Finite Group Databasebeta

← #27153240

Subgroups — #27153240

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×SD16×D97C211yesmaximal, characteristic, complemented
2Q8×C22⋊D97C211yesmaximal, characteristic, complemented
3#20261373C211yesmaximal, characteristic
4#20261128C211yesmaximal, characteristic, complemented
5#20260981C211yesmaximal, characteristic, complemented
6#20260973C211yesmaximal, characteristic, complemented
7#20260959C211yesmaximal, characteristic, complemented
8#20260938C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260889C211yesmaximal, characteristic, complemented
11#20260884C211yesmaximal, characteristic, complemented
12#20260874C211yesmaximal, characteristic, complemented
13#20260836C211yesmaximal, characteristic, complemented
14#20260804C211yesmaximal, characteristic, complemented
15C2×#137×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095501C2211yescharacteristic, complemented
19C2×#13095500C2211yescharacteristic
20C2×#13095499C2211yescharacteristic
21C2×#13095498C2211yescharacteristic
22C2×#13095498C2211yescharacteristic
23SD16×D97—24no—
24C2×C8×D97C2211yescharacteristic
25#15690808—21no—
26#15690808C2211yescharacteristic, complemented
27#15690799C2211yescharacteristic
28#15690793—21no—
29#15690789—21no—
30#15690789C2211yescharacteristic, complemented
31#15690786—21no—
32#15690786C2211yescharacteristic
33#15690786C2211yescharacteristic
34#15690784C2211yescharacteristic
35#15690782—21no—
36#15690778—21no—
37Q8×Dic97C2211yescharacteristic, complemented
38D4×Dic97C2211yescharacteristic
39C4×C22⋊D97—21no—
40C4×C22⋊D97C2211yescharacteristic
41C2×#13095488—21no—
42#15690744C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690734C2211yescharacteristic, complemented
47#15690734C2211yescharacteristic, complemented
48#15690732C2211yescharacteristic
49#15690731C2211yescharacteristic
50C22⋊#11715895C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic
52#15690720C2211yescharacteristic, complemented
53C2×#41×C97C2211yescharacteristic
54C22×Q8×C97C2211yescharacteristic, complemented
55#137×C97—24no—
56C2×Q16×C97C2211yescharacteristic, complemented
57C2×Q16×C97C2211yescharacteristic, complemented
58C2×SD16×C97C2211yescharacteristic
59C2×SD16×C97C2211yescharacteristic
60C2×M16×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62C2×Dic194—21no—
63C2×Dic194C2311yescharacteristic
64Q8×D97—22no—
65Q8×D97—41no—
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97D411yescharacteristic
70D4×D97—22no—
71D4×D97—41no—
72C22×Dic97—21no—
73C2×C4×D97—21no—
74C2×C4×D97C2311yescharacteristic
75#13095501—41no—
76#13095500—22no—
77#13095499—22no—
78#13095498—22no—
79#13095498—41no—
80C2×#11715895C2311yescharacteristic
81C2×#11715895C2311yescharacteristic
82C8×D97—22no—
83C4⋊Dic97—21no—
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2311yescharacteristic
86#13095490—21no—
87#13095490—21no—
88#13095490—21no—
89#13095490—21no—
90#13095490—21no—
91#13095490D411yescharacteristic
92C22⋊Dic97—21no—
93C22⋊Dic97—21no—
94C22⋊Dic97D411yescharacteristic
95#13095488—21no—
96#13095488—21no—
97#13095488—21no—
98#13095488—21no—
99#13095488—21no—
100#13095488D411yescharacteristic
101C4×Dic97—21no—
102C4×Dic97C2311yescharacteristic
103#41×C97—22no—
104#41×C97—41no—
105C2×Q8×C97—21no—
