Finite Group Databasebeta

← #27155829

Subgroups — #27155829

Order 12416.

992 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261463C211yesmaximal, characteristic, complemented
2#20261462C212yesmaximal, complemented
3#20261458C211yesmaximal, characteristic, complemented
4#20261457C212yesmaximal, complemented
5#20261426C212yesmaximal, complemented
6#20261417C214yesmaximal, complemented
7#20261400C212yesmaximal, complemented
8#20261400C211yesmaximal, characteristic, complemented
9#20261346C212yesmaximal, complemented
10#20261344C214yesmaximal, complemented
11#20261290C211yesmaximal, characteristic, complemented
12#20261289C212yesmaximal, complemented
13#20261272C211yesmaximal, characteristic
14#20261270C212yesmaximal
15#20261211C212yesmaximal, complemented
16C4×Q8×D97C211yesmaximal, characteristic, complemented
17#557×C97C211yesmaximal, characteristic, complemented
18C2×Q8×D97C2211yescharacteristic
19C2×Q8×D97C2212yes—
20#15690810C2211yescharacteristic
21#15690809C2212yes—
22#15690809C2214yes—
23#15690808C2212yes—
24#15690805C2214yes—
25#15690805C2214yes—
26#15690805C2214yes—
27#15690805C2212yes—
28#15690804C2214yes—
29#15690802C2212yes—
30#15690801C2211yescharacteristic
31#15690801C2212yes—
32#15690800C2214yes—
33#15690800C2214yes—
34#15690800C2214yes—
35#15690800C2214yes—
36#15690799C2211yescharacteristic
37#15690798C2212yes—
38#15690795C2212yes—
39#15690795C2214yes—
40#15690790C2212yes—
41#15690788C2214yes—
42#15690788C2214yes—
43#15690788C2214yes—
44#15690786C2212yes—
45#15690786C2212yes—
46#15690786C2212yes—
47#15690786C2211yescharacteristic
48#15690785C2211yescharacteristic
49Q8×Dic97C2211yescharacteristic
50Q8×Dic97C2212yes—
51C4×Dic194C2211yescharacteristic
52C4×Dic194C2212yes—
53C4×Dic194C2212yes—
54C4×Dic194C2212yes—
55C4×Dic194C2214yes—
56#15690775C2214yes—
57#15690775C2211yescharacteristic
58#15690775C2212yes—
59#15690775C2212yes—
60#15690775C2212yes—
61C4×D388C2212yes—
62C4×D388C2211yescharacteristic
63#15690772C2214yes—
64#15690772C2212yes—
65#15690768C2212yes—
66#15690768C2212yes—
67#15690768C2214yes—
68#15690767C2214yes—
69#15690767C2214yes—
70#15690767C2212yes—
71C4⋊C4×D97C2212yes—
72C4⋊C4×D97C2211yescharacteristic
73C42×D97C2212yes—
74C42×D97C2211yescharacteristic
75C4⋊Q8×C97C2211yescharacteristic
76C4⋊Q8×C97C2212yes—
77#125×C97C2214yes—
78#125×C97C2212yes—
79C4×Q8×C97C2212yes—
80C4×Q8×C97C2211yescharacteristic
81C4×Q8×C97C2212yes—
82C4×Q8×C97C2211yescharacteristic
83C2×Dic194C2312yes—
84C2×Dic194C2314yes—
85C2×Dic194C2312yes—
86C2×Dic194C2311yescharacteristic
87Q8×D97—24no—
88Q8×D97—28no—
89C2×D388C2312yes—
90C2×D388C2311yescharacteristic
91C2×C4×D97C2311yescharacteristic
92C2×C4×D97C2312yes—
93C2×C4×D97C2311yescharacteristic
94C2×C4×D97C2312yes—
95C2×C4×D97C2311yescharacteristic
96C2×C4×D97C2314yes—
97C2×C4×D97C2312yes—
98C4⋊Dic97C2311yescharacteristic
99C4⋊Dic97C2312yes—
100C4⋊Dic97C2312yes—
101C4⋊Dic97C2314yes—
102C4⋊Dic97C2312yes—
103C4⋊Dic97C2311yescharacteristic
104C4⋊Dic97C2312yes—
105#13095490C2312yes—
106#13095490C2312yes—
107#13095490C2312yes—
108#13095490C2312yes—
109#13095490C2312yes—
110#13095490C2314yes—
111#13095490C2314yes—
112#13095490C2314yes—
113#13095490C2314yes—
114#13095490C2314yes—
115#13095490C2314yes—
116#13095488C2312yes—
117#13095488C2312yes—
118#13095488C2312yes—
119#13095488C2314yes—
