Finite Group Databasebeta

← #27155386

Subgroups — #27155386

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×#15690858C211yesmaximal, characteristic, complemented
2C22×Q8×D97C211yesmaximal, characteristic, complemented
3#20261400C213yesmaximal, complemented
4Q8×C22⋊D97C211yesmaximal, characteristic, complemented
5#20261398C213yesmaximal, complemented
6#20261397C211yesmaximal, characteristic, complemented
7#20261396C211yesmaximal, characteristic, complemented
8#20261395C213yesmaximal, complemented
9#20261394C213yesmaximal, complemented
10#20261310C213yesmaximal, complemented
11#20261308C213yesmaximal, complemented
12#20261248C213yesmaximal, complemented
13#20261244C213yesmaximal, complemented
14#20261211C211yesmaximal, characteristic, complemented
15D4×Q8×C97C211yesmaximal, characteristic, complemented
16#15690858—28no—
17C2×#13095512C2213yes—
18C2×#13095512C2211yescharacteristic, complemented
19C2×#13095511C2213yes—
20C2×#13095511C2213yescomplemented
21C22×Dic194C2213yes—
22C2×Q8×D97—28no—
23C2×Q8×D97C2213yes—
24C2×Q8×D97C2211yescharacteristic
25C2×Q8×D97C2213yes—
26C2×Q8×D97C2211yescharacteristic
27C2×Q8×D97C2213yescomplemented
28C22×C4×D97C2213yes—
29#15690809C2216yescomplemented
30#15690808C2213yescomplemented
31#15690799C2213yescomplemented
32#15690798C2213yescomplemented
33#15690796C2216yescomplemented
34#15690795C2213yes—
35#15690795C2216yes—
36#15690792C2213yes—
37#15690790C2213yescomplemented
38#15690789C2213yescomplemented
39#15690788C2213yes—
40#15690788C2216yescomplemented
41#15690788C2216yescomplemented
42#15690787C2213yes—
43#15690786C2213yes—
44#15690786C2213yescomplemented
45#15690785C2213yes—
46#15690784C2213yes—
47#15690783C2213yes—
48#15690782C2213yes—
49#15690780C2213yes—
50#15690778C2211yescharacteristic
51Q8×Dic97C2211yescharacteristic, complemented
52C4×Dic194C2213yescomplemented
53#15690775C2213yescomplemented
54C4×D388C2213yes—
55#15690772C2216yes—
56C4×C22⋊D97C2213yes—
57C4×C22⋊D97C2213yes—
58#15690770C2216yes—
59#15690768C2213yes—
60#15690767C2213yes—
61C22×Q8×C97C2211yescharacteristic
62C22×Q8×C97C2211yescharacteristic, complemented
63C4⋊Q8×C97C2213yescomplemented
64C22⋊Q8×C97C2213yes—
65C22⋊Q8×C97C2213yescomplemented
66C4×Q8×C97C2211yescharacteristic, complemented
67C4×D4×C97C2213yes—
68C23×D97C2311yescharacteristic
69#13095512—24no—
70#13095512—212no—
71#13095511—212no—
72#13095511—212no—
73C2×Dic194—212no—
74C2×Dic194C2313yes—
75C2×Dic194C2316yes—
76C2×Dic194C2313yes—
77C2×Dic194C2313yes—
78Q8×D97—24no—
79Q8×D97—212no—
80Q8×D97—212no—
81Q8×D97—212no—
82Q8×D97D418yescomplemented
83C2×D388C2313yes—
84C2×D388C2313yes—
85C2×D388C2313yes—
86C2×C22⋊D97C2313yes—
87C2×C22⋊D97C2311yescharacteristic
88C2×C22⋊D97C2311yescharacteristic
89C2×C22⋊D97C2313yes—
90C22×Dic97C2313yes—
91C2×C4×D97—212no—
92C2×C4×D97C2316yes—
93C2×C4×D97C2313yes—
94C2×C4×D97C2313yes—
95C2×C4×D97C2313yes—
96C2×C4×D97C2313yes—
97C2×C4×D97C2313yes—
98C4⋊Dic97C2313yes—
99C4⋊Dic97C2313yes—
100#13095490C2313yes—
