Finite Group Databasebeta

← #27180761

Subgroups — #27180761

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×#15692287C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265690C211yesmaximal, characteristic
4C22⋊Q8×D101C211yesmaximal, characteristic, complemented
5#20265433C211yesmaximal, characteristic, complemented
6#20265425C211yesmaximal, characteristic, complemented
7#20265423C211yesmaximal, characteristic, complemented
8#20265417C211yesmaximal, characteristic, complemented
9#20265415C211yesmaximal, characteristic, complemented
10#20265409C211yesmaximal, characteristic, complemented
11#20265407C211yesmaximal, characteristic, complemented
12#20265216C211yesmaximal, characteristic, complemented
13#20265184C211yesmaximal, characteristic, complemented
14#20265177C211yesmaximal, characteristic, complemented
15#470×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19#15692287—24no—
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108063C2211yescharacteristic, complemented
22C2×#13108061C2211yescharacteristic, complemented
23C2×#13108058C2211yescharacteristic
24C2×#13108057C2211yescharacteristic
25C2×#13108057C2211yescharacteristic
26M16×D101—24no—
27C2×C8×D101C2211yescharacteristic
28C2×C8×D101C2211yescharacteristic
29#15692240—21no—
30#15692235C2211yescharacteristic
31#15692234C2211yescharacteristic
32#15692233—21no—
33#15692232—21no—
34#15692231C2211yescharacteristic
35#15692231C2211yescharacteristic
36#15692230C2211yescharacteristic
37#15692229C2211yescharacteristic
38#15692225—21no—
39#15692217—21no—
40#15692215—21no—
41#15692213C2211yescharacteristic
42C4⋊C4×D101—21no—
43C4⋊C4×D101C2211yescharacteristic
44C22⋊C4×D101—21no—
45#15692189C2211yescharacteristic, complemented
46#15692188C2211yescharacteristic, complemented
47#15692186C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692179C2211yescharacteristic, complemented
50#15692179C2211yescharacteristic, complemented
51#15692178C2211yescharacteristic, complemented
52#15692177C2211yescharacteristic, complemented
53#15692175C2211yescharacteristic, complemented
54C2×Q16×C101C2211yescharacteristic, complemented
55C2×M16×C101C2211yescharacteristic
56C22⋊Q8×C101C2211yescharacteristic
57C22⋊Q8×C101C2211yescharacteristic
58#106×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60#103×C101C2211yescharacteristic, complemented
61C23×D101D411yescharacteristic
62#13108075—22no—
63#13108075—22no—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66Q8×D101—22no—
67Q8×D101—22no—
68C2×D404—21no—
69C2×D404—21no—
70C2×D404C2311yescharacteristic
71C2×C22⋊D101—21no—
72C22×Dic101D411yescharacteristic
73C2×C4×D101—22no—
74C2×C4×D101—22no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101—21no—
78C2×C4×D101D411yescharacteristic
79C2×C4×D101D411yescharacteristic
80C2×C4×D101C2311yescharacteristic
81C2×C4×D101C2311yescharacteristic
82#13108064—22no—
83#13108063—22no—
84#13108061—22no—
85#13108058—22no—
86#13108057—22no—
87#13108057D414yescomplemented
88C2×#11717579C2311yescharacteristic
89C2×#11717579C2311yescharacteristic
90C8×D101—22no—
91C8×D101—22no—
92C4⋊Dic101—21no—
93C4⋊Dic101C2311yescharacteristic
94C4⋊Dic101C2311yescharacteristic
95#13108053—21no—
96#13108053—21no—
97#13108053—21no—
98#13108053—21no—
99C22⋊Dic101—21no—
100#13108051—21no—
101#13108051—21no—
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106C4×Dic101—21no—
107C2×Q8×C101C2311yescharacteristic
108C2×Q8×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110Q16×C101—22no—
111M16×C101—22no—
112C2×C8×C101C2311yescharacteristic
113C2×C8×C101C2311yescharacteristic
114C4⋊C4×C101—21no—
115C4⋊C4×C101—21no—
116C4⋊C4×C101C2311yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
118C22⋊C4×C101—21no—
119C22⋊C4×C101—21no—
120C23×C101C2×D411yescharacteristic
121C22×D101—22no—
122C22×D101—22no—
123C22×D101—21no—
124C22×D101—41no—
125C22×D101C2×D411yescharacteristic
126C22×D101C2×D411yescharacteristic
127Q8×C101—21no—
128Q8×C101—21no—
129Q8×C101—21no—
130Q8×C101—21no—
131C2×C4×C101—21no—
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—
140C2×Dic101—22no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101C2×D411yescharacteristic
145C2×Dic101C2×D411yescharacteristic
146D404—21no—
147D404—21no—
148D404—42no—
149D404—42no—
150C4×D101—22no—
151C4×D101—21no—
152C4×D101—21no—
153C4×D101—22no—
154C4×D101—42no—
155C4×D101—42no—
156C4×D101—42no—
157C4×D101C2×D412yes—
158C4×D101#4112yes—
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×C101C22×D411yescharacteristic
