Finite Group Databasebeta

← #27153201

Subgroups — #27153201

Order 12416.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261356C211yesmaximal, characteristic, complemented
2#20261025C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic, complemented
4#20261020C211yesmaximal, characteristic, complemented
5#20261017C211yesmaximal, characteristic, complemented
6#20260991C211yesmaximal, characteristic, complemented
7#20260986C211yesmaximal, characteristic
8#20260966C211yesmaximal, characteristic, complemented
9#20260962C211yesmaximal, characteristic, complemented
10#20260957C211yesmaximal, characteristic, complemented
11#20260954C211yesmaximal, characteristic, complemented
12#20260950C211yesmaximal, characteristic, complemented
13#20260947C211yesmaximal, characteristic, complemented
14#20260762C211yesmaximal, characteristic, complemented
15#542×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095499C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×#13095495C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22#15690810C2211yescharacteristic
23#15690805—21no—
24#15690798C2211yescharacteristic
25#15690796—21no—
26#15690792—21no—
27#15690789C2211yescharacteristic
28#15690788—21no—
29#15690784C2211yescharacteristic
30C4×Dic194C2211yescharacteristic
31C4×D388C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34#15690741C2211yescharacteristic
35#15690741C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37C4⋊#11715895C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39C4⋊#11715895C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690734C2211yescharacteristic
42#15690733C2211yescharacteristic
43#15690733C2211yescharacteristic
44#15690731C2211yescharacteristic
45#15690731C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic
48C4⋊Q8×C97C2211yescharacteristic
49#126×C97—21no—
50#124×C97—21no—
51#124×C97C2211yescharacteristic
52#123×C97—21no—
53C22⋊Q8×C97—21no—
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58#117×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194C2311yescharacteristic
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C4×D97—21no—
64#13095501—41no—
65#13095499—41no—
66#13095498—41no—
67#13095496—41no—
68#13095495—41no—
69C2×#11715895—41no—
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C4⋊Dic97—21no—
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80#13095488—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84C4×Dic97—21no—
85C2×Q8×C97—21no—
86C2×Q8×C97C2311yescharacteristic
87C2×D4×C97—21no—
88C2×D4×C97C2311yescharacteristic
89C22×C4×C97—21no—
90C22×C4×C97C2311yescharacteristic
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97D411yescharacteristic
95C4⋊C4×C97D411yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97—81no—
114D4×C97—21no—
115D4×C97—21no—
116D4×C97—81no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—81no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—81no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133D388—21no—
134D388—21no—
135C4×D97—81no—
136Dic194—21no—
137Dic194—21no—
138Dic194—81no—
139#11715895—21no—
140#11715895—21no—
141#11715895—21no—
142#11715895—21no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97—21no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97—41no—
149C22×C97C22×D411yescharacteristic
150C2×D97—41no—
151C2×D97—41no—
152C2×D97—41no—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97—41no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97H(5,𝔽2)11yescharacteristic
162C4×C972−1+411yescharacteristic
163Dic97—41no—
164Dic97—41no—
165Dic97—41no—
166C2×C97—41no—
167C2×C97—81no—
168C2×C97#53711yescharacteristic
169C2×C97#57711yescharacteristic
170C2×C97#57711yescharacteristic
171D97—81no—
172#3263—971nomaximal, Sylow
173C97#326311yescharacteristic, complemented, Sylow
