Finite Group Databasebeta

← #27150450

Subgroups — #27150450

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690818C211yesmaximal, characteristic, complemented
2C2×#15690816C211yesmaximal, characteristic, complemented
3#20261282C211yesmaximal, characteristic
4#20260809C211yesmaximal, characteristic, complemented
5#20260808C211yesmaximal, characteristic, complemented
6#20260807C211yesmaximal, characteristic, complemented
7#20260806C211yesmaximal, characteristic, complemented
8#20260805C211yesmaximal, characteristic, complemented
9#20260804C211yesmaximal, characteristic, complemented
10#20260803C211yesmaximal, characteristic, complemented
11#20260757C211yesmaximal, characteristic, complemented
12#20260700C211yesmaximal, characteristic, complemented
13#20260700C211yesmaximal, characteristic, complemented
14#97×D97C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095511C2211yescharacteristic
18#15690818—24no—
19#15690816—24no—
20C2×#13095495C2211yescharacteristic
21C2×#13095494C2211yescharacteristic
22C2×#13095494C2211yescharacteristic
23C2×#13095494C2211yescharacteristic
24C2×#13095494C2211yescharacteristic
25C22×#11715895C2211yescharacteristic
26C2×C8×D97C2211yescharacteristic
27C2×C8×D97C2211yescharacteristic
28C2×C8×D97C2211yescharacteristic
29C2×C8×D97C2211yescharacteristic
30#15690809C411yescharacteristic
31#15690806C411yescharacteristic
32#15690796C411yescharacteristic
33#15690796C411yescharacteristic
34#15690785C411yescharacteristic
35#15690783C411yescharacteristic
36#15690783C411yescharacteristic
37#15690781C411yescharacteristic
38C4×D388C2211yescharacteristic
39C4×C22⋊D97C2211yescharacteristic
40C4×C22⋊D97C2211yescharacteristic
41C42×D97C2211yescharacteristic
42C4⋊#11715895C2211yescharacteristic
43#15690739C2211yescharacteristic
44#15690739C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47#15690722C2211yescharacteristic
48#15690722C2211yescharacteristic
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690721C2211yescharacteristic
52#15690720C2211yescharacteristic
53#15690720C2211yescharacteristic
54C2×M16×C97C2211yescharacteristic
55C22×C8×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic
57C4⋊C8×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59C22⋊C8×C97C2211yescharacteristic
60#97×C97C2211yescharacteristic
61#13095511—22no—
62#13095511—22no—
63#13095511—22no—
64#13095511—22no—
65C2×Dic194C2×C411yescharacteristic
66C2×Dic194C2×C411yescharacteristic
67C2×D388C2×C411yescharacteristic
68C2×D388C2×C411yescharacteristic
69C2×D388C2×C411yescharacteristic
70C2×C22⋊D97C2×C411yescharacteristic
71C2×C22⋊D97C2×C411yescharacteristic
72C2×C22⋊D97C2×C411yescharacteristic
73C2×C22⋊D97C2×C411yescharacteristic
74C2×C22⋊D97C2×C411yescharacteristic
75C2×C22⋊D97C2×C411yescharacteristic
76C2×C4×D97C2×C411yescharacteristic
77C2×C4×D97C2×C411yescharacteristic
78C2×C4×D97C2311yescharacteristic
79C2×C4×D97C2311yescharacteristic
80C2×C4×D97C2311yescharacteristic
81#13095495—22no—
82#13095494—22no—
83#13095494—22no—
84#13095494—22no—
85#13095494—22no—
86C2×#11715895—22no—
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C2×#11715895C2311yescharacteristic
90C2×#11715895C2311yescharacteristic
91C8×D97—22no—
92C8×D97—22no—
93C8×D97—22no—
94C8×D97D414yescomplemented
95C4⋊Dic97C2×C411yescharacteristic
96#13095490C2×C411yescharacteristic
97#13095490C2×C411yescharacteristic
98C22⋊Dic97C2×C411yescharacteristic
99C22⋊Dic97C2×C411yescharacteristic
100#13095488C2×C411yescharacteristic
