Finite Group Databasebeta

← #27149966

Subgroups — #27149966

Order 12416.

366 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20260677C211yesmaximal, characteristic, complemented
2#20260676C211yesmaximal, characteristic, complemented
3#20260672C211yesmaximal, characteristic
4#20260667C211yesmaximal, characteristic, complemented
5#20260664C211yesmaximal, characteristic
6#20260664C211yesmaximal, characteristic
7#20260662C211yesmaximal, characteristic
8#20260659C211yesmaximal, characteristic
9#20260656C211yesmaximal, characteristic, complemented
10#20260656C211yesmaximal, characteristic, complemented
11#20260648C211yesmaximal, characteristic
12#20260632C211yesmaximal, characteristic, complemented
13#20260627C211yesmaximal, characteristic
14#20260622C211yesmaximal, characteristic, complemented
15#401×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic
17C2×C4⋊Dic97C2211yescharacteristic
18C2×C4⋊Dic97C2211yescharacteristic
19C2×C4⋊Dic97C2211yescharacteristic
20C2×C4⋊Dic97C2211yescharacteristic
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095488C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#95×C97C2211yescharacteristic
45#95×C97C2211yescharacteristic
46#95×C97C2211yescharacteristic
47#95×C97C2211yescharacteristic
48#95×C97C2211yescharacteristic
49#95×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51C23×D97C2311yescharacteristic
52C2×Dic194—41no—
53C2×Dic194—41no—
54C2×Dic194—41no—
55C22×Dic97C2311yescharacteristic
56C22×Dic97C2311yescharacteristic
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C4⋊Dic97—41no—
63C4⋊Dic97—41no—
64C4⋊Dic97—41no—
65C4⋊Dic97—41no—
66C4⋊Dic97—41no—
67#13095490—22no—
68#13095490—22no—
69#13095490—41no—
70#13095490—41no—
71#13095490—41no—
72#13095488—22no—
73#13095488—41no—
74#13095488—41no—
75#13095488—41no—
76#13095488—41no—
77#13095488—41no—
78#13095488—41no—
79C4×Dic97—41no—
80C22×C4×C97C2311yescharacteristic
81C22×C4×C97C2311yescharacteristic
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C22×C4×C97C2311yescharacteristic
85C22×C4×C97C2311yescharacteristic
86C22×C4×C97C2311yescharacteristic
87C23×C97C2411yescharacteristic
88C22×D97—21no—
89C22×D97—21no—
90C22×D97—21no—
91C22×D97—21no—
92C22×D97—21no—
93C22×D97—21no—
94C22×D97—21no—
95C2×C4×C97—21no—
96C2×C4×C97—21no—
97C2×C4×C97—21no—
98C2×C4×C97—21no—
99C2×C4×C97—21no—
100C2×C4×C97—21no—
101C2×C4×C97—21no—
102C2×C4×C97—21no—
103C2×C4×C97—21no—
104C2×C4×C97—21no—
105C2×C4×C97—21no—
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×Dic97—21no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97#4112yes—
137Dic194—44no—
138C22×C97C2×#4111yescharacteristic
139C22×C97H(5,𝔽2)11yescharacteristic
140C22×C97H(5,𝔽2)11yescharacteristic
141C22×C97H(5,𝔽2)11yescharacteristic
142C22×C972−1+411yescharacteristic
143C22×C972−1+411yescharacteristic
144C22×C972−1+411yescharacteristic
145C2×D97—41no—
146C2×D97—41no—
147C2×D97—41no—
148C2×D97—41no—
149C2×D97—41no—
150C2×D97—41no—
151C2×D97—41no—
152C4×C97—41no—
153C4×C97—41no—
154C4×C97—41no—
155C4×C97—41no—
156C4×C97—41no—
157C4×C97—41no—
158C4×C97—41no—
159Dic97—22no—
160Dic97—41no—
161Dic97—41no—
162Dic97—41no—
163Dic97—41no—
164Dic97—41no—
165Dic97—41no—
166C2×C97#53911yescharacteristic
167C2×C97#54211yescharacteristic
168C2×C97#54211yescharacteristic
169C2×C97#56011yescharacteristic
170C2×C97#56311yescharacteristic
171C2×C97#56311yescharacteristic
172C2×C97#56311yescharacteristic
173D97—81no—
174#2838—971nomaximal, Sylow
175C97#283811yescharacteristic, complemented, Sylow