120#13095488C2314yes—
121#13095488C2314yes—
122#13095488C2314yes—
123#13095488C2312yes—
124#13095488C2312yes—
125C4×Dic97C2311yescharacteristic
126C4×Dic97C2312yes—
127C4×Dic97C2312yes—
128C4×Dic97C2311yescharacteristic
129C4×Dic97C2311yescharacteristic
130C4×Dic97C2312yes—
131C4×Dic97C2314yes—
132C4×Dic97C2314yes—
133C4×Dic97C2314yes—
134C2×Q8×C97C2311yescharacteristic
135C2×Q8×C97C2312yes—
136C2×Q8×C97C2311yescharacteristic
137C4⋊C4×C97C2311yescharacteristic
138C4⋊C4×C97C2312yes—
139C4⋊C4×C97C2314yes—
140C4⋊C4×C97C2312yes—
141C4⋊C4×C97C2312yes—
142C4⋊C4×C97C2314yes—
143C4⋊C4×C97C2311yescharacteristic
144C4⋊C4×C97C2314yes—
145C4⋊C4×C97C2312yes—
146C42×C97C2311yescharacteristic
147C42×C97C2312yes—
148C42×C97C2314yes—
149C42×C97C2312yes—
150C22×D97C2412yes—
151C22×D97C2411yescharacteristic
152Q8×C97—22no—
153Q8×C97—24no—
154Q8×C97—22no—
155C2×C4×C97C2412yes—
156C2×C4×C97C2412yes—
157C2×C4×C97C2412yes—
158C2×C4×C97C2411yescharacteristic
159C2×C4×C97C2412yes—
160C2×C4×C97C2414yes—
161C2×C4×C97C2411yescharacteristic
162C2×C4×C97C2411yescharacteristic
163C2×Dic97C2411yescharacteristic
164C2×Dic97C2411yescharacteristic
165C2×Dic97C2414yes—
166C2×Dic97C2412yes—
167C2×Dic97C2411yescharacteristic
168C2×Dic97C2412yes—
169C2×Dic97C2412yes—
170D388—24no—
171D388—22no—
172C4×D97—24no—
173C4×D97—22no—
174C4×D97—22no—
175C4×D97—24no—
176C4×D97—22no—
177C4×D97—24no—
178C4×D97—28no—
179Dic194—24no—
180Dic194—24no—
181Dic194—28no—
182Dic194#4114yes—
183C22×C97C2511yescharacteristic
184C2×D97—21no—
185C2×D97—21no—
186C2×D97—22no—
187C2×D97—22no—
188C2×D97—22no—
189C2×D972−1+412yes—
190C4×C97—22no—
191C4×C97—24no—
192C4×C97—22no—
193C4×C97—21no—
194C4×C97—22no—
195C4×C97—22no—
196C4×C97C2×#4112yes—
197C4×C972−1+412yes—
198Dic97—21no—
199Dic97—22no—
200Dic97—24no—
201Dic97—21no—
202Dic97—22no—
203Dic97—21no—
204Dic97C2×#4114yes—
205C2×C97C22×#4111yescharacteristic
206C2×C97C2×2−1+411yescharacteristic
207C2×C97C2×2−1+411yescharacteristic
208D97—22no—
209D97—42no—
210#3523—971nomaximal, Sylow
211C97#352311yescharacteristic, complemented, Sylow
212#557—972no—
213#557—971no—
214#557D9711yescharacteristic, complemented
215#556—972no—
216#556—971no—
217#552—974no—
218#552—972no—
219D4×Q8—972no—
220D4×Q8—971no—
221#541—972no—
222#541—974no—
223#533—971no—
224#533—972no—
225#529—972no—
226#529—971no—
227#519—972no—
228C2×C4×Q8—971no—
229C22×Q8—971no—
230C22×Q8—972no—
231C4⋊Q8—972no—
232C4⋊Q8—972no—
233C4⋊Q8—974no—
234C4⋊Q8—971no—
235C4⋊Q8C2×D9711yescharacteristic
236C4⋊Q8C2×D9712yes—
237#126—972no—
238#126—974no—
239#126—974no—
240#126—974no—
241#126—974no—
242#125—971no—
243#125—974no—
244#125—972no—
245#125—974no—
246#125—974no—
247#125—974no—
248#125—972no—
249#125C2×D9712yes—
250#125C2×D9714yes—
251#124—971no—
252#124—972no—
253#123—972no—
254#123—974no—
255C22⋊Q8—971no—
256C22⋊Q8—972no—
257C22⋊Q8—972no—
258C22⋊Q8—972no—
259C22⋊Q8—972no—
260C22⋊Q8—974no—
261C22⋊Q8—974no—
262C22⋊Q8—974no—
263#121—971no—
264C4×Q8—971no—
265C4×Q8—972no—
266C4×Q8—971no—
267C4×Q8—972no—
268C4×Q8—974no—
269C4×Q8—972no—
270C4×Q8—972no—
271C4×Q8—971no—
272C4×Q8—972no—
273C4×Q8—974no—
274C4×Q8—972no—
275C4×Q8—972no—
276C4×Q8C2×D9711yescharacteristic
277C4×Q8C2×D9712yes—
278C4×Q8C2×D9711yescharacteristic
279C4×Q8C2×D9712yes—
280C4×D4—974no—
281C4×D4—972no—
282C4×D4—972no—
283C4×D4—971no—
