Finite Group Databasebeta

← #27145023

Subgroups — #27145023

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic194C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D97C211yesmaximal, characteristic
3#20260677C211yesmaximal, characteristic, complemented
4#20260676C211yesmaximal, characteristic, complemented
5#20260664C211yesmaximal, characteristic, complemented
6#20260663C211yesmaximal, characteristic, complemented
7#20260651C211yesmaximal, characteristic
8#20260650C211yesmaximal, characteristic
9#20260648C211yesmaximal, characteristic, complemented
10#20260647C211yesmaximal, characteristic, complemented
11C4×#13095488C211yesmaximal, characteristic, complemented
12C4×#13095488C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
14#20260581C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C4×Dic194—24no—
20C4×C22⋊D97—24no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46C23×C4×C97C2211yescharacteristic
47C2×C22⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50C2×C42×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53C23×D97C2311yescharacteristic
54C2×Dic194—22no—
55C2×Dic194—22no—
56C2×Dic194—22no—
57C2×C22⋊D97—22no—
58C2×C22⋊D97—22no—
59C2×C22⋊D97—22no—
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—
74C22⋊Dic97—22no—
75C22⋊Dic97—22no—
76C22⋊Dic97—22no—
77#13095488—22no—
78#13095488—22no—
79#13095488—22no—
80#13095488—22no—
81#13095488—22no—
82C4×Dic97—22no—
83C4×Dic97—22no—
84C4×Dic97D414yes—
85C24×C97C2311yescharacteristic
86C22×C4×C97—22no—
87C22×C4×C97—22no—
88C22×C4×C97—22no—
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C22×C4×C97C2311yescharacteristic
94C22×C4×C97C2311yescharacteristic
95C22⋊C4×C97—22no—
96C22⋊C4×C97—22no—
97C42×C97—22no—
98C42×C97—22no—
99C23×C97—21no—
100C23×C97—21no—
101C23×C97—21no—
102C23×C97—21no—
103C23×C97—21no—
104C23×C97—21no—
105C23×C97#4112yes—
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—44no—
129C2×C4×C97C2×D412yes—
130C2×C4×C97#4112yes—
131C2×C4×C97#4112yes—
132C2×C4×C97#4112yes—
133C22⋊D97—28no—
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×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×Dic97C2×D412yes—
153C2×Dic97C2×D412yes—
154C2×Dic97#4112yes—
155C4×D97—44no—
156Dic194—28no—
157C22×C97—22no—
158C22×C97—22no—
159C22×C97—22no—
160C22×C97—41no—
161C22×C97—41no—
162C22×C97—41no—
163C22×C97—41no—
164C22×C97C22×D411yescharacteristic
165C22×C97C2×#4111yescharacteristic
166C22×C97C2×#4111yescharacteristic
167C22×C97C2×#4111yescharacteristic
168C22×C97C2×#4111yescharacteristic
169C22×C97C2×#4111yescharacteristic
170C22×C97C2×#4111yescharacteristic
171C2×D97—22no—
172C2×D97—22no—
173C2×D97—22no—
174C2×D97—41no—
175C2×D97—41no—
176C2×D97—41no—
177C2×D97—41no—
178C4×C97—22no—
179C4×C97—41no—
180C4×C97—41no—
181C4×C97—41no—
182C4×C97—41no—
183C4×C97#12414yes—
184Dic97—22no—
185Dic97—22no—
186Dic97—22no—
187Dic97—41no—
188Dic97—41no—
189Dic97—41no—
190Dic97—41no—
191C2×C97—42no—
192C2×C97#52511yescharacteristic
193C2×C97#52511yescharacteristic
194C2×C97C2×#12411yescharacteristic
195C2×C97#52911yescharacteristic
196C2×C97#52911yescharacteristic
197C2×C97#52911yescharacteristic
198C2×C97#52911yescharacteristic
199D97—42no—
200#2398—971nomaximal, Sylow
201C97#239811yescharacteristic, complemented, Sylow
202C2×C4×Q8—971no—
203C2×C4×D4—971no—
204#400—971no—
205#400—971no—
206#397—971no—
207#397—971no—
208#394—971no—
209#394—971no—
210#393—971no—
211#393—971no—
212C4×C22⋊C4—971no—
213C4×C22⋊C4—971no—
214C4×C22⋊C4—971no—
215C4×C22⋊C4D9711yescharacteristic, complemented
