Finite Group Databasebeta

← #27145012

Subgroups — #27145012

Order 12416.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic194C211yesmaximal, characteristic, complemented
2C2×#15690767C211yesmaximal, characteristic
3#20260676C211yesmaximal, characteristic, complemented
4#20260670C211yesmaximal, characteristic
5#20260669C211yesmaximal, characteristic, complemented
6#20260666C211yesmaximal, characteristic, complemented
7#20260659C211yesmaximal, characteristic
8#20260657C211yesmaximal, characteristic
9#20260655C211yesmaximal, characteristic, complemented
10#20260648C211yesmaximal, characteristic
11C4×#13095490C211yesmaximal, characteristic, complemented
12C4×#13095490C211yesmaximal, characteristic, complemented
13C4×#13095488C211yesmaximal, characteristic
14C4×#13095488C211yesmaximal, characteristic
15C4×C4⋊C4×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C4×Dic194—24no—
19#15690767—24no—
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46C2×C4⋊C4×C97C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C42×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50C2×C42×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53C23×D97C2311yescharacteristic
54C2×Dic194—22no—
55C2×Dic194—22no—
56C2×Dic194—22no—
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C2×C4×D97—22no—
65C2×C4×D97—22no—
66C2×C4×D97Q814yescomplemented
67C4⋊Dic97—22no—
68#13095490—22no—
69#13095490—22no—
70#13095490—22no—
71#13095490—22no—
72#13095490—22no—
73#13095490—22no—
74#13095490—22no—
75#13095490—22no—
76#13095488—22no—
77#13095488—22no—
78#13095488—22no—
79#13095488—22no—
80#13095488—22no—
81#13095488—22no—
82C4×Dic97—22no—
83C4×Dic97—22no—
84C4×Dic97D414yes—
85C22×C4×C97C2311yescharacteristic
86C22×C4×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C4⋊C4×C97—22no—
93C4⋊C4×C97—22no—
94C42×C97—22no—
95C42×C97—22no—
96C42×C97—22no—
97C23×C97C2411yescharacteristic
98C22×D97—21no—
99C22×D97—21no—
100C22×D97—21no—
101C22×D97—21no—
102C22×D97—21no—
103C22×D97—21no—
104C22×D97C2×Q812yes—
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×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97C2×D412yes—
122C2×C4×C97C2×Q812yes—
123C2×C4×C97#4112yes—
124C2×C4×C97#4112yes—
125C2×C4×C97#4112yes—
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—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97C2×D412yes—
143C2×Dic97C2×D412yes—
144C2×Dic97C2×Q812yes—
145C2×Dic97#4112yes—
146C2×Dic97#4112yes—
147C4×D97—28no—
148Dic194—28no—
149C22×C97C22×D411yescharacteristic
150C22×C97C22×Q811yescharacteristic
151C22×C97C2×#4111yescharacteristic
152C22×C97C2×#4111yescharacteristic
153C22×C97C2×#4111yescharacteristic
154C22×C97C2×#4111yescharacteristic
155C22×C97C2×#4111yescharacteristic
156C2×D97—22no—
157C2×D97—22no—
158C2×D97—22no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—41no—
163C4×C97—22no—
164C4×C97—22no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97C22⋊Q814yes—
170Dic97—22no—
171Dic97—22no—
172Dic97—22no—
173Dic97—22no—
174Dic97—22no—
175Dic97—41no—
176Dic97—41no—
177C2×C97C2×C22⋊Q811yescharacteristic
178C2×C97#52511yescharacteristic
179C2×C97#52911yescharacteristic
180C2×C97#52911yescharacteristic
181C2×C97#53211yescharacteristic
182C2×C97#53311yescharacteristic
183C2×C97#53311yescharacteristic
184D97—42no—
185#2396—971nomaximal, Sylow
186C97#239611yescharacteristic, complemented, Sylow
187C2×C4×Q8—971no—
188C2×#117—971no—
189#400—971no—
190#399—971no—
191#398—971no—
192#398—971no—
193#396—971no—
194#396—971no—
195#395—971no—
196#393—971no—
197C4×C4⋊C4—971no—
198C4×C4⋊C4—971no—
199C4×C4⋊C4D9711yescharacteristic, complemented
