Finite Group Databasebeta

← #30652380

Subgroups — #30652380

Order 16256.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861257C211yesmaximal, characteristic, complemented
2C22×Q8×D127C211yesmaximal, characteristic, complemented
3#22550541C213yesmaximal, complemented
4Q8×C22⋊D127C211yesmaximal, characteristic, complemented
5#22550539C213yesmaximal, complemented
6#22550538C211yesmaximal, characteristic, complemented
7#22550537C211yesmaximal, characteristic, complemented
8#22550536C213yesmaximal, complemented
9#22550535C213yesmaximal, complemented
10#22550451C213yesmaximal, complemented
11#22550449C213yesmaximal, complemented
12#22550389C213yesmaximal, complemented
13#22550385C213yesmaximal, complemented
14#22550352C211yesmaximal, characteristic, complemented
15D4×Q8×C127C211yesmaximal, characteristic, complemented
16#16861257—28no—
17C2×#14523249C2213yes—
18C2×#14523249C2211yescharacteristic, complemented
19C2×#14523248C2213yes—
20C2×#14523248C2213yescomplemented
21C22×Dic254C2213yes—
22C2×Q8×D127—28no—
23C2×Q8×D127C2213yes—
24C2×Q8×D127C2211yescharacteristic
25C2×Q8×D127C2213yes—
26C2×Q8×D127C2211yescharacteristic
27C2×Q8×D127C2213yescomplemented
28C22×C4×D127C2213yes—
29#16861208C2216yescomplemented
30#16861207C2213yescomplemented
31#16861198C2213yescomplemented
32#16861197C2213yescomplemented
33#16861195C2216yescomplemented
34#16861194C2213yes—
35#16861194C2216yes—
36#16861191C2213yes—
37#16861189C2213yescomplemented
38#16861188C2213yescomplemented
39#16861187C2213yes—
40#16861187C2216yescomplemented
41#16861187C2216yescomplemented
42#16861186C2213yes—
43#16861185C2213yes—
44#16861185C2213yescomplemented
45#16861184C2213yes—
46#16861183C2213yes—
47#16861182C2213yes—
48#16861181C2213yes—
49#16861179C2213yes—
50#16861177C2211yescharacteristic
51Q8×Dic127C2211yescharacteristic, complemented
52C4×Dic254C2213yescomplemented
53#16861174C2213yescomplemented
54C4×D508C2213yes—
55#16861171C2216yes—
56C4×C22⋊D127C2213yes—
57C4×C22⋊D127C2213yes—
58(C22⋊D127)⋊C4C2216yes—
59#16861167C2213yes—
60#16861166C2213yes—
61C22×Q8×C127C2211yescharacteristic
62C22×Q8×C127C2211yescharacteristic, complemented
63C4⋊Q8×C127C2213yescomplemented
64C22⋊Q8×C127C2213yes—
65C22⋊Q8×C127C2213yescomplemented
66C4×Q8×C127C2211yescharacteristic, complemented
67C4×D4×C127C2213yes—
68C23×D127C2311yescharacteristic
69#14523249—212no—
70#14523249—24no—
71#14523248—212no—
72#14523248—212no—
73C2×Dic254—212no—
74C2×Dic254C2313yes—
75C2×Dic254C2316yes—
76C2×Dic254C2313yes—
77C2×Dic254C2313yes—
78Q8×D127—212no—
79Q8×D127—212no—
80Q8×D127—212no—
81Q8×D127—24no—
82Q8×D127D418yescomplemented
83C2×D508C2313yes—
84C2×D508C2313yes—
85C2×D508C2313yes—
86C2×C22⋊D127C2311yescharacteristic
87C2×C22⋊D127C2313yes—