216#376—971no—
217C22×Q8—971no—
218C22×D4—971no—
219C23×C4—971no—
220C23×C4C2×D9711yescharacteristic
221C4×Q8—1944no—
222C4×D4—1944no—
223C2×C4⋊C4—971no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4—971no—
226C2×C4⋊C4—971no—
227C2×C22⋊C4—971no—
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×C42—971no—
238C2×C42—971no—
239C2×C42—971no—
240C2×C42C2×D9711yescharacteristic
241C2×C42C2×D9711yescharacteristic
242#95—971no—
243#95—971no—
244#95—971no—
245#95—971no—
246#95—971no—
247#95—971no—
248#95—971no—
249#95—971no—
250#95—971no—
251#95—971no—
252#95C2×D9711yescharacteristic
253#95C2×D9711yescharacteristic
254C24—971no—
255C24C22×D9711yescharacteristic
256C2×Q8—1942no—
257C2×Q8—1942no—
258C2×Q8—1942no—
259C2×D4—1942no—
260C2×D4—1942no—
261C2×D4—1942no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—971no—
270C22×C4—971no—
271C22×C4—971no—
272C22×C4—1942no—
273C22×C4—1942no—
274C22×C4—1942no—
275C22×C4C22×D9711yescharacteristic
276C22×C4C22×D9711yescharacteristic
277C22×C4C22×D9711yescharacteristic
278C22×C4C22×D9711yescharacteristic
279C22×C4C22×D9711yescharacteristic
280C22×C4C22×D9711yescharacteristic
281C4⋊C4—1942no—
282C4⋊C4—1942no—
283C4⋊C4—1942no—
284C4⋊C4—1942no—
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—
294C22⋊C4—1942no—
295C42—22no—
296C42—22no—
297C42—974no—
298C42—1942no—
299C42—1942no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—972no—
307C23—1941no—
308C23—1941no—
309C23—1941no—
310C23—1941no—
311C23—1941no—
312C23—1941no—
313C23#1309551112yes—
314C23C23×D9711yescharacteristic
315Q8—1948no—
316D4—1948no—
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—44no—
332C2×C4—972no—
333C2×C4—972no—
334C2×C4—972no—
335C2×C4—3884no—
336C2×C4—1941no—
337C2×C4—1941no—
338C2×C4—1941no—
339C2×C4—1941no—
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×C4D4×D9712yes—
355C2×C4#1309551012yes—
356C2×C4#1309551112yes—
357C2×C4#1309551112yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3881no—
366C22—3881no—
367C22—3881no—
368C22—3881no—
369C22—1942no—
370C22—1942no—
371C22—1942no—
372C22C2×D4×D9711yescharacteristic
373C22#41×D9711yescharacteristic
374C22#41×D9711yescharacteristic
375C22C2×#1309551011yescharacteristic
376C22C2×#1309551111yescharacteristic
377C22C2×#1309551111yescharacteristic
378C22C2×#1309551111yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3881no—
385C4—3881no—
386C4—3881no—
387C4—3881no—
388C4—1942no—
389C4—1942no—
390C4—1942no—
391C4#1569079614yes—
392C2—42no—
393C2—3882no—
394C2#2026124411yescharacteristic
395C2#2026124411yescharacteristic
396C2C2×#1569079611yescharacteristic
397C2#2026126311yescharacteristic
398C2#2026126511yescharacteristic
399C2#2026126711yescharacteristic
400C2#2026126911yescharacteristic
subgroupG/H
centerC22×C4C22×D97
commutator, socle, cosocleC23×C97C24
FrattiniC23C23×D97
FittingC4×C22⋊C4×C97C2
prime radicalC22×C22⋊D97C22
hypercenterC4×C22⋊C4D97
hypocenterC97#2398
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic194C2semidirect
3. #20260677C2semidirect
4. #20260676C2semidirect
5. #20260664C2semidirect
6. #20260663C2semidirect
9. #20260648C2semidirect
10. #20260647C2semidirect
11. C4×#13095488C2semidirect
12. C4×#13095488C2semidirect
13. C4×C22⋊Dic97C2semidirect
14. #20260581C2semidirect
15. C4×C22⋊C4×C97C2semidirect
201. C97#2398semidirect
215. C4×C22⋊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×C4×C22⋊D972
#202606772
#202606762
#202606642
#202606632
#202606512
#202606502
#202606482
#202606472
C4×#130954882
C4×C22⋊Dic972
#202605812
C4×C22⋊C4×C972
#239897