176#401D9711yescharacteristic, complemented
177#400—971no—
178#400—971no—
179#399—971no—
180#398—971no—
181#397—971no—
182#397—971no—
183#397—971no—
184#396—971no—
185#395—971no—
186#395—971no—
187#393—971no—
188#389—971no—
189#388—971no—
190#387—971no—
191C22×Q8—971no—
192C2×C4⋊C4—971no—
193C2×C4⋊C4—971no—
194C2×C4⋊C4—971no—
195C2×C4⋊C4—971no—
196C2×C4⋊C4—971no—
197C2×C4⋊C4—971no—
198C2×C4⋊C4—971no—
199C2×C4⋊C4—971no—
200C2×C4⋊C4—971no—
201C2×C4⋊C4—971no—
202C2×C22⋊C4—971no—
203C2×C22⋊C4—971no—
204C2×C22⋊C4—971no—
205C2×C22⋊C4—971no—
206C2×C22⋊C4—971no—
207C2×C22⋊C4—971no—
208C2×C22⋊C4—971no—
209C2×C42—971no—
210#95—971no—
211#95—971no—
212#95—971no—
213#95—971no—
214#95—971no—
215#95—971no—
216#95—971no—
217#95—971no—
218#95—971no—
219#95C2×D9711yescharacteristic
220#95C2×D9711yescharacteristic
221#95C2×D9711yescharacteristic
222#95C2×D9711yescharacteristic
223#95C2×D9711yescharacteristic
224#95C2×D9711yescharacteristic
225#95C2×D9711yescharacteristic
226C24—971no—
227C2×Q8—3881no—
228C2×Q8—3881no—
229C2×Q8—3881no—
230C22×C4—971no—
231C22×C4—971no—
232C22×C4—971no—
233C22×C4—971no—
234C22×C4—971no—
235C22×C4—971no—
236C22×C4—971no—
237C22×C4C22×D9711yescharacteristic
238C22×C4C22×D9711yescharacteristic
239C22×C4C22×D9711yescharacteristic
240C22×C4C22×D9711yescharacteristic
241C22×C4C22×D9711yescharacteristic
242C22×C4C22×D9711yescharacteristic
243C22×C4C22×D9711yescharacteristic
244C4⋊C4—3881no—
245C4⋊C4—3881no—
246C4⋊C4—3881no—
247C4⋊C4—3881no—
248C4⋊C4—3881no—
249C4⋊C4—3881no—
250C4⋊C4—3881no—
251C4⋊C4—3881no—
252C4⋊C4—1942no—
253C4⋊C4—1942no—
254C22⋊C4—3881no—
255C22⋊C4—3881no—
256C22⋊C4—3881no—
257C22⋊C4—3881no—
258C22⋊C4—3881no—
259C22⋊C4—3881no—
260C22⋊C4—1942no—
261C42—3881no—
262C23—1941no—
263C23—1941no—
264C23—1941no—
265C23—1941no—
266C23—1941no—
267C23—1941no—
268C23—1941no—
269C23C23×D9711yescharacteristic
270Q8—3884no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—972no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1941no—
312C2×C4—1941no—
313C22—3881no—
314C22—3881no—
315C22—3881no—
316C22—3881no—
317C22—3881no—
318C22—3881no—
319C22—3881no—
320C22#41×D9711yescharacteristic
321C22#1569085611yescharacteristic
322C22#1569085611yescharacteristic
323C22#1569085611yescharacteristic
324C22#1569085711yescharacteristic
325C22#1569085711yescharacteristic
326C22#1569085711yescharacteristic
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—3881no—
335C4—3881no—
336C4—3881no—
337C4—3881no—
338C4—3881no—
339C4—3881no—
340C4—1942no—
341C2—7761no—
342C2#2026132611yescharacteristic
343C2#2026135111yescharacteristic
344C2#2026135111yescharacteristic
345C2#2026147611yescharacteristic
346C2#2026148711yescharacteristic
347C2#2026148711yescharacteristic
348C2#2026148711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#401×C97C2
prime radicalC23×D97C23
hypercenter#401D97
hypocenterC97#2838
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #401

Complements

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

NH ≅ G/Ntype
1. #20260677C2semidirect
2. #20260676C2semidirect
4. #20260667C2semidirect
9. #20260656C2semidirect
10. #20260656C2semidirect
12. #20260632C2semidirect
14. #20260622C2semidirect
15. #401×C97C2semidirect
175. C97#2838semidirect
176. #401D97semidirect

Maximal subgroups

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

subgroupindex
#202606772
#202606762
#202606722
#202606672
#202606642
#202606622
#202606592
#202606562
#202606482
#202606322
#202606272
#202606222
#401×C972
#283897