Finite Group Databasebeta

← #27155382

Subgroups — #27155382

Order 12416.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690857C211yesmaximal, characteristic, complemented
2C22×Q8×D97C211yesmaximal, characteristic, complemented
3Q8×D388C211yesmaximal, characteristic, complemented
4D4×Dic194C213yesmaximal, complemented
5Q8×C22⋊D97C211yesmaximal, characteristic, complemented
6#20261398C213yesmaximal, complemented
7#20261393C213yesmaximal, complemented
8#20261392C211yesmaximal, characteristic, complemented
9#20261391C213yesmaximal, complemented
10#20261308C213yesmaximal, complemented
11#20261305C213yesmaximal, complemented
12#20261245C213yesmaximal, complemented
13#20261244C213yesmaximal, complemented
14#20261211C211yesmaximal, characteristic, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16#15690857—28no—
17C2×#13095511C2213yes—
18C2×#13095510C2213yes—
19C2×#13095510C2213yescomplemented
20C22×Dic194C2213yes—
21C22×Dic194C2213yescomplemented
22C2×Q8×D97—28no—
23C2×Q8×D97C2211yescharacteristic
24C2×Q8×D97C2213yes—
25C2×Q8×D97C2213yes—
26C2×Q8×D97C2211yescharacteristic
27C2×Q8×D97C2211yescharacteristic, complemented
28C22×C4×D97C2213yes—
29#15690810C2213yescomplemented
30#15690809C2216yescomplemented
31#15690808C2213yescomplemented
32#15690797C2213yescomplemented
33#15690796C2216yescomplemented
34#15690795C2213yes—
35#15690792C2213yes—
36#15690792C2216yes—
37#15690790C2213yes—
38#15690790C2213yescomplemented
39#15690789C2213yes—
40#15690789C2213yescomplemented
41#15690788C2213yes—
42#15690787C2213yes—
43#15690787C2216yescomplemented
44#15690787C2216yescomplemented
45#15690786C2213yescomplemented
46#15690783C2213yes—
47#15690782C2213yes—
48#15690781C2213yes—
49#15690780C2211yescharacteristic
50#15690778C2213yes—
51Q8×Dic97C2211yescharacteristic, complemented
52C4×Dic194C2213yescomplemented
53#15690775C2213yescomplemented
54C4×D388C2213yes—
55D4×Dic97C2216yes—
56C4×C22⋊D97C2213yes—
57C4×C22⋊D97C2213yes—
58#15690770C2216yes—
59#15690768C2213yes—
60#15690767C2213yes—
61C2×#41×C97C2211yescharacteristic, complemented
62C22×Q8×C97C2211yescharacteristic
63#124×C97C2213yescomplemented
64C22⋊Q8×C97C2213yescomplemented
65#121×C97C2213yes—
66C4×Q8×C97C2211yescharacteristic, complemented
67C4×D4×C97C2213yes—
68C23×D97C2311yescharacteristic
69#13095511—212no—
70#13095510—212no—
71#13095510—212no—
72C2×Dic194—212no—
73C2×Dic194—212no—
74C2×Dic194C2313yes—
75C2×Dic194C2316yes—
76C2×Dic194C2313yes—
77C2×Dic194C2313yes—
78C2×Dic194C2313yes—
79Q8×D97—24no—
80Q8×D97—24no—
81Q8×D97—212no—
82Q8×D97—212no—
83Q8×D97D418yescomplemented
84C2×D388C2311yescharacteristic
85C2×D388C2311yescharacteristic
