Finite Group Databasebeta

← #27182643

Subgroups — #27182643

Order 12928.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692286C211yesmaximal, characteristic, complemented
2C22×#13108061C211yesmaximal, characteristic, complemented
3#20265823C211yesmaximal, characteristic
4#20265569C211yesmaximal, characteristic, complemented
5#20265425C211yesmaximal, characteristic, complemented
6#20265416C211yesmaximal, characteristic, complemented
7#20265399C211yesmaximal, characteristic, complemented
8#20265379C211yesmaximal, characteristic, complemented
9#20265337C211yesmaximal, characteristic, complemented
10#20265333C211yesmaximal, characteristic, complemented
11#20265327C211yesmaximal, characteristic, complemented
12#20265316C211yesmaximal, characteristic, complemented
13#20265275C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15D4×Q8×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×D404C2211yescharacteristic
18#15692286—24no—
19C2×#13108064C2211yescharacteristic, complemented
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108061—24no—
23C2×#13108061C2211yescharacteristic
24C2×#13108061C2211yescharacteristic
25C2×#13108061C2211yescharacteristic
26C2×#13108061C2211yescharacteristic
27C2×#13108061C2211yescharacteristic
28C2×#13108061C2211yescharacteristic
29C2×#13108058C2211yescharacteristic, complemented
30C22×#11717579C2211yescharacteristic
31#15692243C2211yescharacteristic
32#15692240—21no—
33#15692234C2211yescharacteristic
34#15692229C2211yescharacteristic
35#15692228—21no—
36#15692223—21no—
37C4×D404C2211yescharacteristic
38C22⋊C4×D101—21no—
39#15692188C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692179C2211yescharacteristic, complemented
44#15692178C2211yescharacteristic, complemented
45#15692175C2211yescharacteristic
46#15692175C2211yescharacteristic
47C22⋊#11717579C2211yescharacteristic
48C22⋊#11717579C2211yescharacteristic, complemented
49#15692166C2211yescharacteristic, complemented
50C22×Q8×C101C2211yescharacteristic
51C22×Q8×C101C2211yescharacteristic, complemented
52C4⋊Q8×C101—21no—
53C4⋊Q8×C101C2211yescharacteristic, complemented
54C22⋊Q8×C101—21no—
55C22⋊Q8×C101—21no—
56C22⋊Q8×C101C2211yescharacteristic
57C22⋊Q8×C101C2211yescharacteristic, complemented
58C4×Q8×C101C2211yescharacteristic, complemented
59C4×D4×C101—21no—
60C4×D4×C101C2211yescharacteristic
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64C2×Dic202C2311yescharacteristic
65C2×D404—22no—
66C2×D404—21no—
67C2×D404—41no—
68C2×D404C2311yescharacteristic
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101—21no—
74#13108064—22no—
75#13108064—22no—
76#13108064—41no—
77#13108061—22no—
78#13108061—22no—
79#13108061—22no—
80#13108061—22no—
81#13108061—41no—
82#13108061D414yescomplemented
83#13108058—22no—
84C2×#11717579—22no—
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90#13108053—21no—
91C22⋊Dic101—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96C2×Q8×C101—21no—
97C2×Q8×C101—21no—
98C2×Q8×C101—22no—
99C2×Q8×C101—22no—
100C2×Q8×C101—41no—
101C2×Q8×C101—41no—
102C2×Q8×C101C2311yescharacteristic
103C2×Q8×C101C2311yescharacteristic
104C2×Q8×C101C2311yescharacteristic
105C2×D4×C101D411yescharacteristic
106C22×C4×C101—21no—
107C22×C4×C101—21no—
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C4⋊C4×C101—21no—
111C4⋊C4×C101—21no—
112C4⋊C4×C101—21no—
113C4⋊C4×C101—21no—
114C4⋊C4×C101—21no—
115C4⋊C4×C101D411yescharacteristic
