Finite Group Databasebeta

← #27180757

Subgroups — #27180757

Order 12928.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692284C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265687C211yesmaximal, characteristic
4C22⋊Q8×D101C211yesmaximal, characteristic, complemented
5#20265435C211yesmaximal, characteristic, complemented
6#20265433C211yesmaximal, characteristic, complemented
7#20265421C211yesmaximal, characteristic, complemented
8#20265419C211yesmaximal, characteristic, complemented
9#20265417C211yesmaximal, characteristic, complemented
10#20265405C211yesmaximal, characteristic, complemented
11#20265403C211yesmaximal, characteristic, complemented
12#20265218C211yesmaximal, characteristic, complemented
13#20265184C211yesmaximal, characteristic, complemented
14#20265175C211yesmaximal, characteristic, complemented
15#468×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19#15692284—24no—
20C2×Dic404C2211yescharacteristic, complemented
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108062C2211yescharacteristic, complemented
23C2×#13108058C2211yescharacteristic
24C2×#13108057C2211yescharacteristic
25C2×#13108057C2211yescharacteristic
26M16×D101—24no—
27C2×C8×D101C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic
29#15692237—21no—
30#15692235C2211yescharacteristic
31#15692235C2211yescharacteristic
32#15692234C2211yescharacteristic
33#15692234C2211yescharacteristic
34#15692233—21no—
35#15692232—21no—
36#15692231C2211yescharacteristic
37#15692226C2211yescharacteristic
38#15692223—21no—
39D4×Dic101—21no—
40#15692215—21no—
41#15692213C2211yescharacteristic
42C4⋊C4×D101—21no—
43C4⋊C4×D101C2211yescharacteristic
44C22⋊C4×D101—21no—
45#15692188C2211yescharacteristic, complemented
46#15692187C2211yescharacteristic, complemented
47#15692186C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692180C2211yescharacteristic, complemented
50#15692179C2211yescharacteristic, complemented
51#15692178C2211yescharacteristic, complemented
52#15692178C2211yescharacteristic, complemented
53#15692176C2211yescharacteristic, complemented
54C2×SD16×C101C2211yescharacteristic, complemented
55C2×M16×C101C2211yescharacteristic
56C22⋊Q8×C101C2211yescharacteristic
57#121×C101C2211yescharacteristic
58#107×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#102×C101C2211yescharacteristic, complemented
61C23×D101D411yescharacteristic
62#13108073—22no—
63#13108073—22no—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101—22no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C22×Dic101—21no—
72C22×Dic101D411yescharacteristic
73C2×C4×D101—22no—
74C2×C4×D101—22no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101D411yescharacteristic
78C2×C4×D101D411yescharacteristic
79C2×C4×D101C2311yescharacteristic
80C2×C4×D101C2311yescharacteristic
81Dic404—22no—
82#13108064—22no—
83#13108062—22no—
84#13108058—22no—
85#13108057—22no—
86#13108057D414yescomplemented
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C8×D101—22no—
90C8×D101—22no—
91C4⋊Dic101—21no—
92C4⋊Dic101—21no—
93C4⋊Dic101C2311yescharacteristic
94C4⋊Dic101C2311yescharacteristic
95#13108053—21no—
96#13108053—21no—
97#13108053—21no—
98#13108053—21no—
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106C4×Dic101—21no—
107C2×Q8×C101C2311yescharacteristic
108C2×D4×C101—21no—
109C2×D4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111SD16×C101—22no—
112M16×C101—22no—
