Finite Group Databasebeta

← #27144994

Subgroups — #27144994

Order 12416.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690770C211yesmaximal, characteristic
2C2×#15690766C211yesmaximal, characteristic, complemented
3#20260676C211yesmaximal, characteristic, complemented
4#20260670C211yesmaximal, characteristic, complemented
5#20260669C211yesmaximal, characteristic, complemented
6#20260664C211yesmaximal, characteristic, complemented
7#20260663C211yesmaximal, characteristic, complemented
8#20260651C211yesmaximal, characteristic
9#20260650C211yesmaximal, characteristic
10C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
11#20260586C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
13#20260582C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15#397×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19#15690770—24no—
20#15690766—24no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38C2×C4×Dic97C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46C2×C4⋊C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53C23×D97C2311yescharacteristic
54C2×C22⋊D97—22no—
55C2×C22⋊D97—22no—
56C2×C22⋊D97—22no—
57C22×Dic97—22no—
58C22×Dic97—22no—
59C22×Dic97D414yes—
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C2×C4×D97—22no—
68C2×C4×D97—22no—
69C2×C4×D97—22no—
70C4⋊Dic97—22no—
71#13095490—22no—
72#13095490—22no—
73#13095490—22no—
74#13095490—22no—
75#13095490—22no—
76C22⋊Dic97—22no—
77C22⋊Dic97—22no—
78#13095488—22no—
79#13095488—22no—
80#13095488—22no—
81#13095488—22no—
82#13095488—22no—
83C4×Dic97—22no—
84C4×Dic97—22no—
85C4×Dic97—22no—
86C4×Dic97—22no—
87C4×Dic97—22no—
88C24×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C22×C4×C97C2311yescharacteristic
95C4⋊C4×C97—22no—
96C22⋊C4×C97—22no—
97C22⋊C4×C97—22no—
98C22⋊C4×C97—22no—
99C23×C97—21no—
100C23×C97—21no—
101C23×C97—21no—
102C23×C97—21no—
103C23×C97—21no—
104C23×C97—21no—
105C23×C97C2×D412yes—
106C23×C97C2411yescharacteristic
107C22×D97—21no—
108C22×D97—21no—
109C22×D97—21no—
110C22×D97—21no—
111C22×D97—21no—
112C22×D97—21no—
113C22×D97#4112yes—
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×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97#4112yes—
132C22⋊D97—28no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—28no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—21no—
149C2×Dic97C2×D412yes—
150C2×Dic97C2×D412yes—
151C2×Dic97#4112yes—
152C2×Dic97#4112yes—
153C2×Dic97#4112yes—
154C2×Dic97#4112yes—
155C4×D97—44no—
156C22×C97—22no—
157C22×C97—22no—
158C22×C97—22no—
159C22×C97—41no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97C22×D411yescharacteristic
164C22×C97C2×#4111yescharacteristic
165C22×C97C2×#4111yescharacteristic
166C22×C97C2×#4111yescharacteristic
167C22×C97C2×#4111yescharacteristic
168C22×C97C2×#4111yescharacteristic
169C22×C97C2×#4111yescharacteristic
170C2×D97—22no—
171C2×D97—22no—
172C2×D97—22no—
173C2×D97—41no—
174C2×D97—41no—
175C2×D97—41no—
176C2×D97—41no—
177C4×C97—22no—
178C4×C97—41no—
179C4×C97—41no—
180C4×C97—41no—
181C4×C97—41no—
182C4×C97—41no—
183Dic97—22no—
184Dic97—22no—
185Dic97—22no—
186Dic97—41no—
187Dic97—41no—
188Dic97—41no—
189Dic97#12314yes—
190C2×C97—42no—
191C2×C97C2×#12311yescharacteristic
192C2×C97#52511yescharacteristic
193C2×C97#52911yescharacteristic
194C2×C97#52911yescharacteristic
195C2×C97#52911yescharacteristic
196C2×C97#52911yescharacteristic
197C2×C97#53211yescharacteristic
198D97—42no—
199#2395—971nomaximal, Sylow
200C97#239511yescharacteristic, complemented, Sylow
201C2×C4×D4—971no—
202C2×#117—971no—
203#400—971no—
204#399—971no—
205#398—971no—
206#397—971no—
207#397—971no—
208#397D9711yescharacteristic, complemented
209#394—971no—
210#394—971no—
211C4×C4⋊C4—971no—
212C4×C4⋊C4—971no—
213C4×C22⋊C4—971no—
214C4×C22⋊C4—971no—
215C4×C22⋊C4—971no—
216C22×D4—971no—
217C23×C4—971no—
218C23×C4—971no—
