Finite Group Databasebeta

← #27153292

Subgroups — #27153292

Order 12416.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690841C211yesmaximal, characteristic, complemented
2C2×#15690830C211yesmaximal, characteristic, complemented
3#20261390C211yesmaximal, characteristic, complemented
4#20261130C211yesmaximal, characteristic, complemented
5#20260979C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260947C211yesmaximal, characteristic, complemented
8#20260945C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260895C211yesmaximal, characteristic, complemented
11#20260891C211yesmaximal, characteristic, complemented
12#20260878C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095511C2211yescharacteristic
18#15690841—24no—
19#15690830—24no—
20C2×#13095501C2211yescharacteristic
21C2×#13095501C2211yescharacteristic
22C2×#13095501C2211yescharacteristic
23C2×#13095501C2211yescharacteristic
24C2×#13095499C2211yescharacteristic
25C2×#13095499C2211yescharacteristic
26C2×#13095498C2211yescharacteristic
27C2×#13095498C2211yescharacteristic
28C2×#13095496C2211yescharacteristic
29C2×#13095495C2211yescharacteristic
30C22×#11715895C2211yescharacteristic
31#15690810C2211yescharacteristic
32#15690792—21no—
33#15690792—21no—
34#15690789C2211yescharacteristic
35#15690789C2211yescharacteristic
36#15690781—21no—
37C4×D388C2211yescharacteristic
38C4⋊C4×D97—21no—
39#15690743C2211yescharacteristic
40C4⋊#11715895C2211yescharacteristic
41#15690734C2211yescharacteristic
42#15690734C2211yescharacteristic
43#15690733C2211yescharacteristic
44#15690733C2211yescharacteristic
45#15690731C2211yescharacteristic
46#15690730C2211yescharacteristic
47C22⋊#11715895C2211yescharacteristic
48C22⋊#11715895C2211yescharacteristic
49#15690721C2211yescharacteristic
50C2×#41×C97C2211yescharacteristic
51C22×Q8×C97C2211yescharacteristic
52#124×C97—21no—
53#124×C97C2211yescharacteristic
54C22⋊Q8×C97—21no—
55C22⋊Q8×C97C2211yescharacteristic
56#121×C97—21no—
57#121×C97C2211yescharacteristic
58C4×Q8×C97C2211yescharacteristic
59C4×D4×C97—21no—
60C4×D4×C97C2211yescharacteristic
61#13095511—22no—
62#13095511—22no—
63#13095511—41no—
64#13095511—41no—
65C2×Dic194C2311yescharacteristic
66C2×Dic194C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73#13095501—22no—
74#13095501—22no—
75#13095501—22no—
76#13095501—41no—
77#13095499—22no—
78#13095499—41no—
79#13095498—22no—
80#13095498D414yescomplemented
81#13095496—22no—
82#13095495—22no—
83C2×#11715895—22no—
84C2×#11715895C2311yescharacteristic
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C4⋊Dic97—21no—
89C4⋊Dic97—21no—
90C4⋊Dic97C2311yescharacteristic
91#13095490—21no—
92#13095490—21no—
93C22⋊Dic97—21no—
94C22⋊Dic97—21no—
95#13095488—21no—
96#41×C97—22no—
97#41×C97—41no—
98C2×Q8×C97—21no—
99C2×Q8×C97—22no—
100C2×Q8×C97—41no—
101C2×Q8×C97C2311yescharacteristic
102C2×Q8×C97C2311yescharacteristic
103C2×D4×C97—21no—
104C2×D4×C97—21no—
105C2×D4×C97D411yescharacteristic
106C2×D4×C97C2311yescharacteristic
107C22×C4×C97—21no—
108C22×C4×C97—21no—
109C22×C4×C97C2311yescharacteristic
110C22×C4×C97C2311yescharacteristic
111C4⋊C4×C97—21no—
112C4⋊C4×C97—21no—
113C4⋊C4×C97D411yescharacteristic
114C4⋊C4×C97C2311yescharacteristic
115C22⋊C4×C97—21no—
116C22⋊C4×C97—21no—
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C22⋊C4×C97D411yescharacteristic
120C22⋊C4×C97D411yescharacteristic
121C42×C97—21no—
122C42×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97C2×D411yescharacteristic
125C23×C97C2×D411yescharacteristic
126C22×D97—21no—
127Q8×C97—21no—
128Q8×C97—21no—
129Q8×C97—21no—
130Q8×C97—42no—
131Q8×C97C2×D412yes—
132D4×C97—21no—
133D4×C97—21no—
134D4×C97—41no—
135D4×C97—42no—
136D4×C97—42no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—42no—
145C2×C4×C97—42no—
146C2×C4×C97C2×D411yescharacteristic
147C2×C4×C97C2×D411yescharacteristic
148C2×C4×C97C2×D411yescharacteristic
149C2×C4×C97C2×D411yescharacteristic
150C2×C4×C97C2411yescharacteristic
151C22⋊D97—42no—
152C22⋊D97—42no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156D388—21no—
157D388C2×D412yes—
158C4×D97—42no—
159C4×D97—42no—
160C4×D97—42no—
161Dic194—21no—
162Dic194—21no—
163Dic194—21no—
164Dic194—21no—
165#11715895—21no—
166#11715895—21no—
167#11715895—21no—
168#11715895C2×D412yes—
169C22×C97—21no—
170C22×C97—21no—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—41no—
176C22×C97—41no—
177C22×C97—41no—
178C22×C97C22×D411yescharacteristic
179C2×D97—22no—
180C2×D97—41no—
181C2×D97—41no—
182C4×C97—21no—
183C4×C97—21no—
184C4×C97—22no—
185C4×C97—21no—
186C4×C97—21no—