113C2×C8×C101C2311yescharacteristic
114C2×C8×C101C2311yescharacteristic
115C4⋊C4×C101—21no—
116C4⋊C4×C101C2311yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
118C22⋊C4×C101—21no—
119C22⋊C4×C101—21no—
120C23×C101—21no—
121C23×C101C2×D411yescharacteristic
122C22×D101—22no—
123C22×D101—22no—
124C22×D101—41no—
125C22×D101C2×D411yescharacteristic
126C22×D101C2×D411yescharacteristic
127Q8×C101—21no—
128Q8×C101—21no—
129D4×C101—21no—
130D4×C101—21no—
131D4×C101—42no—
132C2×C4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101C2×D411yescharacteristic
138C2×C4×C101C2411yescharacteristic
139C22⋊D101—42no—
140C22⋊D101—42no—
141C2×Dic101—22no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—42no—
147C2×Dic101C2×D411yescharacteristic
148C2×Dic101C2×D411yescharacteristic
149C4×D101—22no—
150C4×D101—21no—
151C4×D101—21no—
152C4×D101—22no—
153C4×D101—42no—
154C4×D101—42no—
155C4×D101C2×D412yes—
156C4×D101#4112yes—
157Dic202—21no—
158Dic202—21no—
159Dic202—21no—
160Dic202—21no—
161Dic202—42no—
162C8×C101—21no—
163C8×C101C2×D412yes—
164#11717579—21no—
165#11717579C2×D412yes—
166C22×C101—21no—
167C22×C101—21no—
168C22×C101—21no—
169C22×C101—41no—
170C22×C101—41no—
171C22×C101—41no—
172C22×C101C22×D411yescharacteristic
173C2×D101—21no—
174C2×D101—21no—
175C2×D101—21no—
176C2×D101—21no—
177C2×D101—44no—
178C2×D101#12112yes—
179C2×D101#13712yescomplemented
180C4×C101—21no—
181C4×C101—41no—
182C4×C101—41no—
183C4×C101—41no—
184C4×C101C22×D411yescharacteristic
185C4×C101C2×#4111yescharacteristic
186Dic101—21no—
187Dic101—41no—
188Dic101—41no—
189Dic101—41no—
190Dic101—41no—
191Dic101#12112yes—
192C2×C101—41no—
193C2×C101—81no—
194C2×C101C2×#12111yescharacteristic
195C2×C101C2×#13711yescharacteristic
196C2×C101#57511yescharacteristic
197D101—22no—
198D101—22no—
199#3056—1011nomaximal, Sylow
200C101#305611yescharacteristic, complemented, Sylow
201C2×#137—1011no—
202C22×M16—1011no—
203#525—1011no—
204C2×C22⋊Q8—1011no—
205#470—1011no—
206#470—1011no—
207#468—1011no—
208#468D10111yescharacteristic, complemented
209#467—1011no—
210#467—1011no—
211#465—1011no—
212#465—1011no—
213#428—1011no—
214#419—1011no—
215#417—1011no—
216C2×#41—1011no—
217C22×Q8—1011no—
218C23×C4—1011no—
219#137—2024no—
220C2×Q16—1011no—
221C2×Q16—1011no—
222C2×SD16—1011no—
223C2×SD16C2×D10111yescharacteristic, complemented
224C2×M16—1011no—
225C2×M16—1011no—
226C2×M16—1011no—
227C2×M16—2024no—
228C2×M16C2×D10111yescharacteristic
229C22×C8—1011no—
230C22×C8—1011no—
231#123—2021no—
232C22⋊Q8—1011no—
233C22⋊Q8—1011no—
234C22⋊Q8—1011no—
235C22⋊Q8—1011no—
236C22⋊Q8—1011no—
237C22⋊Q8—2021no—
238C22⋊Q8—2021no—
239C22⋊Q8C2×D10111yescharacteristic
240#121—1011no—
241#121C2×D10111yescharacteristic
242#120—2021no—
243C4×D4—2021no—
244C4×D4—2021no—
245#117—1011no—
246C2×C4⋊C4—1011no—
247C2×C4⋊C4—2021no—
248C2×C22⋊C4—2021no—
249#107—1011no—
250#107C2×D10111yescharacteristic, complemented
251#106—1011no—
252#106—1011no—
253#103—1011no—
254#103—1011no—
255#103—1011no—
256#103—1011no—
257#103—1011no—
258#103C2×D10111yescharacteristic, complemented
259#102—1011no—
260#102C2×D10111yescharacteristic, complemented
261C24—1011no—
262#41—2022no—
263#41—2022no—
264C2×Q8—1011no—
265C2×Q8—1011no—
266C2×Q8—2021no—
267C2×Q8—2022no—
268C2×Q8—2022no—
269C2×Q8C22×D10111yescharacteristic
270C2×D4—21no—
271C2×D4—2021no—
272C2×D4—2021no—
273C2×D4C22×D10111yescharacteristic
274C22×C4—1011no—
275C22×C4—1011no—
276C22×C4—1011no—