174#542—971no—
175#542D9711yescharacteristic, complemented
176#476—971no—
177#476—971no—
178#475—971no—
179#475—971no—
180#470—971no—
181#469—971no—
182#465—971no—
183#465—971no—
184#464—971no—
185#464—971no—
186#463—971no—
187#463—971no—
188#424—971no—
189C2×Q16—971no—
190C2×SD16—971no—
191C2×SD16—971no—
192C2×D8—971no—
193C2×M16—971no—
194C22×C8—971no—
195C4⋊Q8—971no—
196C4⋊Q8C2×D9711yescharacteristic
197#126—21no—
198#126—1941no—
199#124—21no—
200#124—971no—
201#124—1941no—
202#124C2×D9711yescharacteristic
203#123—21no—
204#123—1941no—
205C22⋊Q8—21no—
206C22⋊Q8—971no—
207C22⋊Q8—1941no—
208C22⋊Q8C2×D9711yescharacteristic
209#121—971no—
210#121C2×D9711yescharacteristic
211C4×Q8—971no—
212C4×Q8C2×D9711yescharacteristic
213C4×D4—971no—
214C4×D4C2×D9711yescharacteristic
215#117C2×D9711yescharacteristic
216#106—971no—
217#106—971no—
218#106—971no—
219#106—971no—
220C4⋊C8—971no—
221C4⋊C8—971no—
222C4⋊C8—971no—
223C4⋊C8—971no—
224#103—971no—
225#103—971no—
226#103—971no—
227#103—971no—
228#102—971no—
229#102—971no—
230#102—971no—
231#102—971no—
232C2×Q8—21no—
233C2×Q8—971no—
234C2×Q8—1941no—
235C2×Q8C22×D9711yescharacteristic
236C2×D4—21no—
237C2×D4—971no—
238C2×D4—1941no—
239C2×D4C22×D9711yescharacteristic
240C22×C4—21no—
241C22×C4—1941no—
242C22×C4C22×D9711yescharacteristic
243Q16—3881no—
244SD16—3881no—
245SD16—3881no—
246D8—3881no—
247M16—3881no—
248C2×C8—971no—
249C2×C8—971no—
250C2×C8—971no—
251C2×C8—971no—
252C2×C8—3881no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—971no—
257C4⋊C4—971no—
258C4⋊C4—1941no—
259C4⋊C4—1941no—
260C4⋊C4—1941no—
261C4⋊C4C22⋊D9711yescharacteristic
262C4⋊C4C22⋊D9711yescharacteristic
263C4⋊C4C22×D9711yescharacteristic
264C4⋊C4C22×D9711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1941no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C22⋊C4C22⋊D9711yescharacteristic
276C22⋊C4C22⋊D9711yescharacteristic
277C42—21no—
278C42—1941no—
279C42C22×D9711yescharacteristic
280C42C22×D9711yescharacteristic
281C23—21no—
282C23—1941no—
283C23C2×C22⋊D9711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—7761no—
288Q8—1941no—
289Q8—1941no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—7761no—
294D4—1941no—
295D4—1941no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—7761no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4C2×C22⋊D9711yescharacteristic
307C2×C4C2×C22⋊D9711yescharacteristic
308C2×C4C2×C22⋊D9711yescharacteristic
309C2×C4C2×C22⋊D9711yescharacteristic
310C2×C4C2×C22⋊D9711yescharacteristic
311C2×C4C23×D9711yescharacteristic
312C8—1941no—
313C8—1941no—
314C8—1941no—
315C8—1941no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—3881no—
323C22—3881no—
324C22—3881no—
325C22C22×C22⋊D9711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—3881no—
335C4—3881no—
336C4—3881no—
337C4#1569085611yescharacteristic
338C4#1569085711yescharacteristic
339C2—41no—
340C2—81no—
341C2—7761no—
342C2#2026131311yescharacteristic
343C2#2026154311yescharacteristic
344C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#542×C97C2
socleC22×C97C22×D4
prime radical#20260986C2
hypercenter#542D97
hypocenterC97#3263
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20261356C2semidirect
2. #20261025C2semidirect
3. #20261022C2semidirect
4. #20261020C2semidirect
5. #20261017C2semidirect
6. #20260991C2semidirect
8. #20260966C2semidirect
9. #20260962C2semidirect
10. #20260957C2semidirect
11. #20260954C2semidirect
12. #20260950C2semidirect
13. #20260947C2semidirect
14. #20260762C2semidirect
15. #542×C97C2semidirect
173. C97#3263semidirect
175. #542D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202610252
#202610222
#202610202
#202610172
#202609912
#202609862
#202609662
#202609622
#202609572
#202609542
#202609502
#202609472
#202607622
#542×C972
#326397