Finite Group Databasebeta

← #27154802

Subgroups — #27154802

Order 12416.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261470C211yesmaximal, characteristic, complemented
2#20261127C211yesmaximal, characteristic, complemented
3#20261125C211yesmaximal, characteristic
4#20261117C211yesmaximal, characteristic, complemented
5#20260957C211yesmaximal, characteristic, complemented
6#20260954C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260947C211yesmaximal, characteristic, complemented
9#20260937C211yesmaximal, characteristic, complemented
10#20260935C211yesmaximal, characteristic, complemented
11#20260866C211yesmaximal, characteristic, complemented
12#20260845C211yesmaximal, characteristic, complemented
13#20260840C211yesmaximal, characteristic, complemented
14C4×#13095498C211yesmaximal, characteristic, complemented
15#556×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20C2×#13095498C2211yescharacteristic
21C2×#13095498C2211yescharacteristic
22C2×#13095498C2211yescharacteristic
23C2×#13095496C2211yescharacteristic
24C2×#13095496C2211yescharacteristic
25#15690807C2211yescharacteristic
26#15690799—21no—
27#15690798C2211yescharacteristic
28#15690798C2211yescharacteristic
29#15690795—21no—
30#15690795—21no—
31C4×Dic194C2211yescharacteristic
32C4×D388C2211yescharacteristic
33C4×D388C2211yescharacteristic
34#15690772—21no—
35#15690741C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37C4⋊#11715895C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39#15690734C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690733C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690730C2211yescharacteristic
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×D388—21no—
61C2×D388—21no—
62C2×D388C2311yescharacteristic
63C2×D388C2311yescharacteristic
64C2×C4×D97—21no—
65C2×C4×D97—21no—
66#13095501—22no—
67#13095501—41no—
68#13095499—41no—
69#13095498—22no—
70#13095498—41no—
71#13095498—41no—
72#13095498—41no—
73#13095496—22no—
74#13095496—41no—
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80#13095490—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—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—
113C22×D97—21no—
114Q8×C97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—21no—
118Q8×C97—41no—
119D4×C97—21no—
120D4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—42no—
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C2×Dic97—21no—
133C2×Dic97—21no—
134D388—21no—
135D388—21no—
136D388—21no—
137D388—21no—
138D388—42no—
139D388—81no—
140C4×D97—81no—
141C4×D97—81no—
142Dic194—21no—
143Dic194—21no—
144#11715895—21no—
145#11715895—21no—
146#11715895—21no—
147#11715895#4112yes—
148C22×C97—41no—
149C22×C97—41no—
150C22×C97—41no—
151C22×C97C22×D411yescharacteristic
152C2×D97—41no—
153C2×D97—41no—
154C2×D97—41no—
155C2×D97—41no—
156C2×D97—41no—
157C2×D97—41no—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97—21no—
163C4×C97—21no—
164C4×C97—41no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97C2×#4111yescharacteristic
168C4×C97H(5,𝔽2)11yescharacteristic
169Dic97—41no—
170Dic97—41no—
171C2×C97—81no—
172C2×C97#55011yescharacteristic
173C2×C97#57511yescharacteristic
174C2×C97#57711yescharacteristic
175D97—81no—
176D97—81no—
177#3401—971nomaximal, Sylow
178C97#340111yescharacteristic, complemented, Sylow
179#559—971no—
180#556D9711yescharacteristic, complemented
181#497—971no—
182#496—971no—
183#495—971no—
184#464—971no—
185#464—971no—
186#463—971no—
187#463—971no—
188#461—971no—
189#460—971no—
190#446—971no—
191#442—971no—
192#441—971no—
193C4×SD16—971no—
194C2×Q16—971no—
195C2×Q16—971no—