200C4×C22⋊C4—971no—
201C4×C22⋊C4—971no—
202C22×Q8—971no—
203C23×C4—971no—
204C4×Q8—1944no—
205#117—1944no—
206C2×C4⋊C4—971no—
207C2×C4⋊C4—971no—
208C2×C4⋊C4—971no—
209C2×C4⋊C4—971no—
210C2×C4⋊C4—971no—
211C2×C4⋊C4—971no—
212C2×C4⋊C4—971no—
213C2×C4⋊C4—971no—
214C2×C4⋊C4—971no—
215C2×C4⋊C4C2×D9711yescharacteristic
216C2×C4⋊C4C2×D9711yescharacteristic
217C2×C22⋊C4—971no—
218C2×C22⋊C4—971no—
219C2×C22⋊C4—971no—
220C2×C22⋊C4—971no—
221C2×C22⋊C4—971no—
222C2×C22⋊C4—971no—
223C2×C42—971no—
224C2×C42—971no—
225C2×C42—971no—
226C2×C42C2×D9711yescharacteristic
227C2×C42C2×D9711yescharacteristic
228C2×C42C2×D9711yescharacteristic
229#95—971no—
230#95—971no—
231#95—971no—
232#95—971no—
233#95—971no—
234#95—971no—
235#95—971no—
236#95—971no—
237#95C2×D9711yescharacteristic
238#95C2×D9711yescharacteristic
239C24—971no—
240C2×Q8—1942no—
241C2×Q8—1942no—
242C2×Q8—1942no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—971no—
248C22×C4—974no—
249C22×C4—971no—
250C22×C4—971no—
251C22×C4—1942no—
252C22×C4—1942no—
253C22×C4C22×D9711yescharacteristic
254C22×C4C22×D9711yescharacteristic
255C22×C4C22×D9711yescharacteristic
256C22×C4C22×D9711yescharacteristic
257C22×C4C22×D9711yescharacteristic
258C22×C4C22×D9711yescharacteristic
259C22×C4C22×D9711yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—1942no—
263C4⋊C4—1942no—
264C4⋊C4—1942no—
265C4⋊C4—1942no—
266C4⋊C4—1942no—
267C4⋊C4—1942no—
268C4⋊C4—1942no—
269C4⋊C4—1942no—
270C4⋊C4—1942no—
271C22⋊C4—1942no—
272C22⋊C4—1942no—
273C22⋊C4—1942no—
274C22⋊C4—1942no—
275C22⋊C4—1942no—
276C22⋊C4—1942no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—974no—
281C42—1942no—
282C42—1942no—
283C23—972no—
284C23—1941no—
285C23—1941no—
286C23—1941no—
287C23—1941no—
288C23—1941no—
289C23—1941no—
290C23C23×D9711yescharacteristic
291Q8—1948no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—972no—
309C2×C4—972no—
310C2×C4—972no—
311C2×C4—972no—
312C2×C4—972no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1948no—
330C2×C4D4×D9712yes—
331C2×C4Q8×D9712yes—
332C2×C4#1309551112yes—
333C2×C4#1309551112yes—
334C2×C4#1309551112yes—
335C22—3881no—
336C22—3881no—
337C22—3881no—
338C22—3881no—
339C22—1942no—
340C22—1942no—
341C22—1942no—
342C22C2×D4×D9711yescharacteristic
343C22C2×Q8×D9711yescharacteristic
344C22#41×D9711yescharacteristic
345C22#41×D9711yescharacteristic
346C22C2×#1309551111yescharacteristic
347C22C2×#1309551111yescharacteristic
348C22C2×#1309551111yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3881no—
356C4—3881no—
357C4—1942no—
358C4—1942no—
359C4—1942no—
360C4—1942no—
361C4—1942no—
362C4#1569078814yes—
363C2—3882no—
364C2C2×#1569078811yescharacteristic
365C2#2026124411yescharacteristic
366C2#2026126311yescharacteristic
367C2#2026126911yescharacteristic
368C2#2026128211yescharacteristic
369C2#2026128911yescharacteristic
370C2#2026128911yescharacteristic
subgroupG/H
centerC22×C4C22×D97
commutator, socle, cosocleC23×C97C24
FrattiniC23C23×D97
FittingC4×C4⋊C4×C97C2
prime radicalC23×D97C23
hypercenterC4×C4⋊C4D97
hypocenterC97#2396
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic194C2semidirect
3. #20260676C2semidirect
5. #20260669C2semidirect
6. #20260666C2semidirect
9. #20260655C2semidirect
11. C4×#13095490C2semidirect
12. C4×#13095490C2semidirect
15. C4×C4⋊C4×C97C2semidirect
66. C2×C4×D97Q8semidirect
186. C97#2396semidirect
199. C4×C4⋊C4D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic1942
C2×#156907672
#202606762
#202606702
#202606692
#202606662
#202606592
#202606572
#202606552
#202606482
C4×#130954902
C4×#130954882
C4×C4⋊C4×C972
#239697