106C2×Q8×C97—22no—
107C2×Q8×C97—41no—
108C2×Q8×C97C2311yescharacteristic
109C2×Q8×C97C2311yescharacteristic
110C2×Q8×C97C2311yescharacteristic
111C2×D4×C97—21no—
112C2×D4×C97C2311yescharacteristic
113C22×C4×C97—21no—
114C22×C4×C97—21no—
115C22×C4×C97C2311yescharacteristic
116Q16×C97—22no—
117Q16×C97—22no—
118SD16×C97—22no—
119SD16×C97D414yescomplemented
120M16×C97—22no—
121C2×C8×C97C2311yescharacteristic
122C2×C8×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97—42no—
127C22×D97—42no—
128C22×D97—81no—
129C22×D97C2×D411yescharacteristic
130Q8×C97—21no—
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97—21no—
134Q8×C97—21no—
135Q8×C97—42no—
136Q8×C97C2×D412yes—
137D4×C97—21no—
138D4×C97—42no—
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—42no—
146C2×C4×C97—42no—
147C2×C4×C97C2×D411yescharacteristic
148C2×C4×C97C2411yescharacteristic
149C22⋊D97—22no—
150C22⋊D97—42no—
151C22⋊D97—81no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—81no—
156C2×Dic97C2×D411yescharacteristic
157C2×Dic97C2×D411yescharacteristic
158C2×Dic97C2×D411yescharacteristic
159D388—21no—
160D388—21no—
161C4×D97—21no—
162C4×D97—21no—
163C4×D97—42no—
164Dic194—21no—
165Dic194—21no—
166Dic194—42no—
167C8×C97—21no—
168C8×C97C2×D412yes—
169#11715895—21no—
170#11715895—21no—
171C22×C97—22no—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—41no—
176C22×C97—41no—
177C22×C97C22×D411yescharacteristic
178C2×D97—21no—
179C2×D97—21no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—84no—
184C2×D97#13712yescomplemented
185C4×C97—22no—
186C4×C97—21no—
187C4×C97—41no—
188C4×C97—41no—
189C4×C97C22×D411yescharacteristic
190C4×C97H(5,𝔽2)11yescharacteristic
191Dic97—21no—
192Dic97—21no—
193Dic97—21no—
194Dic97—41no—
195Dic97—41no—
196Dic97—41no—
197C2×C97—42no—
198C2×C97—41no—
199C2×C97D4211yescharacteristic
200C2×C97C2×#13711yescharacteristic
201C2×C97#57711yescharacteristic
202D97—22no—
203D97—81no—
204#3266—971nomaximal, Sylow
205C97#326611yescharacteristic, complemented, Sylow
206C2×#137D9711yescharacteristic, complemented
207C22×SD16—971no—
208D4×Q8—971no—
209#546—971no—
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×Q8C2×D9711yescharacteristic, complemented
224C22×D4—971no—
225#137—24no—
226C2×Q16—971no—
227C2×Q16C2×D9711yescharacteristic, complemented
228C2×Q16C2×D9711yescharacteristic, complemented
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—1944no—
234C2×SD16C2×D9711yescharacteristic
235C2×SD16C2×D9711yescharacteristic
236C2×M16C2×D9711yescharacteristic, complemented
237C22×C8—971no—
238C4⋊Q8—971no—
239C4⋊Q8—1941no—
240#124—971no—
241#123—1941no—
242C22⋊Q8—971no—
243C22⋊Q8—971no—
244C22⋊Q8—971no—
245C22⋊Q8—1941no—
246C22⋊Q8—1941no—
247#121—971no—
248#121—1941no—
249#120—1941no—
250C4×Q8—971no—
251C4×D4—971no—
252C4×D4—971no—
253C4×D4—1941no—
254C2×C22⋊C4—1941no—
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—1941no—
267#41—22no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—971no—
273C2×Q8—3881no—
274C2×Q8—1941no—
275C2×Q8—1942no—
276C2×Q8C22×D9711yescharacteristic
277C2×Q8C22×D9711yescharacteristic
278C2×Q8C22×D9711yescharacteristic
279C2×D4—21no—
280C2×D4—971no—
281C2×D4—971no—
282C2×D4—3881no—
283C2×D4—1941no—
284C2×D4—1941no—
285C2×D4—1942no—
286C2×D4C22×D9711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—971no—
