Finite Group Databasebeta

← #27153242

Subgroups — #27153242

Order 12416.

494 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×#15690840C211yesmaximal, characteristic, complemented
2C22×#13095499C211yesmaximal, characteristic, complemented
3D4×Dic194C211yesmaximal, characteristic, complemented
4#20261130C211yesmaximal, characteristic, complemented
5#20260979C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260954C211yesmaximal, characteristic, complemented
8#20260940C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260890C211yesmaximal, characteristic, complemented
11#20260886C211yesmaximal, characteristic, complemented
12#20260873C211yesmaximal, characteristic, complemented
13#20260835C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15#546×C97C211yesmaximal, characteristic
16C22×Dic194C2211yescharacteristic
17C22×Dic194C2211yescharacteristic, complemented
18#15690840—24no—
19C2×#13095501C2211yescharacteristic, complemented
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095499—24no—
23C2×#13095499C2211yescharacteristic
24C2×#13095499C2211yescharacteristic
25C2×#13095499C2211yescharacteristic
26C2×#13095499C2211yescharacteristic
27C2×#13095499C2211yescharacteristic
28C2×#13095499C2211yescharacteristic
29C2×#13095495C2211yescharacteristic, complemented
30C22×#11715895C2211yescharacteristic
31#15690810C2211yescharacteristic, complemented
32#15690809—21no—
33#15690789C2211yescharacteristic
34#15690789C2211yescharacteristic, complemented
35#15690787—21no—
36#15690787—21no—
37C4×Dic194C2211yescharacteristic, complemented
38D4×Dic97—21no—
39#15690743C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690734C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690733C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690731C2211yescharacteristic
46#15690731C2211yescharacteristic
47C22⋊#11715895C2211yescharacteristic
48C22⋊#11715895C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50C2×#41×C97C2211yescharacteristic
51C22×D4×C97C2211yescharacteristic
52#124×C97C2211yescharacteristic
53#123×C97—21no—
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97—21no—
56#121×C97C2211yescharacteristic
57#120×C97—21no—
58C4×D4×C97C2211yescharacteristic
59C4×D4×C97C2211yescharacteristic
60C2×C22⋊C4×C97—21no—
61C2×Dic194—22no—
62C2×Dic194—22no—
63C2×Dic194—21no—
64C2×Dic194—21no—
65C2×Dic194—41no—
66C2×Dic194—41no—
67C2×Dic194C2311yescharacteristic
68C2×Dic194C2311yescharacteristic
69C2×Dic194C2311yescharacteristic
70C22×Dic97—21no—
71C22×Dic97—21no—
72#13095501—22no—
73#13095501—22no—
74#13095501—41no—
75#13095499—22no—
76#13095499—22no—
77#13095499—22no—
78#13095499—22no—
79#13095499—41no—
80#13095499D414yescomplemented
81#13095495—22no—
82C2×#11715895—22no—
83C2×#11715895C2311yescharacteristic
84C2×#11715895C2311yescharacteristic
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C4⋊Dic97—21no—
88C4⋊Dic97—21no—
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91#13095490—21no—
92#13095490—21no—
93C22⋊Dic97—21no—
94C22⋊Dic97—21no—
95C4×Dic97—21no—
96C24×C97—21no—
97#41×C97—22no—
98#41×C97—41no—
99C2×Q8×C97C2311yescharacteristic
100C2×D4×C97—22no—
101C2×D4×C97—21no—
102C2×D4×C97—21no—
103C2×D4×C97—21no—
104C2×D4×C97—41no—
105C2×D4×C97D411yescharacteristic
106C2×D4×C97C2311yescharacteristic
107C2×D4×C97C2311yescharacteristic
108C22×C4×C97—21no—
109C22×C4×C97—21no—
110C22×C4×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
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×C97—21no—
120C22⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C42×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—42no—
126C23×C97—42no—
127C23×C97—81no—
128C23×C97C2×D411yescharacteristic
129C23×C97C2×D411yescharacteristic
130Q8×C97—21no—
131Q8×C97—21no—
132D4×C97—22no—
133D4×C97—21no—
134D4×C97—21no—
135D4×C97—21no—
136D4×C97—42no—
137D4×C97—42no—
138D4×C97—81no—
139D4×C97C2×D412yes—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—42no—
147C2×C4×C97—81no—
148C2×C4×C97C2×D411yescharacteristic
149C2×C4×C97C2×D411yescharacteristic
150C2×C4×C97C2×D411yescharacteristic
151C2×C4×C97C2×D411yescharacteristic
152C2×C4×C97C2411yescharacteristic
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
157C2×Dic97—42no—
158C2×Dic97—42no—
159Dic194—21no—
160Dic194—21no—
161Dic194—21no—
162Dic194—21no—
163Dic194—21no—
164Dic194—42no—
165Dic194—42no—
166Dic194C2×D412yes—
167#11715895—21no—
168#11715895—21no—
169#11715895—21no—
170#11715895C2×D412yes—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—22no—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—41no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97—84no—
183C22×C97#13712yescomplemented
184C22×C97C22×D411yescharacteristic
185C4×C97—21no—
186C4×C97—21no—
187C4×C97—21no—
188C4×C97—21no—
189C4×C97—41no—
190C4×C97—41no—
191C4×C97C22×D411yescharacteristic
192C4×C97H(5,𝔽2)11yescharacteristic