86C2×C22⋊D97C2313yes—
87C2×C22⋊D97C2311yescharacteristic
88C2×C22⋊D97C2313yes—
89C2×C22⋊D97C2313yes—
90C22×Dic97C2316yes—
91C22×Dic97C2313yes—
92C2×C4×D97—212no—
93C2×C4×D97C2313yes—
94C2×C4×D97C2313yes—
95C2×C4×D97C2313yes—
96C2×C4×D97C2313yes—
97C2×C4×D97C2313yes—
98C4⋊Dic97C2313yes—
99C4⋊Dic97C2316yes—
100C4⋊Dic97C2313yes—
101#13095490C2313yes—
102#13095490C2313yes—
103#13095490C2316yes—
104#13095490C2313yes—
105#13095490C2313yes—
106C22⋊Dic97C2313yes—
107C22⋊Dic97C2313yes—
108C22⋊Dic97C2316yes—
109#13095488C2311yescharacteristic
110#13095488C2313yes—
111#13095488C2313yes—
112#13095488C2311yescharacteristic
113#13095488C2313yes—
114#13095488C2313yes—
115C4×Dic97C2313yes—
116C4×Dic97C2316yes—
117#41×C97—24no—
118C2×Q8×C97—24no—
119C2×Q8×C97C2313yes—
120C2×Q8×C97C2311yescharacteristic
121C2×D4×C97C2313yes—
122C2×D4×C97C2313yes—
123C22×C4×C97C2313yes—
124C22×C4×C97C2313yes—
125C4⋊C4×C97C2313yes—
126C4⋊C4×C97C2313yes—
127C22⋊C4×C97C2316yes—
128C22⋊C4×C97C2311yescharacteristic
129C22⋊C4×C97C2313yes—
130C42×C97C2313yes—
131C23×C97C2413yes—
132C23×C97C2411yescharacteristic
133C22×D97—22no—
134C22×D97—22no—
135C22×D97—22no—
136C22×D97C2411yescharacteristic
137C22×D97C2411yescharacteristic
138C22×D97C2411yescharacteristic
139Q8×C97—26no—
140Q8×C97C2×D414yes—
141D4×C97—26no—
142D4×C97—26no—
143C2×C4×C97—26no—
144C2×C4×C97—26no—
145C2×C4×C97C2411yescharacteristic
146C2×C4×C97C2413yes—
147C2×C4×C97C2413yes—
148C2×C4×C97C2411yescharacteristic
149C2×C4×C97C2413yes—
150C22⋊D97—22no—
151C22⋊D97—26no—
152C22⋊D97—26no—
153C22⋊D97—26no—
154C2×Dic97—26no—
155C2×Dic97—212no—
156C2×Dic97C2411yescharacteristic
157C2×Dic97C2413yes—
158C2×Dic97C2413yes—
159C2×Dic97C2413yes—
160C2×Dic97C2413yes—
161D388—22no—
162D388—22no—
163C4×D97—26no—
164C4×D97—26no—
165C4×D97—26no—
166C4×D97—26no—
167C4×D97C2×D4112yes—
168Dic194—26no—
169Dic194—26no—
170Dic194—212no—
171Dic194—26no—
172Dic194C2×D4112yes—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—23no—
176C22×C97—23no—
177C22×C97—23no—
178C22×C972−1+412yescomplemented
179C22×C97C2511yescharacteristic
180C2×D97—28no—
181C2×D97—21no—
182C2×D97—21no—
183C2×D97—21no—
184C2×D97—21no—
185C2×D97—21no—
186C2×D97—21no—
187C2×D97—21no—
188C2×D97C22×D412yes—
189C2×D972−1+412yescomplemented
190C4×C97—21no—
191C4×C97—23no—
192C4×C97—21no—
193C4×C97—23no—
194C4×C97C22×D416yes—
195Dic97—23no—
196Dic97—23no—
197Dic97—23no—
198Dic97—21no—
199Dic97C22×D416yes—
200C2×C97—22no—
201C2×C97—43no—
202C2×C97C23×D411yescharacteristic
203C2×C97C2×2−1+411yescharacteristic
204C2×C97C2×2−1+411yescharacteristic
205D97—22no—
206D97—22no—
207D97—41no—
208#3486—971nomaximal, Sylow