101#13095488C2×C411yescharacteristic
102#13095488C2×C411yescharacteristic
103#13095488C2×C411yescharacteristic
104C4×Dic97C2×C411yescharacteristic
105C4×Dic97C2×C411yescharacteristic
106C4×Dic97C2311yescharacteristic
107C2×D4×C97C2×C411yescharacteristic
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110M16×C97—22no—
111C2×C8×C97—22no—
112C2×C8×C97C2311yescharacteristic
113C2×C8×C97C2311yescharacteristic
114C2×C8×C97C2311yescharacteristic
115C2×C8×C97C2311yescharacteristic
116C4⋊C4×C97C2×C411yescharacteristic
117C22⋊C4×C97C2×C411yescharacteristic
118C22⋊C4×C97C2×C411yescharacteristic
119C42×C97C2311yescharacteristic
120C23×C97C22×C411yescharacteristic
121C23×C97C22×C411yescharacteristic
122C22×D97C22×C411yescharacteristic
123C22×D97C22×C411yescharacteristic
124C22×D97C22×C411yescharacteristic
125D4×C97—41no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97C22×C411yescharacteristic
131C2×C4×C97C22×C411yescharacteristic
132C2×C4×C97C22×C411yescharacteristic
133C2×C4×C97C22×C411yescharacteristic
134C2×C4×C97C2411yescharacteristic
135C22⋊D97—22no—
136C22⋊D97—22no—
137C22⋊D97—22no—
138C22⋊D97—22no—
139C22⋊D97—41no—
140C22⋊D97—41no—
141C2×Dic97C22×C411yescharacteristic
142C2×Dic97C22×C411yescharacteristic
143C2×Dic97C22×C411yescharacteristic
144C2×Dic97C22×C411yescharacteristic
145C2×Dic97C22×C411yescharacteristic
146D388—22no—
147D388—22no—
148D388—41no—
149C4×D97—21no—
150C4×D97—21no—
151C4×D97—21no—
152C4×D97—21no—
153C4×D97—41no—
154C4×D97—41no—
155C4×D97C2×D412yes—
156C4×D97#4112yes—
157Dic194—22no—
158Dic194—22no—
159C8×C97—21no—
160C8×C97—21no—
161C8×C97—21no—
162C8×C97C2×D412yes—
163#11715895—21no—
164#11715895—21no—
165#11715895—21no—
166#11715895C2×D412yes—
167C22×C97—21no—
168C22×C97—21no—
169C22×C97—21no—
170C22×C97—21no—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97C23×C411yescharacteristic
174C2×D97—21no—
175C2×D97—21no—
176C2×D97—21no—
177C2×D97—21no—
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—21no—
181C2×D97—21no—
182C2×D97C4×D412yes—
183C4×C97—21no—
184C4×C97—21no—
185C4×C97—21no—
186C4×C97—21no—
187C4×C97C22×D411yescharacteristic
188C4×C97C2×#4111yescharacteristic
189Dic97—21no—
190Dic97—21no—
191Dic97—21no—
192Dic97—21no—
193Dic97C4×D412yes—
194C2×C97—41no—
195C2×C97—41no—
196C2×C97C2×C4×D411yescharacteristic
197C2×C97#56811yescharacteristic
198C2×C97#56811yescharacteristic
199D97—22no—
200D97—41no—
201D97—41no—
202#2932—971nomaximal, Sylow
203C97#293211yescharacteristic, complemented, Sylow
204C2×#131—971no—
205C2×#131—971no—
206#532—971no—
207#435—971no—
208#435—971no—
209#435—971no—
210#435—971no—
211#435—971no—
212#435—971no—
213#435—971no—
214#435D9711yescharacteristic, complemented
215#423—971no—
216#408—971no—
217#408—971no—
218C2×#97—971no—
219C2×#41—971no—
220C2×#41—971no—
221#131—1944no—
222#131—1944no—
223C2×M16—971no—
224C2×M16—971no—
225C2×M16—971no—
226C2×M16—971no—
227C2×M16—971no—
228C2×M16C2×D9711yescharacteristic
229C22×C8—971no—
230C22×C8—971no—
231C22×C8—971no—
232C22×C8—971no—
233C22×C8—971no—
234C22×C8C2×D9711yescharacteristic
235C4⋊Q8—971no—
236#127—971no—
237#124—971no—
238#124—971no—
239#121—971no—
240#121—971no—
241#121—971no—
242#121—971no—
243C4×D4—971no—
244C4×D4—971no—
245C4×D4—971no—
246C4×D4C2×D9711yescharacteristic
247C2×C42—971no—
248C4⋊C8—971no—
249C4⋊C8—971no—
250C4⋊C8—971no—
251C4⋊C8C2×D9711yescharacteristic
252C22⋊C8—971no—
253C22⋊C8—971no—
254C22⋊C8—971no—
255C22⋊C8—971no—
256C22⋊C8—971no—
