Finite Group Databasebeta

← #27154811

Subgroups — #27154811

Order 12416.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261458C211yesmaximal, characteristic, complemented
2#20261130C211yesmaximal, characteristic, complemented
3#20261127C211yesmaximal, characteristic, complemented
4#20261117C211yesmaximal, characteristic
5#20260957C211yesmaximal, characteristic, complemented
6#20260954C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic
8#20260947C211yesmaximal, characteristic
9#20260945C211yesmaximal, characteristic, complemented
10#20260942C211yesmaximal, characteristic, complemented
11#20260866C211yesmaximal, characteristic, complemented
12#20260835C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14C4×#13095501C211yesmaximal, characteristic, complemented
15#556×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095501C2211yescharacteristic
19C2×#13095501C2211yescharacteristic
20C2×#13095499C2211yescharacteristic
21C2×#13095499C2211yescharacteristic
22C2×#13095498C2211yescharacteristic
23C2×#13095498C2211yescharacteristic
24C2×#13095496C2211yescharacteristic
25#15690810C2211yescharacteristic
26#15690805—21no—
27#15690805—21no—
28#15690798C2211yescharacteristic
29#15690798C2211yescharacteristic
30#15690786—21no—
31C4×Dic194C2211yescharacteristic
32C4×Dic194C2211yescharacteristic
33C4×D388C2211yescharacteristic
34#15690768—21no—
35#15690743C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37C4⋊#11715895C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39#15690734C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690733C2211yescharacteristic
42#15690733C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690730C2211yescharacteristic
45#15690721C2211yescharacteristic
46#15690721C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48C4⋊Q8×C97C2211yescharacteristic
49#126×C97—21no—
50#126×C97—21no—
51#124×C97C2211yescharacteristic
52#124×C97C2211yescharacteristic
53C22⋊Q8×C97—21no—
54C4×Q8×C97C2211yescharacteristic
55C4×Q8×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58#117×C97—21no—
59C2×Dic194C2311yescharacteristic
60C2×Dic194C2311yescharacteristic
61C2×D388C2311yescharacteristic
62C2×C4×D97—21no—
63#13095501—22no—
64#13095501—41no—
65#13095501—41no—
66#13095501—41no—
67#13095499—22no—
68#13095499—41no—
69#13095498—22no—
70#13095498—41no—
71#13095496—41no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C4⋊Dic97—21no—
77C4⋊Dic97—21no—
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84#13095488—21no—
85#13095488—21no—
86C4×Dic97—21no—
87C4×Dic97—21no—
88C2×Q8×C97D411yescharacteristic
89C2×Q8×C97C2311yescharacteristic
90C2×Q8×C97C2311yescharacteristic
91C2×D4×C97C2311yescharacteristic
92C22×C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97—21no—
97C4⋊C4×C97D411yescharacteristic
98C4⋊C4×C97D411yescharacteristic
99C4⋊C4×C97D411yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C42×C97—21no—
107C42×C97—21no—
108C42×C97C2311yescharacteristic
109C42×C97C2311yescharacteristic
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C22×D97—21no—
113Q8×C97—21no—
114Q8×C97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—41no—
118D4×C97—21no—
119D4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—42no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134D388—21no—
135D388—21no—
136C4×D97—42no—
137Dic194—21no—
138Dic194—21no—
139Dic194—21no—
140Dic194—21no—
141#11715895—21no—
142#11715895—21no—
143#11715895—21no—
144#11715895#4112yes—
145C22×C97—41no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97C22×D411yescharacteristic
149C2×D97—41no—
150C2×D97—41no—
151C2×D97—41no—
152C4×C97—21no—
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×C97C2×#4111yescharacteristic
162C4×C97H(5,𝔽2)11yescharacteristic
163Dic97—41no—
164Dic97—41no—
165Dic97—41no—
