Finite Group Databasebeta

← #27182837

Subgroups — #27182837

Order 12928.

504 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×#15692275C211yesmaximal, characteristic, complemented
2C2×SD16×D101C211yesmaximal, characteristic, complemented
3#20265837C211yesmaximal, characteristic, complemented
4#20265815C211yesmaximal, characteristic, complemented
5#20265562C211yesmaximal, characteristic, complemented
6#20265409C211yesmaximal, characteristic, complemented
7#20265408C211yesmaximal, characteristic, complemented
8#20265402C211yesmaximal, characteristic, complemented
9#20265394C211yesmaximal, characteristic, complemented
10#20265331C211yesmaximal, characteristic, complemented
11#20265324C211yesmaximal, characteristic, complemented
12#20265312C211yesmaximal, characteristic, complemented
13#20265268C211yesmaximal, characteristic, complemented
14#20265240C211yesmaximal, characteristic, complemented
15#452×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C2×Q8×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692275—24no—
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108063C2211yescharacteristic
22C2×#13108062C2211yescharacteristic
23C2×#13108061C2211yescharacteristic
24C2×#13108061C2211yescharacteristic, complemented
25SD16×D101—24no—
26C2×D808C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C22×#11717579C2211yescharacteristic, complemented
29C2×C8×D101C2211yescharacteristic
30#15692244C2211yescharacteristic
31#15692244C2211yescharacteristic, complemented
32#15692241—21no—
33#15692238—21no—
34#15692233—21no—
35#15692231C2211yescharacteristic
36#15692230C2211yescharacteristic
37#15692229C2211yescharacteristic
38#15692228—21no—
39#15692227—21no—
40#15692223—21no—
41#15692220C2211yescharacteristic, complemented
42D4×Dic101C2211yescharacteristic
43#15692217—21no—
44C4×C22⋊D101C2211yescharacteristic
45C2×#13108051—21no—
46#15692186C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692179C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692175C2211yescharacteristic
52#15692167C2211yescharacteristic
53C8×Dic101C2211yescharacteristic, complemented
54C2×#41×C101C2211yescharacteristic
55C2×Q16×C101C2211yescharacteristic, complemented
56C2×SD16×C101C2211yescharacteristic
57C22⋊Q8×C101C2211yescharacteristic, complemented
58#103×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66Q8×D101—22no—
67Q8×D101—41no—
68C2×D404—21no—
69C2×D404C2311yescharacteristic
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101D411yescharacteristic
75D4×D101—22no—
76D4×D101—41no—
77C22×Dic101—21no—
78C2×C4×D101—21no—
79C2×C4×D101—21no—
80C2×C4×D101C2311yescharacteristic
81#13108064—22no—
82#13108063—22no—
83#13108062D414yescomplemented
84#13108061—22no—
85#13108061—22no—
86D808—41no—
87#13108059—22no—
88C2×#11717579—22no—
89C2×#11717579C2311yescharacteristic
90C2×#11717579C2311yescharacteristic
91C8×D101—22no—
92C4⋊Dic101C2311yescharacteristic
93#13108053—21no—
94#13108053—21no—
95#13108053D411yescharacteristic
96C22⋊Dic101—21no—
97C22⋊Dic101D411yescharacteristic
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051D411yescharacteristic
106C4×Dic101—21no—
107C4×Dic101C2311yescharacteristic
108#41×C101—22no—
109#41×C101—41no—
110C2×Q8×C101C2311yescharacteristic
111C2×Q8×C101C2311yescharacteristic
112C2×D4×C101—21no—
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116Q16×C101—41no—
117SD16×C101—22no—
118C2×C8×C101C2311yescharacteristic