116C4⋊C4×C101C2311yescharacteristic
117C22⋊C4×C101—21no—
118C22⋊C4×C101—21no—
119C22⋊C4×C101D411yescharacteristic
120C22⋊C4×C101D411yescharacteristic
121C42×C101—21no—
122C42×C101C2311yescharacteristic
123C23×C101C2×D411yescharacteristic
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—42no—
128C22×D101—42no—
129C22×D101—81no—
130Q8×C101—21no—
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—21no—
134Q8×C101—21no—
135Q8×C101—42no—
136Q8×C101—42no—
137Q8×C101C2×D412yes—
138D4×C101—22no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—42no—
148C2×C4×C101—42no—
149C2×C4×C101C2×D411yescharacteristic
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2×D411yescharacteristic
152C2×C4×C101C2×D411yescharacteristic
153C2×C4×C101C2411yescharacteristic
154C22⋊D101—42no—
155C22⋊D101—81no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158D404—21no—
159D404—21no—
160D404—21no—
161D404—42no—
162D404C2×D412yes—
163C4×D101—42no—
164C4×D101—81no—
165Dic202—21no—
166Dic202—21no—
167#11717579—21no—
168#11717579—21no—
169#11717579—21no—
170#11717579C2×D412yes—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101#13712yescomplemented
177C22×C101C22×D411yescharacteristic
178C2×D101—22no—
179C2×D101—41no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—41no—
184C2×D101—84no—
185C4×C101—21no—
186C4×C101—21no—
187C4×C101—22no—
188C4×C101—21no—
189C4×C101—21no—
190C4×C101—41no—
191C4×C101—41no—
192C4×C101—41no—
193C4×C101C22×D411yescharacteristic
194C4×C101H(5,𝔽2)11yescharacteristic
195Dic101—41no—
196Dic101—41no—
197C2×C101—22no—
198C2×C101—41no—
199C2×C101D4211yescharacteristic
200C2×C101C2×#13711yescharacteristic
201C2×C101#57711yescharacteristic
202D101—42no—
203D101—81no—
204#3266—1011nomaximal, Sylow
205C101#326611yescharacteristic, complemented, Sylow
206C2×#137—1011no—
207C22×SD16—1011no—
208D4×Q8D10111yescharacteristic, complemented
209#546—1011no—
210#497—1011no—
211#468—1011no—
212#467—1011no—
213#464—1011no—
214#461—1011no—
215#452—1011no—
216#451—1011no—
217#450—1011no—
218#448—1011no—
219#440—1011no—
220#435—1011no—
221C2×#41—1011no—
222C22×Q8C2×D10111yescharacteristic
223C22×Q8C2×D10111yescharacteristic, complemented
224C22×D4—1011no—
225#137—2024no—
226C2×Q16—1011no—
227C2×Q16—1011no—
228C2×Q16—1011no—
229C2×SD16—1011no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×SD16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—2024no—
236C2×M16—1011no—
237C22×C8—1011no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D10111yescharacteristic, complemented
240#124—1011no—
241#123—2021no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1011no—
245C22⋊Q8C2×D10111yescharacteristic
246C22⋊Q8C2×D10111yescharacteristic, complemented
247#121—1011no—
248#121—2021no—
249#120—2021no—
250C4×Q8C2×D10111yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1011no—
253C4×D4C2×D10111yescharacteristic
254C2×C22⋊C4—2021no—
255#107—1011no—
256C4⋊C8—1011no—
257#103—1011no—
258#103—1011no—
259#103—1011no—
260#103—1011no—
261#102—1011no—
262#102—1011no—
263C22⋊C8—1011no—
264C22⋊C8—1011no—
265#97—1011no—
266C24—2021no—
267#41—4041no—
268#41—2022no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1011no—
276C2×Q8C22×D10111yescharacteristic
277C2×Q8C22×D10111yescharacteristic
278C2×Q8C22×D10111yescharacteristic
279C2×D4—1011no—
280C2×D4—1011no—
281C2×D4—4041no—
282C2×D4—2022no—
283C2×D4—2021no—
284C2×D4—2021no—
285C2×D4—2021no—
