Finite Group Databasebeta

← #27144196

Subgroups — #27144196

Order 12416.

652 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690770C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3C2×#15690766C211yesmaximal, characteristic, complemented
4#20260635C211yesmaximal, characteristic
5#20260635C211yesmaximal, characteristic
6#20260632C211yesmaximal, characteristic, complemented
7#20260630C211yesmaximal, characteristic, complemented
8#20260628C211yesmaximal, characteristic, complemented
9#20260626C211yesmaximal, characteristic, complemented
10#20260624C211yesmaximal, characteristic, complemented
11#20260586C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
13#20260582C211yesmaximal, characteristic, complemented
14C42×Dic97C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19#15690770C418yescomplemented
20#15690768—24no—
21#15690766—24no—
22C2×C4⋊Dic97C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38C2×C4×Dic97C2211yescharacteristic
39C2×C4×Dic97C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41C2×C4×Dic97C2211yescharacteristic
42C2×C4×Dic97C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C42×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C2×C22⋊D97—22no—
56C2×C22⋊D97—22no—
57C2×C22⋊D97C2×C414yes—
58C22×Dic97—22no—
59C22×Dic97—22no—
60C22×Dic97C2×C414yes—
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C2×C4×D97—22no—
69C2×C4×D97—22no—
70C2×C4×D97C2×C414yes—
71C4⋊Dic97—22no—
72#13095490—22no—
73#13095490—22no—
74#13095490—22no—
75#13095490—22no—
76#13095490C2×C414yes—
77C22⋊Dic97—22no—
78C22⋊Dic97—22no—
79#13095488—22no—
80#13095488—22no—
81#13095488—22no—
82#13095488—22no—
83#13095488C2×C414yes—
84C4×Dic97—22no—
85C4×Dic97—22no—
86C4×Dic97—22no—
87C4×Dic97—22no—
88C4×Dic97—22no—
89C4×Dic97—22no—
90C4×Dic97C2×C414yes—
91C4×Dic97D414yescomplemented
92C24×C97C2311yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C22×C4×C97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97—22no—
100C22⋊C4×C97—22no—
101C22⋊C4×C97—22no—
102C22⋊C4×C97C2×C414yes—
103C42×C97—22no—
104C23×C97—21no—
105C23×C97—21no—
106C23×C97—21no—
107C23×C97—21no—
108C23×C97—21no—
109C23×C97—21no—
110C23×C97C22×C412yes—
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—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—21no—
132C2×C4×C97—21no—
133C2×C4×C97C22×C412yes—
134C2×C4×C97C22×C412yes—
135C2×C4×C97C2×D412yes—
136C2×C4×C97#4112yes—
137C22⋊D97—28no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—28no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—21no—
149C2×Dic97—21no—
150C2×Dic97—21no—
151C2×Dic97C22×C412yes—
152C2×Dic97C22×C412yes—
153C2×Dic97C22×C412yes—
154C2×Dic97C2×D412yes—
155C2×Dic97C2×D412yes—
156C2×Dic97#4112yes—
157C2×Dic97#4112yes—
158C2×Dic97#4112yes—
159C2×Dic97#4112yes—
160C4×D97—28no—
161C22×C97—22no—
162C22×C97—22no—
163C22×C97—22no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—41no—
167C22×C97—41no—
168C22×C97C23×C411yescharacteristic
169C22×C97C22×D411yescharacteristic
170C22×C97C2×#4111yescharacteristic
171C22×C97C2×#4111yescharacteristic
172C22×C97C2×#4111yescharacteristic
173C22×C97C2×#4111yescharacteristic
174C22×C97C2×#4111yescharacteristic
175C2×D97—22no—
176C2×D97—22no—
177C2×D97—22no—
178C2×D97—41no—
179C2×D97—41no—
180C2×D97—41no—
181C2×D97—41no—
182C4×C97—22no—
183C4×C97—22no—
184C4×C97—22no—
185C4×C97—22no—
186C4×C97—41no—
187C4×C97—41no—
188Dic97—22no—
189Dic97—22no—
190Dic97—22no—
191Dic97—41no—
192Dic97—41no—
193Dic97#11714yescomplemented
194Dic97C4×D414yes—
195C2×C97—42no—
196C2×C97C2×#11711yescharacteristic
197C2×C97C2×C4×D411yescharacteristic
198C2×C97C4×#4111yescharacteristic
199C2×C97#52911yescharacteristic
200C2×C97#52911yescharacteristic
201C2×C97#53211yescharacteristic
202C2×C97#53211yescharacteristic
203D97—42no—
204#2293—971nomaximal, Sylow
205C97#229311yescharacteristic, complemented, Sylow
206C2×C4×D4—971no—
207C2×#117—971no—
208C2×#117—971no—
209#390—971no—
210#390—971no—
211#389—971no—
212#389—971no—
213#388—971no—
214#388—971no—
