Finite Group Databasebeta

← #27144724

Subgroups — #27144724

Order 12416.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690793C211yesmaximal, characteristic
2C2×C4×C22⋊D97C211yesmaximal, characteristic
3C2×#15690769C211yesmaximal, characteristic, complemented
4#20260632C211yesmaximal, characteristic, complemented
5#20260627C211yesmaximal, characteristic, complemented
6#20260624C211yesmaximal, characteristic, complemented
7#20260622C211yesmaximal, characteristic, complemented
8#20260617C211yesmaximal, characteristic, complemented
9#20260614C211yesmaximal, characteristic
10#20260609C211yesmaximal, characteristic, complemented
11#20260608C211yesmaximal, characteristic, complemented
12#20260586C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
14#20260582C211yesmaximal, characteristic, complemented
15C2×#95×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18#15690793C418yescomplemented
19C4×C22⋊D97—24no—
20#15690769—24no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×C4⋊Dic97C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38C2×C4×Dic97C2211yescharacteristic
39C2×C4×Dic97C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C23×C4×C97C2211yescharacteristic
48C23×C4×C97C2211yescharacteristic
49C23×C4×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C2×C22⋊D97—22no—
56C2×C22⋊D97—22no—
57C2×C22⋊D97C2×C414yes—
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C2×C4×D97—22no—
66C2×C4×D97—22no—
67C2×C4×D97—41no—
68C4⋊Dic97—22no—
69C4⋊Dic97—22no—
70#13095490—22no—
71#13095490—22no—
72#13095490—41no—
73#13095490C2×C414yescomplemented
74#13095490C2×C414yescomplemented
75C22⋊Dic97—22no—
76C22⋊Dic97C2×C414yescomplemented
77C22⋊Dic97D414yes—
78#13095488—22no—
79#13095488—22no—
80#13095488—41no—
81#13095488C2×C414yescomplemented
82#13095488C2×C414yescomplemented
83C4×Dic97—22no—
84C4×Dic97—22no—
85C4×Dic97—22no—
86C4×Dic97—22no—
87C4×Dic97—41no—
88C24×C97C2311yescharacteristic
89C22×C4×C97—22no—
90C22×C4×C97—22no—
91C22×C4×C97—22no—
92C22×C4×C97—22no—
93C22×C4×C97—22no—
94C22×C4×C97—22no—
95C22×C4×C97—41no—
96C22×C4×C97—41no—
97C22×C4×C97C2×C414yescomplemented
98C22×C4×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C23×C97—21no—
105C23×C97—21no—
106C23×C97—21no—
107C23×C97—21no—
108C23×C97C22×C412yes—
109C23×C97C2×D412yes—
110C23×C97#4112yes—
111C23×C97C2411yescharacteristic
112C22×D97—21no—
113C22×D97—21no—
114C22×D97—21no—
115C22×D97—21no—
116C22×D97—21no—
117C22×D97—21no—
118C22×D97C22×C412yes—
119C2×C4×C97—28no—
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—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—44no—
135C2×C4×C97—44no—
136C2×C4×C97C22×C412yes—
137C2×C4×C97C22×C412yes—
138C2×C4×C97#4112yes—
139C2×C4×C97#4112yes—
140C22⋊D97—28no—
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×Dic97—21no—
150C2×Dic97—21no—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
157C2×Dic97C22×C412yes—
158C2×Dic97C22×C412yes—
159C2×Dic97C22×C412yes—
160C2×Dic97C2×D412yes—
161C2×Dic97C2×D412yes—
162C4×D97—44no—
163C22×C97—22no—
164C22×C97—22no—
165C22×C97—22no—
166C22×C97—22no—
167C22×C97—22no—
168C22×C97—22no—
169C22×C97C4×D414yes—
170C22×C97C23×C411yescharacteristic
171C22×C97C22×D411yescharacteristic
172C22×C97C2×#4111yescharacteristic
173C22×C97C2×#4111yescharacteristic
174C22×C97C2×#4111yescharacteristic
175C22×C97H(5,𝔽2)11yescharacteristic
176C22×C972−1+411yescharacteristic
177C2×D97—22no—
178C2×D97—22no—
179C2×D97—22no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—41no—
184C4×C97—22no—
185C4×C97—22no—
186C4×C97—22no—
187C4×C97—22no—
188C4×C97—41no—
189C4×C97—41no—
190Dic97—22no—
191Dic97—22no—
192Dic97—22no—
193Dic97—22no—
194Dic97—22no—
195Dic97—41no—
196Dic97—41no—
197C2×C97—24no—
198C2×C97C2×C4×D411yescharacteristic
199C2×C97C4×#4111yescharacteristic
200C2×C97#52011yescharacteristic
201C2×C97#54611yescharacteristic
202C2×C97#54711yescharacteristic
203C2×C97#55111yescharacteristic
204C2×C97#55211yescharacteristic
205D97—42no—
206#2359—971nomaximal, Sylow
207C97#235911yescharacteristic, complemented, Sylow
208C2×#123—971no—
209C2×C4×D4—971no—
210C2×#117—971no—
211#389—971no—
212#388—971no—
213#388—971no—
214#387—971no—
215#387—971no—
216#386—971no—
217#385—971no—
218#385—971no—
219C4×C4⋊C4—971no—
220C4×C22⋊C4—971no—
221C4×C22⋊C4—971no—