119C2×C8×C101C2311yescharacteristic
120C4⋊C4×C101—21no—
121C4⋊C4×C101C2311yescharacteristic
122C22⋊C4×C101—21no—
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—42no—
128C22×D101—42no—
129C22×D101—81no—
130C22×D101C2×D411yescharacteristic
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—21no—
134Q8×C101—21no—
135D4×C101—21no—
136D4×C101—42no—
137D4×C101C2×D412yes—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—42no—
144C2×C4×C101C2×D411yescharacteristic
145C2×C4×C101C2411yescharacteristic
146C22⋊D101—22no—
147C22⋊D101—42no—
148C22⋊D101—42no—
149C22⋊D101—81no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101—81no—
153C2×Dic101C2×D411yescharacteristic
154C2×Dic101C2×D411yescharacteristic
155C2×Dic101C2×D411yescharacteristic
156D404—21no—
157D404—21no—
158D404—21no—
159D404—21no—
160D404—42no—
161C4×D101—21no—
162C4×D101—21no—
163C4×D101—42no—
164C4×D101—42no—
165Dic202—21no—
166Dic202—42no—
167Dic202C2×D412yes—
168C8×C101—21no—
169C8×C101—21no—
170#11717579—21no—
171#11717579C2×D412yes—
172C22×C101—21no—
173C22×C101—22no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101C22×D411yescharacteristic
179C2×D101—21no—
180C2×D101—21no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—41no—
184C2×D101—41no—
185C2×D101—41no—
186C2×D101—41no—
187C2×D101—84no—
188C2×D101#13512yescomplemented
189C4×C101—21no—
190C4×C101—41no—
191C4×C101—41no—
192C4×C101—41no—
193C4×C101C22×D411yescharacteristic
194C4×C101H(5,𝔽2)11yescharacteristic
195Dic101—22no—
196Dic101—21no—
197Dic101—21no—
198Dic101—21no—
199Dic101—41no—
200C2×C101—42no—
201C2×C101—41no—
202C2×C101D4211yescharacteristic
203C2×C101C2×#13511yescharacteristic
204C2×C101#57711yescharacteristic
205D101—22no—
206D101—81no—
207D101—81no—
208#3282—1011nomaximal, Sylow
209C101#328211yescharacteristic, complemented, Sylow
210C2×#135—1011no—
211C22×SD16—1011no—
212#548—1011no—
213#546—1011no—
214#495—1011no—
215#465—1011no—
216#465—1011no—
217#464—1011no—
218#463—1011no—
219#452D10111yescharacteristic, complemented
220#451—1011no—
221#449—1011no—
222#447—1011no—
223C4×SD16—1011no—
224C8×D4—1011no—
225C2×#41—1011no—
226C2×#41C2×D10111yescharacteristic
227C22×Q8—1011no—
228C22×D4—1011no—
229#135—2024no—
230C2×Q16—1011no—
231C2×Q16C2×D10111yescharacteristic, complemented
232C2×SD16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16—2024no—
237C2×SD16C2×D10111yescharacteristic
238C2×D8—1011no—
239C2×D8—1011no—
240C22×C8—1011no—
241C22×C8—1011no—
242#124—1011no—
243#124—1011no—
244#124—2021no—
245#123—2021no—
246C22⋊Q8—1011no—
247C22⋊Q8—2021no—
248C22⋊Q8C2×D10111yescharacteristic, complemented
249#121—1011no—
250#121—1011no—
251#121—2021no—
252#121—2021no—
253#120—2021no—
254C4×Q8—1011no—
255C4×D4—1011no—
256C4×D4—1011no—
257C4×D4—2021no—
258C2×C22⋊C4—2021no—
259#106—1011no—
260C4⋊C8—1011no—
261#103—1011no—
262#103—1011no—
263#103C2×D10111yescharacteristic, complemented
264#102—1011no—
265#102—1011no—
266#102C2×D10111yescharacteristic
267C22⋊C8—1011no—
268C22⋊C8C2×D10111yescharacteristic, complemented
269C4×C8—1011no—
270C24—2021no—
271#41—22no—
272#41—41no—
273#41—4041no—
274#41—2022no—
275C2×Q8—1011no—
276C2×Q8—4041no—
277C2×Q8—2021no—
278C2×Q8—2022no—
279C2×Q8C22×D10111yescharacteristic
280C2×Q8C22×D10111yescharacteristic
281C2×D4—21no—
282C2×D4—1011no—
283C2×D4—1011no—
284C2×D4—1011no—
285C2×D4—4041no—
286C2×D4—2021no—
287C2×D4—2021no—
288C2×D4—2021no—
289C2×D4—2021no—
290C2×D4—2022no—
