Finite Group Databasebeta

← #27179832

Subgroups — #27179832

Order 12928.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692263C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265686C211yesmaximal, characteristic
4#20265256C211yesmaximal, characteristic, complemented
5#20265253C211yesmaximal, characteristic, complemented
6#20265253C211yesmaximal, characteristic, complemented
7#20265251C211yesmaximal, characteristic, complemented
8#20265249C211yesmaximal, characteristic, complemented
9#20265247C211yesmaximal, characteristic, complemented
10#20265245C211yesmaximal, characteristic, complemented
11#20265202C211yesmaximal, characteristic, complemented
12#20265142C211yesmaximal, characteristic, complemented
13#20265139C211yesmaximal, characteristic, complemented
14#20265131C211yesmaximal, characteristic, complemented
15#436×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18#15692263—24no—
19C2×#13108058C2211yescharacteristic
20C2×#13108057C2211yescharacteristic
21C2×#13108057C2211yescharacteristic
22C2×#13108057C2211yescharacteristic
23C2×#13108057C2211yescharacteristic
24C2×#13108057C2211yescharacteristic, complemented
25M16×D101—24no—
26C22×#11717579C2211yescharacteristic, complemented
27C2×C8×D101C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic
29C2×C8×D101C2211yescharacteristic, complemented
30#15692240C411yescharacteristic
31#15692237C411yescharacteristic
32#15692233C411yescharacteristic
33#15692232C411yescharacteristic
34#15692228C411yescharacteristic
35#15692227C411yescharacteristic
36#15692225C411yescharacteristic
37#15692223C411yescharacteristic
38C4×D404C2211yescharacteristic
39C4×C22⋊D101C2211yescharacteristic
40C4×C22⋊D101C2211yescharacteristic
41#15692212C2211yescharacteristic
42C4⋊#11717579C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692184C2211yescharacteristic, complemented
45C22⋊#11717579C2211yescharacteristic
46C22⋊#11717579C2211yescharacteristic, complemented
47#15692167C2211yescharacteristic
48#15692167C2211yescharacteristic
49#15692167C2211yescharacteristic, complemented
50#15692167C2211yescharacteristic, complemented
51#15692166C2211yescharacteristic, complemented
52#15692165C2211yescharacteristic, complemented
53C8×Dic101C2211yescharacteristic, complemented
54C2×M16×C101C2211yescharacteristic
55C2×M16×C101C2211yescharacteristic, complemented
56C4×D4×C101C2211yescharacteristic
57C4⋊C8×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic
59C22⋊C8×C101C2211yescharacteristic, complemented
60C4×C8×C101C2211yescharacteristic, complemented
61C23×D101C2×C411yescharacteristic
62#13108074—22no—
63#13108074—22no—
64C2×Dic202C2×C411yescharacteristic
65C2×D404C2×C411yescharacteristic
66C2×D404C2×C411yescharacteristic
67C2×C22⋊D101C2×C411yescharacteristic
68C2×C22⋊D101C2×C411yescharacteristic
69C2×C22⋊D101C2×C411yescharacteristic
70C2×C22⋊D101C2×C411yescharacteristic
71C22×Dic101C2×C411yescharacteristic
72C2×C4×D101—22no—
73C2×C4×D101—22no—
74C2×C4×D101C2×C411yescharacteristic
75C2×C4×D101C2×C411yescharacteristic
76C2×C4×D101C2311yescharacteristic
77C2×C4×D101C2311yescharacteristic
78C2×C4×D101C2311yescharacteristic
79#13108058—22no—
80#13108057—22no—
81#13108057—22no—
82#13108057—22no—
83#13108057—22no—
84#13108057—22no—
85C2×#11717579—22no—
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C2×#11717579C2311yescharacteristic
90C8×D101—22no—
91C8×D101—22no—
92C8×D101D414yescomplemented
93C4⋊Dic101C2×C411yescharacteristic
94#13108053C2×C411yescharacteristic
95#13108053C2×C411yescharacteristic
96#13108053C2×C411yescharacteristic
97#13108053C2×C411yescharacteristic
98C22⋊Dic101C2×C411yescharacteristic
99C22⋊Dic101C2×C411yescharacteristic
100#13108051C2×C411yescharacteristic
101#13108051C2×C411yescharacteristic
102#13108051C2×C411yescharacteristic
103#13108051C2×C411yescharacteristic
104#13108051C2×C411yescharacteristic
105#13108051C2×C411yescharacteristic