290C22×C4—1941no—
291C22×C4—1941no—
292C22×C4C22×D9711yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—3881no—
296SD16—22no—
297SD16—3881no—
298SD16—1942no—
299SD16—1942no—
300SD16—1942no—
301SD16C22⋊D9714yescomplemented
302M16—22no—
303C2×C8—971no—
304C2×C8—971no—
305C2×C8—1942no—
306C2×C8C22×D9711yescharacteristic
307C2×C8C22×D9711yescharacteristic
308C4⋊C4—971no—
309C4⋊C4—971no—
310C4⋊C4—971no—
311C4⋊C4—1941no—
312C4⋊C4—1941no—
313C4⋊C4—1941no—
314C4⋊C4—1941no—
315C4⋊C4—1941no—
316C4⋊C4—1941no—
317C22⋊C4—971no—
318C22⋊C4—971no—
319C22⋊C4—1941no—
320C22⋊C4—1941no—
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C42—971no—
327C42—1941no—
328C23—21no—
329C23—7761no—
330C23—971no—
331C23—3882no—
332C23—3882no—
333C23—1941no—
334C23D4×D9711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—3882no—
342Q8—1941no—
343Q8—1941no—
344Q8C2×C22⋊D9712yes—
345D4—21no—
346D4—42no—
347D4—7761no—
348D4—3882no—
349D4—1942no—
350D4—1941no—
351D4—1941no—
352D4C2×C22⋊D9712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—7761no—
361C2×C4—971no—
362C2×C4—971no—
363C2×C4—971no—
364C2×C4—3882no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4—1941no—
368C2×C4—1941no—
369C2×C4—1941no—
370C2×C4D4×D9711yescharacteristic
371C2×C4C23×D9711yescharacteristic
372C8—21no—
373C8—1941no—
374C8—1941no—
375C8C2×C22⋊D9712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—7764no—
383C22—972no—
384C22—3881no—
385C22—3881no—
386C22—3881no—
387C22—1941no—
388C22—1941no—
389C22C2×D4×D9711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—3881no—
395C4—3881no—
396C4—3881no—
397C4—1941no—
398C4—1941no—
399C4—1941no—
400C4C22×C22⋊D9711yescharacteristic
401C4#1569085511yescharacteristic
402C2—41no—
403C2—42no—
404C2—7761no—
405C2—1942no—
406C2D4×C22⋊D9711yescharacteristic
407C2#137×D9711yescharacteristic
408C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radical#20261373C2
hypercenterC2×#137D97
hypocenterC97#3266
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D97C2semidirect
2. Q8×C22⋊D97C2semidirect
4. #20261128C2semidirect
5. #20260981C2semidirect
6. #20260973C2semidirect
7. #20260959C2semidirect
8. #20260938C2semidirect
9. #20260897C2semidirect
10. #20260889C2semidirect
11. #20260884C2semidirect
12. #20260874C2semidirect
13. #20260836C2semidirect
14. #20260804C2semidirect
15. C2×#137×C97C2semidirect
18. C2×#13095501C22semidirect
26. #15690808C22semidirect
30. #15690789C22semidirect
37. Q8×Dic97C22semidirect
42. #15690744C22semidirect
43. #15690739C22semidirect
44. #15690735C22semidirect
45. #15690734C22semidirect
46. #15690734C22semidirect
47. #15690734C22semidirect
50. C22⋊#11715895C22semidirect
52. #15690720C22semidirect
54. C22×Q8×C97C22semidirect
56. C2×Q16×C97C22semidirect
57. C2×Q16×C97C22semidirect
60. C2×M16×C97C22semidirect
119. SD16×C97D4semidirect
184. C2×D97#137semidirect
205. C97#3266semidirect
206. C2×#137D97semidirect
223. C22×Q8C2×D97semidirect
227. C2×Q16C2×D97semidirect
228. C2×Q16C2×D97semidirect
236. C2×M16C2×D97semidirect
301. SD16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D972
Q8×C22⋊D972
#202613732
#202611282
#202609812
#202609732
#202609592
#202609382
#202608972
#202608892
#202608842
#202608742
#202608362
#202608042
C2×#137×C972
#326697