209C97#348611yescharacteristic, complemented, Sylow
210C2×2−1+4—971no—
211C23×Q8—971no—
212D4×Q8—971no—
213D4×Q8—973no—
214D4×Q8—971no—
215D4×Q8—973no—
216#548—971no—
217#548—973no—
218#548—973no—
219#548D9711yescharacteristic, complemented
220#536—973no—
221#536—973no—
222#525—973no—
223#525—973no—
224#519—971no—
2252−1+4—1948no—
226C2×#41—973no—
227C2×#41—973no—
228C2×#41—973no—
229C2×#41C2×D9711yescharacteristic, complemented
230C22×Q8—971no—
231C22×Q8—973no—
232C22×Q8—973no—
233C22×Q8—971no—
234C22×Q8—971no—
235C22×Q8—973no—
236C22×Q8—973no—
237C22×Q8—1948no—
238C22×Q8C2×D9711yescharacteristic
239C23×C4—973no—
240C4⋊Q8—973no—
241C4⋊Q8—976no—
242C4⋊Q8—973no—
243#124—973no—
244#124—976no—
245#124C2×D9713yescomplemented
246#123—973no—
247#123—976no—
248#123—973no—
249C22⋊Q8—973no—
250C22⋊Q8—973no—
251C22⋊Q8—973no—
252C22⋊Q8—973no—
253C22⋊Q8—973no—
254C22⋊Q8—973no—
255C22⋊Q8—973no—
256C22⋊Q8—976no—
257C22⋊Q8—976no—
258C22⋊Q8C2×D9713yescomplemented
259#121—973no—
260#121—973no—
261#121—973no—
262#121C2×D9713yes—
263#120—973no—
264#120—971no—
265C4×Q8—971no—
266C4×Q8—973no—
267C4×Q8—973no—
268C4×Q8C2×D9711yescharacteristic, complemented
269C4×D4—976no—
270C4×D4—973no—
271C4×D4—973no—
272C4×D4—976no—
273C4×D4—973no—
274C4×D4C2×D9713yes—
275#117—973no—
276#117—973no—
277C24—971no—
278#41—24no—
279#41—19412no—
280#41—19412no—
281#41—19412no—
282C2×Q8—24no—
283C2×Q8—978no—
284C2×Q8—973no—
285C2×Q8—976no—
286C2×Q8—973no—
287C2×Q8—973no—
288C2×Q8—973no—
289C2×Q8—19412no—
290C2×Q8—19412no—
291C2×Q8—19412no—
292C2×Q8—19412no—
293C2×Q8—1944no—
294C2×Q8—1944no—
295C2×Q8C22×D9713yes—
296C2×Q8C22×D9711yescharacteristic
297C2×D4—973no—
298C2×D4—971no—
299C2×D4—973no—
300C2×D4—973no—
301C2×D4—971no—
302C2×D4—971no—
303C2×D4C22×D9713yes—
304C2×D4C22×D9713yes—
305C22×C4—973no—
306C22×C4—973no—
307C22×C4—976no—
308C22×C4—973no—
309C22×C4—973no—
310C22×C4—973no—
311C22×C4—973no—
312C22×C4—19412no—
313C22×C4C22×D9713yes—
314C22×C4C22×D9713yes—
315C4⋊C4—973no—
316C4⋊C4—973no—
317C4⋊C4—973no—
318C4⋊C4—976no—
319C4⋊C4—973no—
320C4⋊C4—973no—
321C4⋊C4—976no—
322C4⋊C4—973no—
323C4⋊C4C22×D9713yes—
324C4⋊C4C22×D9713yes—
325C22⋊C4—973no—
326C22⋊C4—973no—
327C22⋊C4—976no—
328C22⋊C4—973no—
329C22⋊C4—971no—
330C22⋊C4—971no—
331C22⋊C4—973no—
332C22⋊C4—973no—
333C22⋊C4—973no—
334C22⋊C4C22×D9716yes—
335C22⋊C4C22×D9711yescharacteristic
336C22⋊C4C22×D9713yes—
337C42—976no—
338C42—973no—
339C42C22×D9713yes—
340C23—971no—
341C23—971no—
342C23—971no—
343C23—1942no—
344C23—1942no—
345C23—1942no—
346C23C23×D9713yes—
347C23C23×D9711yescharacteristic
348Q8—26no—
349Q8—9712no—
350Q8—19412no—
351Q8—1946no—
352Q8—1946no—
353Q8—1946no—