222C2×#95D9711yescharacteristic, complemented
223C22×D4—971no—
224C23×C4—971no—
225C23×C4C2×D9711yescharacteristic
226C23×C4C2×D9711yescharacteristic
227C23×C4C2×D9711yescharacteristic
228#123—978no—
229C4×D4—1944no—
230#117—1944no—
231C2×C4⋊C4—971no—
232C2×C4⋊C4—971no—
233C2×C4⋊C4—971no—
234C2×C4⋊C4—971no—
235C2×C4⋊C4—971no—
236C2×C4⋊C4—971no—
237C2×C4⋊C4—971no—
238C2×C22⋊C4—971no—
239C2×C22⋊C4—971no—
240C2×C22⋊C4—971no—
241C2×C22⋊C4—971no—
242C2×C22⋊C4—971no—
243C2×C22⋊C4—971no—
244C2×C22⋊C4—971no—
245C2×C22⋊C4—971no—
246C2×C42—971no—
247C2×C42—971no—
248C2×C42—971no—
249C2×C42—971no—
250C2×C42—971no—
251#95—971no—
252#95—971no—
253#95—971no—
254#95—971no—
255#95—971no—
256#95—971no—
257#95C2×D9711yescharacteristic
258#95C2×D9711yescharacteristic
259#95C2×D9711yescharacteristic
260#95C2×D9711yescharacteristic
261C24—971no—
262C24C22×D9711yescharacteristic
263C2×D4—974no—
264C2×D4—1942no—
265C2×D4—1942no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—971no—
275C22×C4—971no—
276C22×C4—971no—
277C22×C4—971no—
278C22×C4—971no—
279C22×C4—971no—
280C22×C4—971no—
281C22×C4—3881no—
282C22×C4—1942no—
283C22×C4—1942no—
284C22×C4C4×D9714yescomplemented
285C22×C4C22×D9711yescharacteristic
286C22×C4C22×D9711yescharacteristic
287C22×C4C22×D9711yescharacteristic
288C22×C4C22×D9711yescharacteristic
289C22×C4C22×D9711yescharacteristic
290C22×C4C22×D9711yescharacteristic
291C4⋊C4—974no—
292C4⋊C4—974no—
293C4⋊C4—3881no—
294C4⋊C4—1942no—
295C4⋊C4—1942no—
296C4⋊C4—1942no—
297C4⋊C4—1942no—
298C22⋊C4—974no—
299C22⋊C4—974no—
300C22⋊C4—974no—
301C22⋊C4—974no—
302C22⋊C4—3881no—
303C22⋊C4—1942no—
304C22⋊C4—1942no—
305C22⋊C4—1942no—
306C42—3881no—
307C42—1942no—
308C42—1942no—
309C42—1942no—
310C42—1942no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—972no—
316C23—1941no—
317C23—1941no—
318C23—1941no—
319C23—1941no—
320C23—1941no—
321C23—1941no—
322C23C2×C4×D9712yes—
323C23D4×D9712yes—
324C23#1309551012yes—
325C23C23×D9711yescharacteristic
326D4—1948no—
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—972no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—972no—
348C2×C4—972no—
349C2×C4—3884no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4—1941no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4—1941no—
358C2×C4—1941no—
359C2×C4—1941no—
360C2×C4—1941no—
361C2×C4—1941no—
362C2×C4—1941no—
363C2×C4—1941no—
364C2×C4—1941no—
365C2×C4—1941no—
366C2×C4C2×C4×D9712yes—
367C2×C4C2×C4×D9712yes—
368C2×C4#1309551112yes—
369C2×C4#1309551112yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—3881no—
377C22—3881no—
378C22—3881no—
379C22—3881no—
380C22—1942no—
381C22—1942no—
382C22—1942no—
383C22#1569077014yes—
384C22C22×C4×D9711yescharacteristic
385C22C2×D4×D9711yescharacteristic
386C22C2×#1309551011yescharacteristic
387C22C2×#1309551111yescharacteristic
388C22C2×#1309551111yescharacteristic
389C22#1569085511yescharacteristic
390C22#1569085811yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—3881no—
398C4—3881no—
399C4—1942no—
400C4—1942no—
401C4—1942no—
402C4—1942no—
403C4—1942no—
404C2—24no—
405C2—3882no—
406C2C2×#1569077011yescharacteristic
407C2C4×#1309551111yescharacteristic
408C2#2026121611yescharacteristic
409C2#2026137511yescharacteristic
410C2#2026138611yescharacteristic
411C2#2026141111yescharacteristic
412C2#2026142011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC2×#95×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenterC2×#95D97
hypocenterC97#2359
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
3. C2×#15690769C2semidirect
4. #20260632C2semidirect
5. #20260627C2semidirect
6. #20260624C2semidirect
7. #20260622C2semidirect
8. #20260617C2semidirect
10. #20260609C2semidirect
11. #20260608C2semidirect
12. #20260586C2semidirect
13. C4×C22⋊Dic97C2semidirect
14. #20260582C2semidirect
15. C2×#95×C97C2semidirect
18. #15690793C4semidirect
73. #13095490C2×C4semidirect
74. #13095490C2×C4semidirect
76. C22⋊Dic97C2×C4semidirect
81. #13095488C2×C4semidirect
82. #13095488C2×C4semidirect
97. C22×C4×C97C2×C4semidirect
207. C97#2359semidirect
222. C2×#95D97semidirect
284. C22×C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907932
C2×C4×C22⋊D972
C2×#156907692
#202606322
#202606272
#202606242
#202606222
#202606172
#202606142
#202606092
#202606082
#202605862
C4×C22⋊Dic972
#202605822
C2×#95×C972
#235997