193Dic97—22no—
194Dic97—41no—
195Dic97—41no—
196Dic97—41no—
197C2×C97—22no—
198C2×C97—41no—
199C2×C97—42no—
200C2×C97—81no—
201C2×C97D4211yescharacteristic
202C2×C97C2×#13711yescharacteristic
203C2×C97#57711yescharacteristic
204#3266—971nomaximal, Sylow
205C97#326611yescharacteristic, complemented, Sylow
206C2×#137—971no—
207C22×SD16—971no—
208D4×Q8—971no—
209#546D9711yescharacteristic
210#497—971no—
211#468—971no—
212#467—971no—
213#464—971no—
214#461—971no—
215#452—971no—
216#451—971no—
217#450—971no—
218#448—971no—
219#440—971no—
220#435—971no—
221C2×#41C2×D9711yescharacteristic
222C22×Q8—971no—
223C22×Q8—971no—
224C22×D4C2×D9711yescharacteristic
225#137—1944no—
226C2×Q16—971no—
227C2×Q16—971no—
228C2×Q16—971no—
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—1944no—
236C2×M16—971no—
237C22×C8—971no—
238C4⋊Q8—971no—
239C4⋊Q8—1941no—
240#124C2×D9711yescharacteristic
241#123—21no—
242C22⋊Q8—971no—
243C22⋊Q8—971no—
244C22⋊Q8—1941no—
245C22⋊Q8—1941no—
246C22⋊Q8C2×D9711yescharacteristic
247#121—21no—
248#121C2×D9711yescharacteristic
249#120—21no—
250C4×Q8—971no—
251C4×D4—1941no—
252C4×D4C2×D9711yescharacteristic
253C4×D4C2×D9711yescharacteristic
254C2×C22⋊C4—21no—
255#107—971no—
256C4⋊C8—971no—
257#103—971no—
258#103—971no—
259#103—971no—
260#103—971no—
261#102—971no—
262#102—971no—
263C22⋊C8—971no—
264C22⋊C8—971no—
265#97—971no—
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—971no—
270C2×Q8—971no—
271C2×Q8—971no—
272C2×Q8—3881no—
273C2×Q8—3881no—
274C2×Q8—1942no—
275C2×Q8—1942no—
276C2×Q8—1941no—
277C2×Q8—1941no—
278C2×Q8C22×D9711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D9711yescharacteristic
285C2×D4C22×D9711yescharacteristic
286C2×D4C22×D9711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1941no—
290C22×C4—1941no—
291C22×C4C22×D9711yescharacteristic
292C22×C4C22×D9711yescharacteristic
293Q16—3881no—
294Q16—1942no—
295Q16—1942no—
296SD16—974no—
297SD16—3881no—
298SD16—1942no—
299SD16—1942no—
300SD16—1942no—
301SD16—1942no—
302M16—1942no—
303C2×C8—971no—
304C2×C8—971no—
305C2×C8—971no—
306C2×C8—971no—
307C2×C8—1942no—
308C4⋊C4—21no—
309C4⋊C4—971no—
310C4⋊C4—1941no—
311C4⋊C4—1941no—
312C4⋊C4—1941no—
313C4⋊C4—1941no—
314C4⋊C4—1941no—
315C4⋊C4C22⋊D9711yescharacteristic
316C4⋊C4C22×D9711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4C22⋊D9711yescharacteristic
325C22⋊C4C22⋊D9711yescharacteristic
326C42—1941no—
327C42C22×D9711yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D9711yescharacteristic
334C23C2×C22⋊D9711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—972no—
338Q8—3882no—
339Q8—3882no—
340Q8—1941no—
341Q8—1941no—
342Q8—1941no—
343Q8—1941no—
344Q8—1941no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D9712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—3882no—
362C2×C4—3882no—
363C2×C4—1941no—
364C2×C4—1941no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4C2×C22⋊D9711yescharacteristic
368C2×C4C2×C22⋊D9711yescharacteristic
369C2×C4C2×C22⋊D9711yescharacteristic
370C2×C4C2×C22⋊D9711yescharacteristic
371C2×C4C23×D9711yescharacteristic
372C8—972no—
373C8—1941no—
374C8—1941no—
375C8—1941no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#1569084012yescomplemented
389C22C22×C22⋊D9711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—3881no—
397C4—3881no—
398C4—3881no—
399C4—1942no—
400C4C2×D4×D9711yescharacteristic
401C4#1569085511yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D9711yescharacteristic
407C2C2×#1569084011yescharacteristic
408C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting, prime radical#546×C97C2
socleC22×C97C22×D4
hypercenter#546D97
hypocenterC97#3266
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#15690840C2semidirect
2. C22×#13095499C2semidirect
3. D4×Dic194C2semidirect
4. #20261130C2semidirect
5. #20260979C2semidirect
6. #20260974C2semidirect
7. #20260954C2semidirect
8. #20260940C2semidirect
9. #20260897C2semidirect
10. #20260890C2semidirect
11. #20260886C2semidirect
12. #20260873C2semidirect
13. #20260835C2semidirect
14. #20260807C2semidirect
17. C22×Dic194C22semidirect
19. C2×#13095501C22semidirect
20. C2×#13095501C22semidirect
21. C2×#13095501C22semidirect
29. C2×#13095495C22semidirect
31. #15690810C22semidirect
34. #15690789C22semidirect
37. C4×Dic194C22semidirect
39. #15690743C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690734C22semidirect
42. #15690733C22semidirect
43. #15690733C22semidirect
44. #15690733C22semidirect
48. C22⋊#11715895C22semidirect
49. #15690721C22semidirect
80. #13095499D4semidirect
183. C22×C97#137semidirect
205. C97#3266semidirect
388. C22#15690840semidirect

Maximal subgroups

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

subgroupindex
C2×#156908402
C22×#130954992
D4×Dic1942
#202611302
#202609792
#202609742
#202609542
#202609402
#202608972
#202608902
#202608862
#202608732
#202608352
#202608072
#546×C972
#326697