170C2×D101—21no—
171C2×D101—21no—
172C2×D101—21no—
173C2×D101—21no—
174C2×D101—44no—
175C2×D101—41no—
176C2×D101—41no—
177C2×D101—41no—
178C2×D101#12112yes—
179C2×D101#13712yescomplemented
180C4×C101—21no—
181C4×C101—41no—
182C4×C101—41no—
183C4×C101—41no—
184C4×C101—41no—
185C4×C101C22×D411yescharacteristic
186C4×C101C2×#4111yescharacteristic
187Dic101—21no—
188Dic101—41no—
189Dic101—41no—
190Dic101—41no—
191Dic101#12112yes—
192C2×C101—41no—
193C2×C101C2×#12111yescharacteristic
194C2×C101C2×#13711yescharacteristic
195C2×C101#57511yescharacteristic
196D101—22no—
197D101—22no—
198D101—81no—
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#470D10111yescharacteristic, complemented
207#468—1011no—
208#468—1011no—
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×Q16C2×D10111yescharacteristic, complemented
222C2×SD16—1011no—
223C2×SD16—1011no—
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—2021no—
237C22⋊Q8—2021no—
238C22⋊Q8C2×D10111yescharacteristic
239C22⋊Q8C2×D10111yescharacteristic
240#121—1011no—
241#121—1011no—
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#107—1011no—
251#106—1011no—
252#106C2×D10111yescharacteristic, complemented
253#103—1011no—
254#103—1011no—
255#103—1011no—
256#103—1011no—
257#103C2×D10111yescharacteristic, complemented
258#103C2×D10111yescharacteristic, complemented
259#102—1011no—
260#102—1011no—
261C24—1011no—
262#41—2022no—
263#41—2022no—
264C2×Q8—1011no—
265C2×Q8—2021no—
266C2×Q8—2022no—
267C2×Q8—2022no—
268C2×Q8C22×D10111yescharacteristic
269C2×Q8C22×D10111yescharacteristic
270C2×D4—1011no—
271C2×D4—2021no—
272C2×D4—2021no—
273C2×D4—2021no—
274C22×C4—1011no—
275C22×C4—1011no—
276C22×C4—1011no—
277C22×C4—1011no—
278C22×C4—1011no—
279C22×C4—2021no—
280C22×C4—2022no—
281C22×C4—2022no—
282C22×C4—2021no—
283C22×C4—2021no—
284C22×C4C22×D10111yescharacteristic
285Q16—22no—
286Q16—2022no—
287SD16—2022no—
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—21no—
301C4⋊C4—1011no—
302C4⋊C4—1011no—
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—1011no—
321C23—1011no—
322C23—4041no—
323C23—2022no—
324C23—2022no—
325C23—2021no—
326C23D4×D10111yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—4042no—
332Q8—2021no—
333Q8—2021no—
334D4—4042no—
335D4—4042no—
336D4—4042no—
337D4—2021no—
338D4—2021no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—1011no—
346C2×C4—1012no—
347C2×C4—1012no—
348C2×C4—1011no—
349C2×C4—4042no—
350C2×C4—4042no—
351C2×C4—4042no—
352C2×C4—2022no—
353C2×C4—2022no—
354C2×C4—2022no—
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—1012no—
370C22—1012no—
371C22—4044no—
372C22—4041no—
373C22—4041no—
374C22—4041no—
375C22—2021no—
376C22—2021no—
377C22—2021no—
378C22—2021no—
379C22C2×D4×D10111yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—1012no—
386C4—4041no—
387C4—4041no—
388C4—4041no—
389C4—2021no—
390C4C2×D4×D10111yescharacteristic
391C4#41×D10111yescharacteristic
392C2—41no—
393C2—8081no—
394C2—2022no—
395C2—2022no—
396C2#121×D10111yescharacteristic
397C2#137×D10111yescharacteristic
398C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#470×C101C2
socleC22×C101C22×D4
prime radical#20265690C2
hypercenter#470D101
hypocenterC101#3056
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#15692287C2semidirect
2. C2×M16×D101C2semidirect
4. C22⋊Q8×D101C2semidirect
5. #20265433C2semidirect
6. #20265425C2semidirect
7. #20265423C2semidirect
8. #20265417C2semidirect
9. #20265415C2semidirect
10. #20265409C2semidirect
11. #20265407C2semidirect
12. #20265216C2semidirect
13. #20265184C2semidirect
14. #20265177C2semidirect
15. #470×C101C2semidirect
20. C2×#13108064C22semidirect
21. C2×#13108063C22semidirect
22. C2×#13108061C22semidirect
45. #15692189C22semidirect
46. #15692188C22semidirect
47. #15692186C22semidirect
48. #15692180C22semidirect
49. #15692179C22semidirect
50. #15692179C22semidirect
51. #15692178C22semidirect
52. #15692177C22semidirect
53. #15692175C22semidirect
54. C2×Q16×C101C22semidirect
58. #106×C101C22semidirect
59. #103×C101C22semidirect
60. #103×C101C22semidirect
87. #13108057D4semidirect
179. C2×D101#137semidirect
200. C101#3056semidirect
206. #470D101semidirect
221. C2×Q16C2×D101semidirect
252. #106C2×D101semidirect
257. #103C2×D101semidirect
258. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922872
C2×M16×D1012
#202656902
C22⋊Q8×D1012
#202654332
#202654252
#202654232
#202654172
#202654152
#202654092
#202654072
#202652162
#202651842
#202651772
#470×C1012
#3056101