291C2×D4C22×D10111yescharacteristic
292C22×C4—21no—
293C22×C4—1011no—
294C22×C4—2021no—
295C22×C4—2021no—
296C22×C4—2021no—
297C22×C4C22×D10111yescharacteristic
298Q16—41no—
299Q16—2022no—
300SD16—22no—
301SD16—1014no—
302SD16—2022no—
303SD16—2022no—
304SD16—2022no—
305D8—4041no—
306D8—2022no—
307C2×C8—1011no—
308C2×C8—1011no—
309C2×C8—2022no—
310C2×C8—2022no—
311C2×C8C22×D10111yescharacteristic
312C2×C8C22×D10111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1011no—
315C4⋊C4—1011no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4C22×D10111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—1011no—
321C22⋊C4—1011no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C22⋊C4—2021no—
330C42—1011no—
331C42—2021no—
332C23—21no—
333C23—8081no—
334C23—1011no—
335C23—4042no—
336C23—4042no—
337C23—2021no—
338C23—2021no—
339C23D4×D10111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—1012no—
345Q8—4042no—
346Q8—2021no—
347D4—21no—
348D4—42no—
349D4—8081no—
350D4—4042no—
351D4—4042no—
352D4—4042no—
353D4—2022no—
354D4—2021no—
355D4—2021no—
356D4—2021no—
357D4—2021no—
358D4D4×D10112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—8081no—
366C2×C4—1011no—
367C2×C4—1011no—
368C2×C4—1011no—
369C2×C4—4042no—
370C2×C4—4042no—
371C2×C4—2021no—
372C2×C4—2021no—
373C2×C4—2021no—
374C2×C4—2021no—
375C2×C4D4×D10111yescharacteristic
376C2×C4C23×D10111yescharacteristic
377C8—21no—
378C8—21no—
379C8—1012no—
380C8—2021no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—8084no—
388C22—1012no—
389C22—4041no—
390C22—4041no—
391C22—4041no—
392C22—4041no—
393C22—4041no—
394C22—4041no—
395C22—2021no—
396C22—2021no—
397C22C2×D4×D10111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—4041no—
403C4—2021no—
404C4—2021no—
405C4—2022no—
406C4—2021no—
407C4C2×D4×D10111yescharacteristic
408C4#1569230011yescharacteristic
409C2—41no—
410C2—42no—
411C2—8081no—
412C2—8081no—
413C2—2022no—
414C2#2026580911yescharacteristic
415C2#135×D10111yescharacteristic
416C2#2026598611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#452×C101C2
socleC22×C101C22×D4
hypercenter#452D101
hypocenterC101#3282
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#15692275C2semidirect
2. C2×SD16×D101C2semidirect
3. #20265837C2semidirect
4. #20265815C2semidirect
5. #20265562C2semidirect
6. #20265409C2semidirect
7. #20265408C2semidirect
8. #20265402C2semidirect
9. #20265394C2semidirect
10. #20265331C2semidirect
11. #20265324C2semidirect
12. #20265312C2semidirect
13. #20265268C2semidirect
14. #20265240C2semidirect
15. #452×C101C2semidirect
16. C2×#13108074C22semidirect
20. C2×#13108064C22semidirect
24. C2×#13108061C22semidirect
28. C22×#11717579C22semidirect
31. #15692244C22semidirect
41. #15692220C22semidirect
46. #15692186C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692179C22semidirect
50. #15692177C22semidirect
53. C8×Dic101C22semidirect
55. C2×Q16×C101C22semidirect
57. C22⋊Q8×C101C22semidirect
58. #103×C101C22semidirect
60. C22⋊C8×C101C22semidirect
83. #13108062D4semidirect
188. C2×D101#135semidirect
209. C101#3282semidirect
219. #452D101semidirect
231. C2×Q16C2×D101semidirect
248. C22⋊Q8C2×D101semidirect
263. #103C2×D101semidirect
268. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922752
C2×SD16×D1012
#202658372
#202658152
#202655622
#202654092
#202654082
#202654022
#202653942
#202653312
#202653242
#202653122
#202652682
#202652402
#452×C1012
#3282101