284#117—974no—
285#117—972no—
286#117—972no—
287#117—972no—
288#117—974no—
289#117—974no—
290C2×C4⋊C4—972no—
291C2×C4⋊C4—971no—
292C2×C42—972no—
293C2×C42—971no—
294C2×Q8—971no—
295C2×Q8—972no—
296C2×Q8—972no—
297C2×Q8—974no—
298C2×Q8—1944no—
299C2×Q8—1948no—
300C2×Q8C22×D9711yescharacteristic
301C2×Q8C22×D9712yes—
302C2×Q8C22×D9711yescharacteristic
303C2×D4—972no—
304C2×D4—971no—
305C22×C4—971no—
306C22×C4—971no—
307C22×C4—972no—
308C22×C4—972no—
309C22×C4—971no—
310C22×C4—974no—
311C22×C4—972no—
312C4⋊C4—972no—
313C4⋊C4—972no—
314C4⋊C4—972no—
315C4⋊C4—974no—
316C4⋊C4—972no—
317C4⋊C4—972no—
318C4⋊C4—974no—
319C4⋊C4—974no—
320C4⋊C4—974no—
321C4⋊C4—974no—
322C4⋊C4—974no—
323C4⋊C4—971no—
324C4⋊C4—972no—
325C4⋊C4—972no—
326C4⋊C4—974no—
327C4⋊C4—972no—
328C4⋊C4—971no—
329C4⋊C4—972no—
330C4⋊C4C22×D9711yescharacteristic
331C4⋊C4C22×D9712yes—
332C4⋊C4C22×D9714yes—
333C4⋊C4C22×D9712yes—
334C4⋊C4C22×D9712yes—
335C4⋊C4C22×D9714yes—
336C4⋊C4C22×D9711yescharacteristic
337C4⋊C4C22×D9714yes—
338C4⋊C4C22×D9712yes—
339C22⋊C4—972no—
340C22⋊C4—972no—
341C22⋊C4—974no—
342C22⋊C4—974no—
343C22⋊C4—972no—
344C22⋊C4—974no—
345C22⋊C4—972no—
346C22⋊C4—974no—
347C22⋊C4—972no—
348C42—972no—
349C42—971no—
350C42—972no—
351C42—971no—
352C42—972no—
353C42—971no—
354C42—974no—
355C42—974no—
356C42—974no—
357C42C22×D9711yescharacteristic
358C42C22×D9712yes—
359C42C22×D9714yes—
360C42C22×D9712yes—
361C23—971no—
362C23—972no—
363Q8—22no—
364Q8—24no—
365Q8—22no—
366Q8—974no—
367Q8—1944no—
368Q8—1948no—
369Q8—1944no—
370D4—1944no—
371D4—1942no—
372C2×C4—972no—
373C2×C4—971no—
374C2×C4—971no—
375C2×C4—974no—
376C2×C4—972no—
377C2×C4—971no—
378C2×C4—972no—
379C2×C4—1948no—
380C2×C4—1944no—
381C2×C4—1944no—
382C2×C4—1942no—
383C2×C4—1944no—
384C2×C4—1942no—
385C2×C4—1942no—
386C2×C4C23×D9712yes—
387C2×C4C23×D9712yes—
388C2×C4C23×D9711yescharacteristic
389C2×C4C23×D9712yes—
390C2×C4C23×D9714yes—
391C2×C4C23×D9712yes—
392C2×C4C23×D9711yescharacteristic
393C2×C4C23×D9711yescharacteristic
394C22—972no—
395C22—1942no—
396C22—1942no—
397C22—1942no—
398C22—1941no—
399C22—1941no—
400C22C24×D9711yescharacteristic
401C4—24no—
402C4—22no—
403C4—22no—
404C4—21no—
405C4—22no—
406C4—22no—
407C4—974no—
408C4—1941no—
409C4—1942no—
410C4—1941no—
411C4—1942no—
412C4—1941no—
413C4—1944no—
414C4#41×D9712yes—
415C4#1569085812yes—
416C2—3882no—
417C2—1942no—
418C2C2×#41×D9711yescharacteristic
419C22−1+4×D9711yescharacteristic
420C2C2×#1569085811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#557×C97C2
prime radical#15690785C22
hypercenter#557D97
hypocenterC97#3523
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #20261463C2semidirect
2. #20261462C2semidirect
3. #20261458C2semidirect
4. #20261457C2semidirect
5. #20261426C2semidirect
6. #20261417C2semidirect
7. #20261400C2semidirect
8. #20261400C2semidirect
9. #20261346C2semidirect
10. #20261344C2semidirect
11. #20261290C2semidirect
12. #20261289C2semidirect
15. #20261211C2semidirect
16. C4×Q8×D97C2semidirect
17. #557×C97C2semidirect
211. C97#3523semidirect
214. #557D97semidirect

Maximal subgroups

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

subgroupindex
#202614632
#202614622
#202614582
#202614572
#202614262
#202614172
#202614002
#202613462
#202613442
#202612902
#202612892
#202612722
#202612702
#202612112
C4×Q8×D972
#557×C972
#352397