187C4×C97—41no—
188C4×C97—41no—
189C4×C97C22×D411yescharacteristic
190C4×C97H(5,𝔽2)11yescharacteristic
191Dic97—41no—
192Dic97—41no—
193Dic97—41no—
194C2×C97—41no—
195C2×C97—41no—
196C2×C97—81no—
197C2×C97D4211yescharacteristic
198C2×C97#57511yescharacteristic
199C2×C97#57811yescharacteristic
200D97—42no—
201#3269—971nomaximal, Sylow
202C97#326911yescharacteristic, complemented, Sylow
203C2×#137—971no—
204C2×#135—971no—
205#548D9711yescharacteristic, complemented
206#547—971no—
207#497—971no—
208#468—971no—
209#467—971no—
210#463—971no—
211#462—971no—
212#452—971no—
213#452—971no—
214#451—971no—
215#449—971no—
216#440—971no—
217#435—971no—
218C2×#41—971no—
219C2×#41—971no—
220C2×#41C2×D9711yescharacteristic
221C22×Q8C2×D9711yescharacteristic
222#137—1944no—
223#135—1944no—
224C2×Q16—971no—
225C2×Q16—971no—
226C2×Q16—971no—
227C2×Q16—971no—
228C2×SD16—971no—
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×D8—971no—
233C2×M16—971no—
234C22×C8—971no—
235C4⋊Q8—971no—
236#124—21no—
237#124C2×D9711yescharacteristic
238#123—1941no—
239#123—1941no—
240C22⋊Q8—21no—
241C22⋊Q8—971no—
242C22⋊Q8—971no—
243C22⋊Q8C2×D9711yescharacteristic
244#121—21no—
245#121—1941no—
246#121C2×D9711yescharacteristic
247C4×Q8C2×D9711yescharacteristic
248C4×D4—21no—
249C4×D4—971no—
250C4×D4C2×D9711yescharacteristic
251C2×C4⋊C4—1941no—
252#107—971no—
253C4⋊C8—971no—
254#103—971no—
255#103—971no—
256#103—971no—
257#103—971no—
258#102—971no—
259#102—971no—
260C22⋊C8—971no—
261C22⋊C8—971no—
262#97—971no—
263#41—22no—
264#41—41no—
265#41—3881no—
266#41—3881no—
267#41—1942no—
268#41—1942no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—971no—
273C2×Q8—971no—
274C2×Q8C22×D9711yescharacteristic
275C2×Q8C22×D9711yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—971no—
279C2×D4—1941no—
280C2×D4—1941no—
281C2×D4C22⋊D9711yescharacteristic
282C2×D4C22×D9711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—1941no—
286C22×C4—1941no—
287C22×C4—1941no—
288C22×C4C22×D9711yescharacteristic
289C22×C4C22×D9711yescharacteristic
290Q16—3881no—
291Q16—1942no—
292Q16—1942no—
293Q16—1942no—
294SD16—974no—
295SD16—3881no—
296SD16—1942no—
297SD16—1942no—
298D8—1942no—
299M16—1942no—
300C2×C8—971no—
301C2×C8—971no—
302C2×C8—971no—
303C2×C8—971no—
304C2×C8—1942no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—971no—
308C4⋊C4—1941no—
309C4⋊C4—1941no—
310C4⋊C4—1941no—
311C4⋊C4—1941no—
312C4⋊C4C22⋊D9711yescharacteristic
313C4⋊C4C22×D9711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—1941no—
319C22⋊C4—1941no—
320C22⋊C4—1941no—
321C22⋊C4C22⋊D9711yescharacteristic
322C22⋊C4C22⋊D9711yescharacteristic
323C42—21no—
324C42C22×D9711yescharacteristic
325C23—21no—
326C23—1941no—
327C23C2×C22⋊D9711yescharacteristic
328C23C2×C22⋊D9711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—1941no—
334Q8—1941no—
335Q8—1941no—
336Q8—1941no—
337Q8D4×D9712yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—972no—
344D4—3882no—
345D4—3882no—
346D4—1941no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—3882no—
357C2×C4—3882no—
358C2×C4—3882no—
359C2×C4—1941no—
360C2×C4—1941no—
361C2×C4—1941no—
362C2×C4C2×C22⋊D9711yescharacteristic
363C2×C4C2×C22⋊D9711yescharacteristic
364C2×C4C2×C22⋊D9711yescharacteristic
365C2×C4C2×C22⋊D9711yescharacteristic
366C2×C4C23×D9711yescharacteristic
367C8—972no—
368C8—1941no—
369C8—1941no—
370C8—1941no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3881no—
381C22—3881no—
382C22—1942no—
383C22C22×C22⋊D9711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—3881no—
392C4—3881no—
393C4—3881no—
394C4C2×D4×D9711yescharacteristic
395C4#1569085511yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—3882no—
400C2D4×C22⋊D9711yescharacteristic
401C2#2026153311yescharacteristic
402C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#548×C97C2
socleC22×C97C22×D4
hypercenter#548D97
hypocenterC97#3269
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
2. C2×#15690830C2semidirect
3. #20261390C2semidirect
4. #20261130C2semidirect
5. #20260979C2semidirect
6. #20260974C2semidirect
7. #20260947C2semidirect
8. #20260945C2semidirect
9. #20260897C2semidirect
10. #20260895C2semidirect
11. #20260891C2semidirect
12. #20260878C2semidirect
13. #20260833C2semidirect
14. #20260807C2semidirect
15. #548×C97C2semidirect
80. #13095498D4semidirect
202. C97#3269semidirect
205. #548D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156908302
#202613902
#202611302
#202609792
#202609742
#202609472
#202609452
#202608972
#202608952
#202608912
#202608782
#202608332
#202608072
#548×C972
#326997