C4⋊C4—1271no—
303C4⋊C4—2541no—
304C4⋊C4—2541no—
305C4⋊C4—2541no—
306C4⋊C4—2541no—
307C4⋊C4—2541no—
308C4⋊C4—2541no—
309C4⋊C4—2541no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C22⋊C4—1271no—
313C22⋊C4—1271no—
314C22⋊C4—2541no—
315C22⋊C4—2541no—
316C22⋊C4—2541no—
317C22⋊C4—2541no—
318C22⋊C4—2541no—
319C42—1271no—
320C42—2541no—
321C23—21no—
322C23—1271no—
323C23D4×D12711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—5082no—
331Q8—2541no—
332Q8—2541no—
333Q8—2541no—
334Q8—2541no—
335Q8C2×C22⋊D12712yes—
336D4—21no—
337D4—42no—
338D4—5082no—
339D4—2542no—
340D4C2×C22⋊D12712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—1271no—
349C2×C4—1271no—
350C2×C4—1271no—
351C2×C4—5082no—
352C2×C4—5082no—
353C2×C4—5082no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4D4×D12711yescharacteristic
361C2×C4C23×D12711yescharacteristic
362C8—21no—
363C8—2541no—
364C8—2541no—
365C8C2×C22⋊D12712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—2541no—
373C22—2541no—
374C22—2541no—
375C22C2×D4×D12711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—1272no—
381C4—5081no—
382C4—5081no—
383C4—5081no—
384C4—5081no—
385C4—2541no—
386C4—2541no—
387C4C22×C22⋊D12711yescharacteristic
388C4#1686125411yescharacteristic
389C2—41no—
390C2—42no—
391C2—5081no—
392C2D4×C22⋊D12711yescharacteristic
393C2#137×D12711yescharacteristic
394C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radical#22550525C2
hypercenterC2×#137D127
hypocenterC127#3276
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×#16861238C2semidirect
2. Q8×C22⋊D127C2semidirect
4. #22550254C2semidirect
5. #22550134C2semidirect
6. #22550131C2semidirect
7. #22550084C2semidirect
8. #22550079C2semidirect
9. #22550038C2semidirect
10. #22550035C2semidirect
11. #22550032C2semidirect
12. #22550030C2semidirect
13. SD16×Dic127C2semidirect
14. #22549952C2semidirect
15. C2×#137×C127C2semidirect
119. SD16×C127D4semidirect
190. Dic127#137semidirect
198. C127#3276semidirect
200. C2×#137D127semidirect
293. SD16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
Q8×C22⋊D1272
#225505252
#225502542
#225501342
#225501312
#225500842
#225500792
#225500382
#225500352
#225500322
#225500302
SD16×Dic1272
#225499522
C2×#137×C1272
#3276127