196C2×SD16—971no—
197C2×SD16—971no—
198C2×SD16—971no—
199C2×SD16—971no—
200C2×SD16—971no—
201C2×D8—971no—
202C2×D8—971no—
203C4⋊Q8C2×D9711yescharacteristic
204#127—971no—
205#126—21no—
206#126—21no—
207#124—971no—
208#124—971no—
209#124—1941no—
210#124C2×D9711yescharacteristic
211#124C2×D9711yescharacteristic
212#123—1941no—
213#123—1941no—
214C22⋊Q8—21no—
215C4×Q8—971no—
216C4×Q8C2×D9711yescharacteristic
217C4×Q8C2×D9711yescharacteristic
218C4×Q8C2×D9711yescharacteristic
219C4×D4—971no—
220C4×D4—971no—
221C4×D4—1941no—
222C4×D4C2×D9711yescharacteristic
223#117—21no—
224#106—971no—
225C4⋊C8—971no—
226C4⋊C8—971no—
227C4⋊C8—971no—
228#103—971no—
229#103—971no—
230#103—971no—
231#102—971no—
232#102—971no—
233#102—971no—
234#97—971no—
235#97—971no—
236C4×C8—971no—
237C2×Q8—971no—
238C2×Q8C22⋊D9711yescharacteristic
239C2×Q8C22×D9711yescharacteristic
240C2×Q8C22×D9711yescharacteristic
241C2×D4—971no—
242C2×D4—971no—
243C2×D4—1941no—
244C2×D4—1941no—
245C2×D4C22×D9711yescharacteristic
246C22×C4—21no—
247C22×C4—1941no—
248C22×C4—1941no—
249Q16—3881no—
250Q16—1942no—
251SD16—3881no—
252SD16—3881no—
253SD16—3881no—
254SD16—3881no—
255SD16—1942no—
256D8—3881no—
257D8—1942no—
258C2×C8—971no—
259C2×C8—971no—
260C2×C8—971no—
261C2×C8—971no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—971no—
267C4⋊C4—1941no—
268C4⋊C4C22⋊D9711yescharacteristic
269C4⋊C4C22⋊D9711yescharacteristic
270C4⋊C4C22⋊D9711yescharacteristic
271C4⋊C4C22×D9711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C42—21no—
284C42—21no—
285C42—1941no—
286C42C22×D9711yescharacteristic
287C42C22×D9711yescharacteristic
288C42C22×D9711yescharacteristic
289C23—21no—
290C23—1941no—
291C23—1941no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—1941no—
298Q8—1941no—
299D4—21no—
300D4—21no—
301D4—7761no—
302D4—3882no—
303D4—1941no—
304D4—1941no—
305D4—1941no—
306D4—1941no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—7761no—
312C2×C4—7761no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4C2×C22⋊D9711yescharacteristic
316C2×C4C2×C22⋊D9711yescharacteristic
317C2×C4C2×C22⋊D9711yescharacteristic
318C2×C4C2×C22⋊D9711yescharacteristic
319C2×C4C2×C22⋊D9711yescharacteristic
320C2×C4C2×C22⋊D9711yescharacteristic
321C2×C4C23×D9711yescharacteristic
322C8—972no—
323C8—1941no—
324C8—1941no—
325C8—1941no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—3881no—
330C22—3881no—
331C22—3881no—
332C22—3881no—
333C22—3881no—
334C22—3881no—
335C22C22×C22⋊D9711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—3881no—
346C4—3881no—
347C4#41×D9711yescharacteristic
348C4#1569085611yescharacteristic
349C2—81no—
350C2—7761no—
351C2—7761no—
352C2#2026140411yescharacteristic
353C2#2026153311yescharacteristic
354C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#556×C97C2
socleC22×C97C22×D4
prime radical#20261125C2
hypercenter#556D97
hypocenterC97#3401
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. #20261470C2semidirect
2. #20261127C2semidirect
4. #20261117C2semidirect
5. #20260957C2semidirect
6. #20260954C2semidirect
7. #20260950C2semidirect
8. #20260947C2semidirect
9. #20260937C2semidirect
10. #20260935C2semidirect
11. #20260866C2semidirect
12. #20260845C2semidirect
13. #20260840C2semidirect
14. C4×#13095498C2semidirect
15. #556×C97C2semidirect
178. C97#3401semidirect
180. #556D97semidirect

Maximal subgroups

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

subgroupindex
#202614702
#202611272
#202611252
#202611172
#202609572
#202609542
#202609502
#202609472
#202609372
#202609352
#202608662
#202608452
#202608402
C4×#130954982
#556×C972
#340197