286C2×D4C22⋊D10111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2021no—
290C22×C4—2021no—
291C22×C4C22×D10111yescharacteristic
292C22×C4C22×D10111yescharacteristic
293Q16—4041no—
294Q16—2022no—
295Q16—2022no—
296SD16—1014no—
297SD16—4041no—
298SD16—2022no—
299SD16—2022no—
300SD16—2022no—
301SD16—2022no—
302M16—2022no—
303C2×C8—1011no—
304C2×C8—1011no—
305C2×C8—1011no—
306C2×C8—1011no—
307C2×C8—2022no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—1011no—
314C4⋊C4—2021no—
315C4⋊C4C22⋊D10111yescharacteristic
316C4⋊C4C22×D10111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2021no—
320C22⋊C4—2021no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4C22⋊D10111yescharacteristic
325C22⋊C4C22⋊D10111yescharacteristic
326C42—21no—
327C42C22×D10111yescharacteristic
328C23—8081no—
329C23—4042no—
330C23—4042no—
331C23—2021no—
332C23—2021no—
333C23C2×C22⋊D10111yescharacteristic
334C23C2×C22⋊D10111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—2021no—
343Q8—2021no—
344Q8D4×D10112yes—
345D4—22no—
346D4—8081no—
347D4—1012no—
348D4—4042no—
349D4—4042no—
350D4—2021no—
351D4—2021no—
352D4—2021no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—8081no—
364C2×C4—4042no—
365C2×C4—2021no—
366C2×C4—2021no—
367C2×C4C2×C22⋊D10111yescharacteristic
368C2×C4C2×C22⋊D10111yescharacteristic
369C2×C4C2×C22⋊D10111yescharacteristic
370C2×C4C2×C22⋊D10111yescharacteristic
371C2×C4C23×D10111yescharacteristic
372C8—1012no—
373C8—2021no—
374C8—2021no—
375C8—2021no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—8084no—
382C22—4041no—
383C22—4041no—
384C22—4041no—
385C22—4041no—
386C22—4041no—
387C22—2022no—
388C22#1569228612yescomplemented
389C22C22×C22⋊D10111yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—4041no—
399C4—4041no—
400C4C2×D4×D10111yescharacteristic
401C4#1569230011yescharacteristic
402C2—22no—
403C2—41no—
404C2—8081no—
405C2—4042no—
406C2D4×C22⋊D10111yescharacteristic
407C2C2×#1569228611yescharacteristic
408C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingD4×Q8×C101C2
socleC22×C101C22×D4
prime radical#20265823C2
hypercenterD4×Q8D101
hypocenterC101#3266
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15692286C2semidirect
2. C22×#13108061C2semidirect
4. #20265569C2semidirect
5. #20265425C2semidirect
6. #20265416C2semidirect
7. #20265399C2semidirect
8. #20265379C2semidirect
9. #20265337C2semidirect
10. #20265333C2semidirect
11. #20265327C2semidirect
12. #20265316C2semidirect
13. #20265275C2semidirect
14. #20265249C2semidirect
15. D4×Q8×C101C2semidirect
19. C2×#13108064C22semidirect
20. C2×#13108064C22semidirect
21. C2×#13108064C22semidirect
29. C2×#13108058C22semidirect
39. #15692188C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692179C22semidirect
42. #15692179C22semidirect
43. #15692179C22semidirect
44. #15692178C22semidirect
48. C22⋊#11717579C22semidirect
49. #15692166C22semidirect
51. C22×Q8×C101C22semidirect
53. C4⋊Q8×C101C22semidirect
57. C22⋊Q8×C101C22semidirect
58. C4×Q8×C101C22semidirect
82. #13108061D4semidirect
176. C22×C101#137semidirect
205. C101#3266semidirect
208. D4×Q8D101semidirect
223. C22×Q8C2×D101semidirect
239. C4⋊Q8C2×D101semidirect
246. C22⋊Q8C2×D101semidirect
250. C4×Q8C2×D101semidirect
388. C22#15692286semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
C22×#131080612
#202658232
#202655692
#202654252
#202654162
#202653992
#202653792
#202653372
#202653332
#202653272
#202653162
#202652752
#202652492
D4×Q8×C1012
#3266101