166C2×C97—81no—
167C2×C97#55011yescharacteristic
168C2×C97#57511yescharacteristic
169C2×C97#57811yescharacteristic
170D97—81no—
171#3402—971nomaximal, Sylow
172C97#340211yescharacteristic, complemented, Sylow
173#556—971no—
174#556D9711yescharacteristic, complemented
175#497—971no—
176#497—971no—
177#495—971no—
178#464—971no—
179#464—971no—
180#463—971no—
181#463—971no—
182#462—971no—
183#462—971no—
184#446—971no—
185#440—971no—
186#440—971no—
187C4×Q16—971no—
188C2×Q16—971no—
189C2×Q16—971no—
190C2×Q16—971no—
191C2×Q16—971no—
192C2×SD16—971no—
193C2×SD16—971no—
194C2×SD16—971no—
195C2×SD16—971no—
196C2×D8—971no—
197C4⋊Q8—971no—
198C4⋊Q8C2×D9711yescharacteristic
199#126—21no—
200#126—21no—
201#126—1941no—
202#126—1941no—
203#124—971no—
204#124—971no—
205#124C2×D9711yescharacteristic
206#124C2×D9711yescharacteristic
207C22⋊Q8—21no—
208C22⋊Q8—1941no—
209C4×Q8—971no—
210C4×Q8—971no—
211C4×Q8C2×D9711yescharacteristic
212C4×Q8C2×D9711yescharacteristic
213C4×Q8C2×D9711yescharacteristic
214C4×D4—971no—
215C4×D4C2×D9711yescharacteristic
216#117—21no—
217#117—1941no—
218#107—971no—
219C4⋊C8—971no—
220C4⋊C8—971no—
221C4⋊C8—971no—
222#103—971no—
223#103—971no—
224#103—971no—
225#103—971no—
226#102—971no—
227#102—971no—
228#97—971no—
229#97—971no—
230C4×C8—971no—
231C2×Q8—971no—
232C2×Q8—971no—
233C2×Q8C22⋊D9711yescharacteristic
234C2×Q8C22×D9711yescharacteristic
235C2×Q8C22×D9711yescharacteristic
236C2×D4—971no—
237C2×D4C22×D9711yescharacteristic
238C22×C4—21no—
239C22×C4—1941no—
240Q16—3881no—
241Q16—3881no—
242Q16—3881no—
243Q16—1942no—
244SD16—3881no—
245SD16—3881no—
246SD16—1942no—
247SD16—1942no—
248D8—3881no—
249C2×C8—971no—
250C2×C8—971no—
251C2×C8—971no—
252C2×C8—971no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—971no—
258C4⋊C4—1941no—
259C4⋊C4—1941no—
260C4⋊C4—1941no—
261C4⋊C4—1941no—
262C4⋊C4C22⋊D9711yescharacteristic
263C4⋊C4C22⋊D9711yescharacteristic
264C4⋊C4C22⋊D9711yescharacteristic
265C4⋊C4C22×D9711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C42—21no—
277C42—21no—
278C42—1941no—
279C42—1941no—
280C42C22×D9711yescharacteristic
281C42C22×D9711yescharacteristic
282C42C22×D9711yescharacteristic
283C23—21no—
284C23—1941no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—41no—
290Q8—1941no—
291Q8—1941no—
292Q8—1941no—
293Q8—1941no—
294D4—21no—
295D4—21no—
296D4—1941no—
297D4—1941no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3882no—
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×C4C2×C22⋊D9711yescharacteristic
312C2×C4C23×D9711yescharacteristic
313C8—972no—
314C8—1941no—
315C8—1941no—
316C8—1941no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—3881no—
321C22—3881no—
322C22—3881no—
323C22C22×C22⋊D9711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3881no—
334C4—3881no—
335C4—3881no—
336C4#41×D9711yescharacteristic
337C4#1569085611yescharacteristic
338C2—81no—
339C2—7761no—
340C2#2026140411yescharacteristic
341C2#2026153311yescharacteristic
342C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#556×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095496C22
hypercenter#556D97
hypocenterC97#3402
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20261458C2semidirect
2. #20261130C2semidirect
3. #20261127C2semidirect
5. #20260957C2semidirect
6. #20260954C2semidirect
9. #20260945C2semidirect
10. #20260942C2semidirect
11. #20260866C2semidirect
12. #20260835C2semidirect
13. #20260833C2semidirect
14. C4×#13095501C2semidirect
15. #556×C97C2semidirect
172. C97#3402semidirect
174. #556D97semidirect

Maximal subgroups

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

subgroupindex
#202614582
#202611302
#202611272
#202611172
#202609572
#202609542
#202609502
#202609472
#202609452
#202609422
#202608662
#202608352
#202608332
C4×#130955012
#556×C972
#340297