88C2×C22⋊D127C2313yes—
89C2×C22⋊D127C2311yescharacteristic
90C22×Dic127C2313yes—
91C2×C4×D127—212no—
92C2×C4×D127C2313yes—
93C2×C4×D127C2313yes—
94C2×C4×D127C2313yes—
95C2×C4×D127C2316yes—
96C2×C4×D127C2313yes—
97C2×C4×D127C2313yes—
98C4⋊Dic127C2313yes—
99C4⋊Dic127C2313yes—
100Dic127⋊C4C2316yes—
101Dic127⋊C4C2313yes—
102Dic127⋊C4C2313yes—
103Dic127⋊C4C2313yes—
104Dic127⋊C4C2313yes—
105C22⋊Dic127C2311yescharacteristic
106C22⋊Dic127C2313yes—
107(C2×D127)⋊C4C2316yes—
108(C2×D127)⋊C4C2313yes—
109(C2×D127)⋊C4C2316yes—
110(C2×D127)⋊C4C2311yescharacteristic
111(C2×D127)⋊C4C2313yes—
112(C2×D127)⋊C4C2313yes—
113(C2×D127)⋊C4C2313yes—
114(C2×D127)⋊C4C2311yescharacteristic
115C4×Dic127C2316yes—
116C4×Dic127C2313yes—
117C2×Q8×C127—24no—
118C2×Q8×C127—24no—
119C2×Q8×C127C2313yes—
120C2×Q8×C127C2313yes—
121C2×Q8×C127C2311yescharacteristic
122C2×D4×C127C2311yescharacteristic
123C22×C4×C127C2313yes—
124C22×C4×C127C2313yes—
125C4⋊C4×C127C2313yes—
126C4⋊C4×C127C2313yes—
127C4⋊C4×C127C2316yes—
128C22⋊C4×C127C2313yes—
129C22⋊C4×C127C2313yes—
130C42×C127C2313yes—
131C22×D127—22no—
132C22×D127—22no—
133C22×D127—22no—
134C22×D127C2413yes—
135C22×D127C2411yescharacteristic
136C22×D127C2411yescharacteristic
137Dic254—26no—
138Dic254—212no—
139Dic254—26no—
140Dic254C2×D4112yes—
141D508—26no—
142D508—26no—
143D508—26no—
144C22⋊D127—26no—
145C22⋊D127—22no—
146C22⋊D127—22no—
147C22⋊D127—26no—
148C2×Dic127—26no—
149C2×Dic127C2411yescharacteristic
150C2×Dic127C2413yes—
151C2×Dic127C2413yes—
152C2×Dic127C2411yescharacteristic
153C2×Dic127C2413yes—
154C4×D127—26no—
155C4×D127—26no—
156C4×D127—26no—
157C4×D127—26no—
158C4×D127—212no—
159C4×D127C2×D4112yes—
160C23×C127C2411yescharacteristic
161C23×C127C2411yescharacteristic
162Q8×C127—26no—
163Q8×C127—26no—
164Q8×C127C2×D414yes—
165D4×C127—22no—
166C2×C4×C127—26no—
167C2×C4×C127—26no—
168C2×C4×C127C2411yescharacteristic
169C2×C4×C127C2413yes—
170C2×C4×C127C2413yes—
171C2×C4×C127C2413yes—
172C2×C4×C127C2413yes—
173C22×C127—21no—
174C22×C127—21no—
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—21no—
178C22×C1272−1+412yescomplemented
179C22×C127C2511yescharacteristic
180C2×D127—28no—
181C2×D127—21no—
182C2×D127—21no—
183C2×D127—21no—
184C2×D127—21no—
185C2×D127—23no—
186C2×D127—23no—
187C2×D127—23no—
188C2×D127C22×D412yes—
189C2×D1272−1+412yescomplemented
190C4×C127—23no—
191C4×C127—23no—
192C4×C127—21no—
193C4×C127—23no—
194C4×C127C22×D416yes—
195Dic127—21no—
196Dic127—23no—
197Dic127—21no—
198Dic127—23no—
199Dic127C22×D416yes—
200C2×C127—22no—
201C2×C127—41no—
202C2×C127C23×D411yescharacteristic
203C2×C127C2×2−1+411yescharacteristic
204C2×C127C2×2−1+411yescharacteristic
205D127—22no—
206D127—22no—
207D127—43no—
208#3486—1271nomaximal, Sylow