354Q8D4×D9714yes—
355D4—26no—
356D4—26no—
357D4—1942no—
358D4—1942no—
359D4—1946no—
360D4—1946no—
361D4—1946no—
362D4—1942no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—973no—
366C2×C4—9712no—
367C2×C4—973no—
368C2×C4—971no—
369C2×C4—973no—
370C2×C4—973no—
371C2×C4—1946no—
372C2×C4—1946no—
373C2×C4—1946no—
374C2×C4—19412no—
375C2×C4—1946no—
376C2×C4—1946no—
377C2×C4C23×D9713yes—
378C2×C4C23×D9711yescharacteristic
379C2×C4C23×D9713yes—
380C2×C4C23×D9711yescharacteristic
381C2×C4C23×D9713yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—972no—
388C22—972no—
389C22—1948no—
390C22—1941no—
391C22—1941no—
392C22—1941no—
393C22—1941no—
394C22—1941no—
395C22—1941no—
396C22—1941no—
397C22#1569085712yescomplemented
398C22C24×D9711yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—976no—
404C4—1943no—
405C4—1943no—
406C4—1943no—
407C4—1941no—
408C4C2×D4×D9716yes—
409C2—22no—
410C2—43no—
411C2—3881no—
412C2—1942no—
413C2—1942no—
414C2C22×D4×D9711yescharacteristic
415C22−1+4×D9711yescharacteristic
416C2C2×#1569085711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#548×C97C2
hypercenter#548D97
hypocenterC97#3486
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#15690857C2semidirect
2. C22×Q8×D97C2semidirect
3. Q8×D388C2semidirect
4. D4×Dic194C2semidirect
5. Q8×C22⋊D97C2semidirect
6. #20261398C2semidirect
7. #20261393C2semidirect
8. #20261392C2semidirect
9. #20261391C2semidirect
10. #20261308C2semidirect
11. #20261305C2semidirect
12. #20261245C2semidirect
13. #20261244C2semidirect
14. #20261211C2semidirect
15. #548×C97C2semidirect
19. C2×#13095510C22semidirect
21. C22×Dic194C22semidirect
27. C2×Q8×D97C22semidirect
29. #15690810C22semidirect
30. #15690809C22semidirect
31. #15690808C22semidirect
32. #15690797C22semidirect
33. #15690796C22semidirect
38. #15690790C22semidirect
40. #15690789C22semidirect
43. #15690787C22semidirect
44. #15690787C22semidirect
45. #15690786C22semidirect
51. Q8×Dic97C22semidirect
52. C4×Dic194C22semidirect
53. #15690775C22semidirect
61. C2×#41×C97C22semidirect
63. #124×C97C22semidirect
64. C22⋊Q8×C97C22semidirect
66. C4×Q8×C97C22semidirect
83. Q8×D97D4semidirect
178. C22×C972−1+4semidirect
189. C2×D972−1+4semidirect
209. C97#3486semidirect
219. #548D97semidirect
229. C2×#41C2×D97semidirect
245. #124C2×D97semidirect
258. C22⋊Q8C2×D97semidirect
268. C4×Q8C2×D97semidirect
397. C22#15690857semidirect

Maximal subgroups

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

subgroupindex
C2×#156908572
C22×Q8×D972
Q8×D3882
D4×Dic1942
Q8×C22⋊D972
#202613982
#202613932
#202613922
#202613912
#202613082
#202613052
#202612452
#202612442
#202612112
#548×C972
#348697