257C22⋊C8—971no—
258C22⋊C8C2×D9711yescharacteristic
259C22⋊C8C2×D9711yescharacteristic
260#97—971no—
261#97—971no—
262#97—971no—
263#97C2×D9711yescharacteristic
264#41—1942no—
265#41—1942no—
266#41—1942no—
267#41—1942no—
268C2×Q8—971no—
269C2×Q8—971no—
270C2×D4—971no—
271C2×D4—971no—
272C2×D4—971no—
273C2×D4—971no—
274C2×D4—971no—
275C2×D4—971no—
276C2×D4—971no—
277C2×D4—971no—
278C2×D4—971no—
279C2×D4C4×D9711yescharacteristic
280C22×C4—971no—
281C22×C4—971no—
282C22×C4—971no—
283C22×C4—971no—
284C22×C4—971no—
285C22×C4C22×D9711yescharacteristic
286C22×C4C22×D9711yescharacteristic
287M16—22no—
288M16—1942no—
289M16—1942no—
290M16—1942no—
291M16—1942no—
292M16—1942no—
293C2×C8—22no—
294C2×C8—971no—
295C2×C8—974no—
296C2×C8—971no—
297C2×C8—971no—
298C2×C8—971no—
299C2×C8—1942no—
300C2×C8—1942no—
301C2×C8—1942no—
302C2×C8—1942no—
303C2×C8C22×D9711yescharacteristic
304C2×C8C22×D9711yescharacteristic
305C2×C8C22×D9711yescharacteristic
306C2×C8C22×D9711yescharacteristic
307C4⋊C4—971no—
308C4⋊C4—971no—
309C4⋊C4—971no—
310C4⋊C4C4×D9711yescharacteristic
311C22⋊C4—971no—
312C22⋊C4—971no—
313C22⋊C4—971no—
314C22⋊C4—971no—
315C22⋊C4—971no—
316C22⋊C4—971no—
317C22⋊C4C4×D9711yescharacteristic
318C22⋊C4C4×D9711yescharacteristic
319C42—971no—
320C42—971no—
321C42—971no—
322C42C22×D9711yescharacteristic
323C23—971no—
324C23—971no—
325C23—971no—
326C23C2×C4×D9711yescharacteristic
327C23C2×C4×D9711yescharacteristic
328Q8—1942no—
329Q8—1942no—
330D4—41no—
331D4—3881no—
332D4—3881no—
333D4—3881no—
334D4—1942no—
335D4—1942no—
336D4—1942no—
337D4—1942no—
338D4—1942no—
339D4—1942no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—971no—
345C2×C4—972no—
346C2×C4—972no—
347C2×C4—971no—
348C2×C4—971no—
349C2×C4—971no—
350C2×C4—971no—
351C2×C4—3881no—
352C2×C4—3881no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4C2×C4×D9711yescharacteristic
358C2×C4C2×C4×D9711yescharacteristic
359C2×C4C2×C4×D9711yescharacteristic
360C2×C4C2×C4×D9711yescharacteristic
361C2×C4C23×D9711yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8—972no—
366C8—1941no—
367C8—1941no—
368C8—1941no—
369C8D4×D9712yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—972no—
377C22—1941no—
378C22—1941no—
379C22—1941no—
380C22—1941no—
381C22—1941no—
382C22—1941no—
383C22—1941no—
384C22—1941no—
385C22C22×C4×D9711yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—972no—
391C4—1941no—
392C4—1941no—
393C4—1941no—
394C4—1941no—
395C4C2×D4×D9711yescharacteristic
396C4#41×D9711yescharacteristic
397C2—41no—
398C2—41no—
399C2—3881no—
400C2—3881no—
401C2—1942no—
402C2C4×D4×D9711yescharacteristic
403C2#2026150011yescharacteristic
404C2#2026150111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#435×C97C2
prime radical#20261282C2
cosocleC2×C4×C97C24
hypercenter#435D97
hypocenterC97#2932
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15690818C2semidirect
2. C2×#15690816C2semidirect
4. #20260809C2semidirect
5. #20260808C2semidirect
6. #20260807C2semidirect
7. #20260806C2semidirect
8. #20260805C2semidirect
9. #20260804C2semidirect
10. #20260803C2semidirect
11. #20260757C2semidirect
12. #20260700C2semidirect
13. #20260700C2semidirect
14. #97×D97C2semidirect
15. #435×C97C2semidirect
94. C8×D97D4semidirect
203. C97#2932semidirect
214. #435D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908182
C2×#156908162
#202612822
#202608092
#202608082
#202608072
#202608062
#202608052
#202608042
#202608032
#202607572
#202607002
#97×D972
#435×C972
#293297