209C127#348611yescharacteristic, complemented, Sylow
210C2×2−1+4—1271no—
211C23×Q8—1271no—
212D4×Q8—1271no—
213D4×Q8—1273no—
214D4×Q8—1273no—
215D4×Q8D12711yescharacteristic, complemented
216#548—1271no—
217#548—1273no—
218#548—1271no—
219#548—1273no—
220#536—1273no—
221#536—1273no—
222#525—1273no—
223#525—1273no—
224#519—1271no—
2252−1+4—2548no—
226C2×#41—1273no—
227C2×#41—1271no—
228C2×#41—1273no—
229C2×#41—1273no—
230C22×Q8—1271no—
231C22×Q8—1273no—
232C22×Q8—1273no—
233C22×Q8—1271no—
234C22×Q8—1273no—
235C22×Q8—1273no—
236C22×Q8—2548no—
237C22×Q8C2×D12711yescharacteristic
238C22×Q8C2×D12711yescharacteristic, complemented
239C23×C4—1273no—
240C4⋊Q8—1273no—
241C4⋊Q8—1276no—
242C4⋊Q8C2×D12713yescomplemented
243#124—1273no—
244#124—1276no—
245#124—1273no—
246#123—1276no—
247#123—1273no—
248#123—1273no—
249C22⋊Q8—1273no—
250C22⋊Q8—1273no—
251C22⋊Q8—1273no—
252C22⋊Q8—1273no—
253C22⋊Q8—1276no—
254C22⋊Q8—1276no—
255C22⋊Q8—1273no—
256C22⋊Q8—1273no—
257C22⋊Q8C2×D12713yes—
258C22⋊Q8C2×D12713yescomplemented
259#121—1273no—
260#121—1273no—
261#121—1273no—
262#121—1273no—
263#120—1271no—
264#120—1273no—
265C4×Q8—1271no—
266C4×Q8—1273no—
267C4×Q8—1273no—
268C4×Q8C2×D12711yescharacteristic, complemented
269C4×D4—1273no—
270C4×D4—1273no—
271C4×D4—1276no—
272C4×D4—1276no—
273C4×D4—1273no—
274C4×D4C2×D12713yes—
275#117—1273no—
276#117—1273no—
277C24—1271no—
278#41—25412no—
279#41—25412no—
280#41—25412no—
281#41—2544no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—1278no—
285C2×Q8—1273no—
286C2×Q8—1276no—
287C2×Q8—1273no—
288C2×Q8—1273no—
289C2×Q8—25412no—
290C2×Q8—25412no—
291C2×Q8—25412no—
292C2×Q8—25412no—
293C2×Q8—2544no—
294C2×Q8C22×D12713yes—
295C2×Q8C22×D12713yes—
296C2×Q8C22×D12711yescharacteristic
297C2×D4—1271no—
298C2×D4—1273no—
299C2×D4—1271no—
300C2×D4—1273no—
301C2×D4—1273no—
302C2×D4—1273no—
303C2×D4—1273no—
304C2×D4C22×D12711yescharacteristic
305C22×C4—1273no—
306C22×C4—1273no—
307C22×C4—1273no—
308C22×C4—1273no—
309C22×C4—1276no—
310C22×C4—1273no—
311C22×C4—1273no—
312C22×C4—25412no—
313C22×C4C22×D12713yes—
314C22×C4C22×D12713yes—
315C4⋊C4—1273no—
316C4⋊C4—1273no—
317C4⋊C4—1276no—
318C4⋊C4—1273no—
319C4⋊C4—1273no—
320C4⋊C4—1273no—
321C4⋊C4—1273no—
322C4⋊C4C22×D12713yes—
323C4⋊C4C22×D12716yes—
324C4⋊C4C22×D12713yes—
325C22⋊C4—1271no—
326C22⋊C4—1273no—
327C22⋊C4—1273no—
328C22⋊C4—1271no—
329C22⋊C4—1273no—
330C22⋊C4—1271no—
331C22⋊C4—1276no—
332C22⋊C4—1273no—
333C22⋊C4—1276no—
334C22⋊C4—1273no—
335C22⋊C4C22×D12713yes—
336C22⋊C4C22×D12713yes—
337C42—1276no—
338C42—1273no—
339C42C22×D12713yes—
340C23—1271no—
341C23—1271no—
342C23—1273no—
343C23—2542no—
344C23—2542no—
345C23—2542no—
346C23C23×D12711yescharacteristic
347C23C23×D12711yescharacteristic
348Q8—26no—
349Q8—26no—
350Q8—12712no—