106C4×Dic101C2311yescharacteristic
107C2×D4×C101C2×C411yescharacteristic
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110M16×C101—22no—
111M16×C101—22no—
112C2×C8×C101C2311yescharacteristic
113C2×C8×C101C2311yescharacteristic
114C2×C8×C101C2311yescharacteristic
115C2×C8×C101C2311yescharacteristic
116C4⋊C4×C101C2×C411yescharacteristic
117C22⋊C4×C101C2×C411yescharacteristic
118C22⋊C4×C101C2×C411yescharacteristic
119C42×C101C2311yescharacteristic
120C23×C101C22×C411yescharacteristic
121C23×C101C22×C411yescharacteristic
122C22×D101—22no—
123C22×D101—41no—
124C22×D101—41no—
125C22×D101C22×C411yescharacteristic
126C22×D101C22×C411yescharacteristic
127C22×D101C22×C411yescharacteristic
128D4×C101—41no—
129C2×C4×C101—21no—
130C2×C4×C101—21no—
131C2×C4×C101—21no—
132C2×C4×C101—21no—
133C2×C4×C101C22×C411yescharacteristic
134C2×C4×C101C22×C411yescharacteristic
135C2×C4×C101C22×C411yescharacteristic
136C2×C4×C101C22×C411yescharacteristic
137C2×C4×C101C2411yescharacteristic
138C22⋊D101—22no—
139C22⋊D101—22no—
140C22⋊D101—41no—
141C22⋊D101—41no—
142C2×Dic101—22no—
143C2×Dic101C22×C411yescharacteristic
144C2×Dic101C22×C411yescharacteristic
145C2×Dic101C22×C411yescharacteristic
146C2×Dic101C22×C411yescharacteristic
147C2×Dic101C22×C411yescharacteristic
148D404—22no—
149D404—41no—
150C4×D101—22no—
151C4×D101—21no—
152C4×D101—21no—
153C4×D101—21no—
154C4×D101—21no—
155C4×D101—22no—
156C4×D101C2×D412yes—
157C4×D101#4112yes—
158Dic202—22no—
159C8×C101—21no—
160C8×C101—21no—
161C8×C101—21no—
162C8×C101C2×D412yes—
163#11717579—21no—
164#11717579—21no—
165#11717579—21no—
166#11717579C2×D412yes—
167C22×C101—21no—
168C22×C101—21no—
169C22×C101—21no—
170C22×C101—21no—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101C23×C411yescharacteristic
174C2×D101—21no—
175C2×D101—21no—
176C2×D101—21no—
177C2×D101—21no—
178C2×D101—21no—
179C2×D101—21no—
180C2×D101—21no—
181C2×D101—21no—
182C2×D101—44no—
183C2×D101C4×D412yes—
184C4×C101—21no—
185C4×C101—21no—
186C4×C101—21no—
187C4×C101—21no—
188C4×C101C22×D411yescharacteristic
189C4×C101C2×#4111yescharacteristic
190Dic101—21no—
191Dic101—21no—
192Dic101—21no—
193Dic101—21no—
194Dic101C4×D412yes—
195C2×C101—41no—
196C2×C101—41no—
197C2×C101C2×C4×D411yescharacteristic
198C2×C101#56811yescharacteristic
199C2×C101#56811yescharacteristic
200D101—22no—
201D101—41no—
202D101—41no—
203#2930—1011nomaximal, Sylow
204C101#293011yescharacteristic, complemented, Sylow
205C2×#131—1011no—
206C22×M16—1011no—
207#525—1011no—
208#436—1011no—
209#436—1011no—
210#436—1011no—
211#436D10111yescharacteristic, complemented
212#435—1011no—
213#435—1011no—
214#435—1011no—
215#435—1011no—
216#424—1011no—
217#408—1011no—
218#407—1011no—
219#405—1011no—
220C2×#41—1011no—
221C23×C4—1011no—
222#131—2024no—
223C2×M16—1011no—
224C2×M16—1011no—
225C2×M16—1011no—
226C2×M16—1011no—
227C2×M16—1011no—
228C2×M16—1011no—
229C2×M16—2024no—
230C2×M16C2×D10111yescharacteristic
231C2×M16C2×D10111yescharacteristic, complemented
232C22×C8—1011no—
233C22×C8—1011no—
234C22×C8—1011no—
235C22×C8—1011no—
236#123—1011no—
237#123—1011no—
238C22⋊Q8—1011no—
239C22⋊Q8—1011no—
240#121—1011no—
241#121—1011no—
242#120—1011no—
243#120—1011no—
244C4×D4—1011no—
245C4×D4—1011no—
246C4×D4—1011no—
247C4×D4C2×D10111yescharacteristic
248#117—1011no—
249C4⋊C8—1011no—
250C4⋊C8—1011no—
251C4⋊C8—1011no—
252C4⋊C8C2×D10111yescharacteristic, complemented
253C22⋊C8—1011no—
254C22⋊C8—1011no—
255C22⋊C8—1011no—
256C22⋊C8—1011no—
257C22⋊C8—1011no—
258C22⋊C8—1011no—
259C22⋊C8C2×D10111yescharacteristic
260C22⋊C8C2×D10111yescharacteristic, complemented
261#97—1011no—
262#97—1011no—
263C4×C8—1011no—
264C4×C8C2×D10111yescharacteristic, complemented
265C24—1011no—
266#41—2022no—
267#41—2022no—
268C2×Q8—1011no—
269C2×D4—1011no—
270C2×D4—1011no—
271C2×D4—1011no—
272C2×D4—1011no—
273C2×D4—1011no—
274C2×D4—1011no—