277C22×C4—1011no—
278C22×C4—1011no—
279C22×C4—2022no—
280C22×C4—2022no—
281C22×C4—2021no—
282C22×C4—2021no—
283C22×C4—2021no—
284C22×C4C22×D10111yescharacteristic
285Q16—2022no—
286Q16—2022no—
287SD16—22no—
288SD16—2022no—
289M16—22no—
290M16—1014no—
291M16—2022no—
292M16—2022no—
293C2×C8—1011no—
294C2×C8—1011no—
295C2×C8—2022no—
296C2×C8—2022no—
297C2×C8C22×D10111yescharacteristic
298C2×C8C22×D10111yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—1011no—
301C4⋊C4—1011no—
302C4⋊C4—2021no—
303C4⋊C4—2021no—
304C4⋊C4—2021no—
305C4⋊C4—2021no—
306C4⋊C4—2021no—
307C4⋊C4—2021no—
308C4⋊C4C22×D10111yescharacteristic
309C4⋊C4C22×D10111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2021no—
313C22⋊C4—2021no—
314C22⋊C4—2021no—
315C22⋊C4—2021no—
316C22⋊C4—2021no—
317C22⋊C4—2021no—
318C22⋊C4—2021no—
319C42—2021no—
320C23—21no—
321C23—1011no—
322C23—1011no—
323C23—4041no—
324C23—2022no—
325C23—2022no—
326C23D4×D10111yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—4042no—
330Q8—2021no—
331Q8—2021no—
332Q8—2021no—
333Q8—2021no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—4042no—
338D4—4042no—
339C2×C4—21no—
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—4042no—
349C2×C4—4042no—
350C2×C4—4042no—
351C2×C4—2022no—
352C2×C4—2022no—
353C2×C4—2022no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4—2021no—
358C2×C4—2021no—
359C2×C4—2021no—
360C2×C4D4×D10111yescharacteristic
361C2×C4C23×D10111yescharacteristic
362C8—21no—
363C8—1012no—
364C8—2021no—
365C8D4×D10112yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—1012no—
373C22—1012no—
374C22—4044no—
375C22—2021no—
376C22—2021no—
377C22—2021no—
378C22—2021no—
379C22C2×D4×D10111yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—1012no—
385C4—4041no—
386C4—4041no—
387C4—4041no—
388C4—4041no—
389C4—2021no—
390C4C2×D4×D10111yescharacteristic
391C4#41×D10111yescharacteristic
392C2—41no—
393C2—81no—
394C2—2022no—
395C2—2022no—
396C2#121×D10111yescharacteristic
397C2#137×D10111yescharacteristic
398C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#468×C101C2
socleC22×C101C22×D4
prime radical#20265687C2
hypercenter#468D101
hypocenterC101#3056
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#15692284C2semidirect
2. C2×M16×D101C2semidirect
4. C22⋊Q8×D101C2semidirect
5. #20265435C2semidirect
6. #20265433C2semidirect
7. #20265421C2semidirect
8. #20265419C2semidirect
9. #20265417C2semidirect
10. #20265405C2semidirect
11. #20265403C2semidirect
12. #20265218C2semidirect
13. #20265184C2semidirect
14. #20265175C2semidirect
15. #468×C101C2semidirect
20. C2×Dic404C22semidirect
21. C2×#13108064C22semidirect
22. C2×#13108062C22semidirect
45. #15692188C22semidirect
46. #15692187C22semidirect
47. #15692186C22semidirect
48. #15692180C22semidirect
49. #15692180C22semidirect
50. #15692179C22semidirect
51. #15692178C22semidirect
52. #15692178C22semidirect
53. #15692176C22semidirect
54. C2×SD16×C101C22semidirect
58. #107×C101C22semidirect
59. #103×C101C22semidirect
60. #102×C101C22semidirect
86. #13108057D4semidirect
179. C2×D101#137semidirect
200. C101#3056semidirect
208. #468D101semidirect
223. C2×SD16C2×D101semidirect
250. #107C2×D101semidirect
258. #103C2×D101semidirect
260. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
C2×M16×D1012
#202656872
C22⋊Q8×D1012
#202654352
#202654332
#202654212
#202654192
#202654172
#202654052
#202654032
#202652182
#202651842
#202651752
#468×C1012
#3056101