215#388—971no—
216#388D9711yescharacteristic, complemented
217C4×C4⋊C4—971no—
218C4×C22⋊C4—971no—
219C4×C22⋊C4—971no—
220C43—971no—
221C22×D4—971no—
222C23×C4—971no—
223C23×C4—971no—
224C4×D4—978no—
225#117—1944no—
226#117—1944no—
227C2×C4⋊C4—971no—
228C2×C4⋊C4—971no—
229C2×C4⋊C4—971no—
230C2×C4⋊C4—971no—
231C2×C4⋊C4—971no—
232C2×C4⋊C4—971no—
233C2×C4⋊C4C2×D9711yescharacteristic
234C2×C22⋊C4—971no—
235C2×C22⋊C4—971no—
236C2×C22⋊C4—971no—
237C2×C22⋊C4—971no—
238C2×C22⋊C4—971no—
239C2×C22⋊C4—971no—
240C2×C22⋊C4—971no—
241C2×C22⋊C4C2×D9711yescharacteristic
242C2×C22⋊C4C2×D9711yescharacteristic
243C2×C22⋊C4C2×D9711yescharacteristic
244C2×C42—971no—
245C2×C42—971no—
246C2×C42—971no—
247C2×C42—971no—
248C2×C42—971no—
249C2×C42—971no—
250C2×C42—971no—
251C2×C42—971no—
252C2×C42C2×D9711yescharacteristic
253#95—971no—
254#95—971no—
255#95—971no—
256#95—971no—
257#95C2×D9711yescharacteristic
258#95C2×D9711yescharacteristic
259C24—971no—
260C24C22×D9711yescharacteristic
261C2×D4—974no—
262C2×D4—1942no—
263C2×D4—1942no—
264C22×C4—971no—
265C22×C4—971no—
266C22×C4—974no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—971no—
270C22×C4—974no—
271C22×C4—971no—
272C22×C4—971no—
273C22×C4—1942no—
274C22×C4—1942no—
275C22×C4—1942no—
276C22×C4—1942no—
277C22×C4C22×D9711yescharacteristic
278C22×C4C22×D9711yescharacteristic
279C22×C4C22×D9711yescharacteristic
280C22×C4C22×D9711yescharacteristic
281C22×C4C22×D9711yescharacteristic
282C22×C4C22×D9711yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—974no—
285C4⋊C4—1942no—
286C4⋊C4—1942no—
287C4⋊C4—1942no—
288C4⋊C4—1942no—
289C4⋊C4—1942no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—974no—
293C22⋊C4—1942no—
294C22⋊C4—1942no—
295C22⋊C4—1942no—
296C22⋊C4—1942no—
297C22⋊C4—1942no—
298C22⋊C4—1942no—
299C22⋊C4C4×D9714yes—
300C42—22no—
301C42—974no—
302C42—974no—
303C42—1942no—
304C42—1942no—
305C42—1942no—
306C42—1942no—
307C42—1942no—
308C42—1942no—
309C23—21no—
310C23—21no—
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—
323C23C23×D9711yescharacteristic
324D4—1948no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
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—972no—
340C2×C4—972no—
341C2×C4—972no—
342C2×C4—972no—
343C2×C4—972no—
344C2×C4—972no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—972no—
348C2×C4—1948no—
349C2×C4—1948no—
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×C4C2×C4×D9712yes—
363C2×C4C2×C4×D9712yes—
364C2×C4D4×D9712yes—
365C2×C4#1309551012yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—3881no—
374C22—3881no—
375C22—3881no—
376C22—3881no—
377C22—1942no—
378C22—1942no—
379C22—1942no—
380C22C22×C4×D9711yescharacteristic
381C22C2×D4×D9711yescharacteristic
382C22#41×D9711yescharacteristic
383C22#41×D9711yescharacteristic
384C22#41×D9711yescharacteristic
385C22#41×D9711yescharacteristic
386C22C2×#1309551011yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—974no—
394C4—974no—
395C4—3881no—
396C4—3881no—
397C4—1942no—
398C4—1942no—
399C4—1942no—
400C2—42no—
401C2—3882no—
402C2#117×D9711yescharacteristic
403C2C4×D4×D9711yescharacteristic
404C2C4×#1309551011yescharacteristic
405C2#2026126211yescharacteristic
406C2#2026126811yescharacteristic
407C2#2026128011yescharacteristic
408C2#2026128411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenter#388D97
hypocenterC97#2293
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. C2×#15690766C2semidirect
6. #20260632C2semidirect
7. #20260630C2semidirect
8. #20260628C2semidirect
9. #20260626C2semidirect
10. #20260624C2semidirect
11. #20260586C2semidirect
12. C22⋊C4×Dic97C2semidirect
13. #20260582C2semidirect
14. C42×Dic97C2semidirect
15. #388×C97C2semidirect
19. #15690770C4semidirect
91. C4×Dic97D4semidirect
193. Dic97#117semidirect
205. C97#2293semidirect
216. #388D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907702
C2×#156907682
C2×#156907662
#202606352
#202606322
#202606302
#202606282
#202606262
#202606242
#202605862
C22⋊C4×Dic972
#202605822
C42×Dic972
#388×C972
#229397