275C2×D4C4×D10111yescharacteristic
276C22×C4—1011no—
277C22×C4—1011no—
278C22×C4—1011no—
279C22×C4—1011no—
280C22×C4—1011no—
281C22×C4—1011no—
282C22×C4—2022no—
283C22×C4—2022no—
284C22×C4C22×D10111yescharacteristic
285C22×C4C22×D10111yescharacteristic
286M16—22no—
287M16—22no—
288M16—2022no—
289M16—2022no—
290M16—2022no—
291M16—2022no—
292M16—2022no—
293M16—2022no—
294C2×C8—1011no—
295C2×C8—1014no—
296C2×C8—1011no—
297C2×C8—1011no—
298C2×C8—1011no—
299C2×C8—2022no—
300C2×C8—2022no—
301C2×C8—2022no—
302C2×C8C22×D10111yescharacteristic
303C2×C8C22×D10111yescharacteristic
304C2×C8C22×D10111yescharacteristic
305C2×C8C22×D10111yescharacteristic
306C4⋊C4—1011no—
307C4⋊C4—1011no—
308C4⋊C4—1011no—
309C4⋊C4—1011no—
310C4⋊C4—1011no—
311C4⋊C4C4×D10111yescharacteristic
312C22⋊C4—1011no—
313C22⋊C4—1011no—
314C22⋊C4—1011no—
315C22⋊C4—1011no—
316C22⋊C4—1011no—
317C22⋊C4—1011no—
318C22⋊C4—1011no—
319C22⋊C4—1011no—
320C22⋊C4C4×D10111yescharacteristic
321C22⋊C4C4×D10111yescharacteristic
322C42—1011no—
323C42C22×D10111yescharacteristic
324C23—1011no—
325C23—1011no—
326C23—1011no—
327C23—4041no—
328C23—4041no—
329C23—2022no—
330C23C2×C4×D10111yescharacteristic
331C23C2×C4×D10111yescharacteristic
332Q8—2022no—
333D4—41no—
334D4—4041no—
335D4—4041no—
336D4—4041no—
337D4—2022no—
338D4—2022no—
339D4—2022no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—1011no—
345C2×C4—1012no—
346C2×C4—1012no—
347C2×C4—1011no—
348C2×C4—1011no—
349C2×C4—1011no—
350C2×C4—1011no—
351C2×C4—2022no—
352C2×C4—2022no—
353C2×C4—2022no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4—2021no—
358C2×C4C2×C4×D10111yescharacteristic
359C2×C4C2×C4×D10111yescharacteristic
360C2×C4C2×C4×D10111yescharacteristic
361C2×C4C2×C4×D10111yescharacteristic
362C2×C4C23×D10111yescharacteristic
363C8—21no—
364C8—21no—
365C8—21no—
366C8—1012no—
367C8—2021no—
368C8—2021no—
369C8—2021no—
370C8D4×D10112yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—1012no—
378C22—4044no—
379C22—2021no—
380C22—2021no—
381C22—2021no—
382C22—2021no—
383C22—2021no—
384C22—2021no—
385C22—2021no—
386C22—2021no—
387C22C22×C4×D10111yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—1012no—
393C4—2021no—
394C4—2021no—
395C4—2021no—
396C4—2021no—
397C4C2×D4×D10111yescharacteristic
398C4#41×D10111yescharacteristic
399C2—41no—
400C2—41no—
401C2—4041no—
402C2—4041no—
403C2—2022no—
404C2C4×D4×D10111yescharacteristic
405C2#2026594211yescharacteristic
406C2#2026594311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D101
commutator, socleC22×C101C23×C4
Fitting#436×C101C2
prime radical#20265686C2
cosocleC2×C4×C101C24
hypercenter#436D101
hypocenterC101#2930
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#15692263C2semidirect
2. C2×M16×D101C2semidirect
4. #20265256C2semidirect
5. #20265253C2semidirect
6. #20265253C2semidirect
7. #20265251C2semidirect
8. #20265249C2semidirect
9. #20265247C2semidirect
10. #20265245C2semidirect
11. #20265202C2semidirect
12. #20265142C2semidirect
13. #20265139C2semidirect
14. #20265131C2semidirect
15. #436×C101C2semidirect
24. C2×#13108057C22semidirect
26. C22×#11717579C22semidirect
29. C2×C8×D101C22semidirect
42. C4⋊#11717579C22semidirect
43. #15692184C22semidirect
44. #15692184C22semidirect
46. C22⋊#11717579C22semidirect
49. #15692167C22semidirect
50. #15692167C22semidirect
51. #15692166C22semidirect
52. #15692165C22semidirect
53. C8×Dic101C22semidirect
55. C2×M16×C101C22semidirect
57. C4⋊C8×C101C22semidirect
59. C22⋊C8×C101C22semidirect
60. C4×C8×C101C22semidirect
92. C8×D101D4semidirect
204. C101#2930semidirect
211. #436D101semidirect
231. C2×M16C2×D101semidirect
252. C4⋊C8C2×D101semidirect
260. C22⋊C8C2×D101semidirect
264. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922632
C2×M16×D1012
#202656862
#202652562
#202652532
#202652512
#202652492
#202652472
#202652452
#202652022
#202651422
#202651392
#202651312
#436×C1012
#2930101