351Q8—25412no—
352Q8—2546no—
353Q8—2546no—
354Q8D4×D12714yes—
355D4—22no—
356D4—2546no—
357D4—2546no—
358D4—2546no—
359D4—2542no—
360D4—2542no—
361D4—2546no—
362D4—2546no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—1273no—
366C2×C4—12712no—
367C2×C4—1271no—
368C2×C4—1273no—
369C2×C4—1273no—
370C2×C4—1271no—
371C2×C4—2546no—
372C2×C4—25412no—
373C2×C4—2546no—
374C2×C4—2546no—
375C2×C4—2546no—
376C2×C4—2546no—
377C2×C4C23×D12713yes—
378C2×C4C23×D12711yescharacteristic
379C2×C4C23×D12713yes—
380C2×C4C23×D12713yes—
381C2×C4C23×D12713yes—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—1272no—
388C22—1272no—
389C22—2548no—
390C22—2543no—
391C22—2543no—
392C22—2543no—
393C22—2541no—
394C22—2541no—
395C22—2541no—
396C22—2541no—
397C22#1686125712yescomplemented
398C22C24×D12711yescharacteristic
399C4—23no—
400C4—23no—
401C4—23no—
402C4—21no—
403C4—1276no—
404C4—2541no—
405C4—2543no—
406C4—2541no—
407C4—2543no—
408C4C2×D4×D12716yes—
409C2—22no—
410C2—41no—
411C2—5083no—
412C2—2542no—
413C2—2542no—
414C2C22×D4×D12711yescharacteristic
415C22−1+4×D12711yescharacteristic
416C2C2×#1686125711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D127
commutator, socle, cosocleC22×C127C25
FittingD4×Q8×C127C2
hypercenterD4×Q8D127
hypocenterC127#3486
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. C22×Q8×D127C2semidirect
3. #22550541C2semidirect
4. Q8×C22⋊D127C2semidirect
5. #22550539C2semidirect
6. #22550538C2semidirect
7. #22550537C2semidirect
8. #22550536C2semidirect
9. #22550535C2semidirect
10. #22550451C2semidirect
11. #22550449C2semidirect
12. #22550389C2semidirect
13. #22550385C2semidirect
14. #22550352C2semidirect
15. D4×Q8×C127C2semidirect
18. C2×#14523249C22semidirect
20. C2×#14523248C22semidirect
27. C2×Q8×D127C22semidirect
29. #16861208C22semidirect
30. #16861207C22semidirect
31. #16861198C22semidirect
32. #16861197C22semidirect
33. #16861195C22semidirect
37. #16861189C22semidirect
38. #16861188C22semidirect
40. #16861187C22semidirect
41. #16861187C22semidirect
44. #16861185C22semidirect
51. Q8×Dic127C22semidirect
52. C4×Dic254C22semidirect
53. #16861174C22semidirect
62. C22×Q8×C127C22semidirect
63. C4⋊Q8×C127C22semidirect
65. C22⋊Q8×C127C22semidirect
66. C4×Q8×C127C22semidirect
82. Q8×D127D4semidirect
178. C22×C1272−1+4semidirect
189. C2×D1272−1+4semidirect
209. C127#3486semidirect
215. D4×Q8D127semidirect
238. C22×Q8C2×D127semidirect
242. C4⋊Q8C2×D127semidirect
258. C22⋊Q8C2×D127semidirect
268. C4×Q8C2×D127semidirect
397. C22#16861257semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
C22×Q8×D1272
#225505412
Q8×C22⋊D1272
#225505392
#225505382
#225505372
#225505362
#225505352
#225504512
#225504492
#225503892
#225503852
#225503522
D4×Q8×C1272
#3486127