219C4×D4—1944no—
220#117—1944no—
221C2×C4⋊C4—971no—
222C2×C4⋊C4—971no—
223C2×C4⋊C4—971no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4—971no—
226C2×C4⋊C4—971no—
227C2×C4⋊C4C2×D9711yescharacteristic
228C2×C22⋊C4—971no—
229C2×C22⋊C4—971no—
230C2×C22⋊C4—971no—
231C2×C22⋊C4—971no—
232C2×C22⋊C4—971no—
233C2×C22⋊C4—971no—
234C2×C22⋊C4—971no—
235C2×C22⋊C4C2×D9711yescharacteristic
236C2×C22⋊C4C2×D9711yescharacteristic
237C2×C22⋊C4C2×D9711yescharacteristic
238C2×C42—971no—
239C2×C42—971no—
240C2×C42—971no—
241C2×C42—971no—
242C2×C42—971no—
243#95—971no—
244#95—971no—
245#95—971no—
246#95—971no—
247#95—971no—
248#95—971no—
249#95—971no—
250#95C2×D9711yescharacteristic
251#95C2×D9711yescharacteristic
252#95C2×D9711yescharacteristic
253C24—971no—
254C24C22×D9711yescharacteristic
255C2×D4—1942no—
256C2×D4—1942no—
257C2×D4—1942no—
258C22×C4—971no—
259C22×C4—974no—
260C22×C4—971no—
261C22×C4—971no—
262C22×C4—971no—
263C22×C4—971no—
264C22×C4—971no—
265C22×C4—971no—
266C22×C4—1942no—
267C22×C4—1942no—
268C22×C4—1942no—
269C22×C4—1942no—
270C22×C4—1942no—
271C22×C4C22×D9711yescharacteristic
272C22×C4C22×D9711yescharacteristic
273C22×C4C22×D9711yescharacteristic
274C22×C4C22×D9711yescharacteristic
275C22×C4C22×D9711yescharacteristic
276C22×C4C22×D9711yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—1942no—
279C4⋊C4—1942no—
280C4⋊C4—1942no—
281C4⋊C4—1942no—
282C4⋊C4—1942no—
283C4⋊C4—1942no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—1942no—
288C22⋊C4—1942no—
289C22⋊C4—1942no—
290C22⋊C4—1942no—
291C22⋊C4—1942no—
292C22⋊C4—1942no—
293C22⋊C4—1942no—
294C42—1942no—
295C42—1942no—
296C42—1942no—
297C42—1942no—
298C42—1942no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—972no—
306C23—1941no—
307C23—1941no—
308C23—1941no—
309C23—1941no—
310C23—1941no—
311C23—1941no—
312C23D4×D9712yes—
313C23C23×D9711yescharacteristic
314D4—1948no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—972no—
333C2×C4—972no—
334C2×C4—972no—
335C2×C4—972no—
336C2×C4—972no—
337C2×C4—972no—
338C2×C4—3884no—
339C2×C4—1948no—
340C2×C4—1941no—
341C2×C4—1941no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4—1941no—
345C2×C4—1941no—
346C2×C4—1941no—
347C2×C4—1941no—
348C2×C4—1941no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4—1941no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4#1309551012yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3881no—
364C22—3881no—
365C22—3881no—
366C22—3881no—
367C22—1942no—
368C22—1942no—
369C22—1942no—
370C22C2×D4×D9711yescharacteristic
371C22#41×D9711yescharacteristic
372C22#41×D9711yescharacteristic
373C22#41×D9711yescharacteristic
374C22#41×D9711yescharacteristic
375C22#41×D9711yescharacteristic
376C22C2×#1309551011yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—974no—
384C4—3881no—
385C4—3881no—
386C4—3881no—
387C4—1942no—
388C4—1942no—
389C4—1942no—
390C2—42no—
391C2—3882no—
392C2#123×D9711yescharacteristic
393C2#2026124311yescharacteristic
394C2#2026126211yescharacteristic
395C2#2026126411yescharacteristic
396C2#2026126811yescharacteristic
397C2#2026126811yescharacteristic
398C2#2026128011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#397×C97C2
prime radicalC22×C22⋊D97C22
hypercenter#397D97
hypocenterC97#2395
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#15690766C2semidirect
3. #20260676C2semidirect
4. #20260670C2semidirect
5. #20260669C2semidirect
6. #20260664C2semidirect
7. #20260663C2semidirect
10. C4⋊C4×Dic97C2semidirect
11. #20260586C2semidirect
12. C22⋊C4×Dic97C2semidirect
13. #20260582C2semidirect
14. #20260582C2semidirect
15. #397×C97C2semidirect
200. C97#2395semidirect
208. #397D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907702
C2×#156907662
#202606762
#202606702
#202606692
#202606642
#202606632
#202606512
#202606502
C4⋊C4×Dic972
#202605862
C22⋊C4×Dic972
#202605822
#397×C972
#239597