101#13095490C2313yes—
102#13095490C2316yes—
103#13095490C2313yes—
104#13095490C2313yes—
105C22⋊Dic97C2311yescharacteristic
106C22⋊Dic97C2313yes—
107#13095488C2311yescharacteristic
108#13095488C2313yes—
109#13095488C2316yes—
110#13095488C2313yes—
111#13095488C2313yes—
112#13095488C2316yes—
113#13095488C2311yescharacteristic
114#13095488C2313yes—
115C4×Dic97C2313yes—
116C4×Dic97C2316yes—
117C2×Q8×C97—24no—
118C2×Q8×C97—24no—
119C2×Q8×C97C2313yes—
120C2×Q8×C97C2313yes—
121C2×Q8×C97C2311yescharacteristic
122C2×D4×C97C2311yescharacteristic
123C22×C4×C97C2313yes—
124C22×C4×C97C2313yes—
125C4⋊C4×C97C2313yes—
126C4⋊C4×C97C2316yes—
127C4⋊C4×C97C2313yes—
128C22⋊C4×C97C2313yes—
129C22⋊C4×C97C2313yes—
130C42×C97C2313yes—
131C23×C97C2411yescharacteristic
132C23×C97C2411yescharacteristic
133C22×D97—22no—
134C22×D97—22no—
135C22×D97—22no—
136C22×D97C2413yes—
137C22×D97C2411yescharacteristic
138C22×D97C2411yescharacteristic
139Q8×C97—26no—
140Q8×C97—26no—
141Q8×C97C2×D414yes—
142D4×C97—22no—
143C2×C4×C97—26no—
144C2×C4×C97—26no—
145C2×C4×C97C2411yescharacteristic
146C2×C4×C97C2413yes—
147C2×C4×C97C2413yes—
148C2×C4×C97C2413yes—
149C2×C4×C97C2413yes—
150C22⋊D97—26no—
151C22⋊D97—22no—
152C22⋊D97—22no—
153C22⋊D97—26no—
154C2×Dic97—26no—
155C2×Dic97C2411yescharacteristic
156C2×Dic97C2413yes—
157C2×Dic97C2413yes—
158C2×Dic97C2411yescharacteristic
159C2×Dic97C2413yes—
160D388—26no—
161D388—26no—
162D388—26no—
163C4×D97—26no—
164C4×D97—26no—
165C4×D97—26no—
166C4×D97—26no—
167C4×D97—212no—
168C4×D97C2×D4112yes—
169Dic194—26no—
170Dic194—212no—
171Dic194—26no—
172Dic194C2×D4112yes—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—21no—
178C22×C972−1+412yescomplemented
179C22×C97C2511yescharacteristic
180C2×D97—28no—
181C2×D97—21no—
182C2×D97—21no—
183C2×D97—21no—
184C2×D97—21no—
185C2×D97—23no—
186C2×D97—23no—
187C2×D97—23no—
188C2×D97C22×D412yes—
189C2×D972−1+412yescomplemented
190C4×C97—23no—
191C4×C97—23no—
192C4×C97—23no—
193C4×C97—21no—
194C4×C97C22×D416yes—
195Dic97—23no—
196Dic97—21no—
197Dic97—21no—
198Dic97—23no—
199Dic97C22×D416yes—
200C2×C97—22no—
201C2×C97—41no—
202C2×C97C23×D411yescharacteristic
203C2×C97C2×2−1+411yescharacteristic
204C2×C97C2×2−1+411yescharacteristic
205D97—22no—
206D97—22no—
207D97—43no—
208#3486—971nomaximal, Sylow
209C97#348611yescharacteristic, complemented, Sylow
210C2×2−1+4—971no—
211C23×Q8—971no—
212D4×Q8—971no—
213D4×Q8—973no—
214D4×Q8—973no—
215D4×Q8D9711yescharacteristic, complemented
216#548—971no—
217#548—973no—
218#548—971no—
219#548—973no—
220#536—973no—
221#536—973no—
222#525—973no—
223#525—973no—
224#519—971no—
2252−1+4—1948no—
226C2×#41—973no—
227C2×#41—971no—
228C2×#41—973no—
229C2×#41—973no—
230C22×Q8—971no—
231C22×Q8—973no—
232C22×Q8—973no—
233C22×Q8—971no—
234C22×Q8—973no—
235C22×Q8—973no—
236C22×Q8—1948no—
237C22×Q8C2×D9711yescharacteristic
238C22×Q8C2×D9711yescharacteristic, complemented
239C23×C4—973no—
240C4⋊Q8—973no—
241C4⋊Q8—976no—
242C4⋊Q8C2×D9713yescomplemented
243#124—973no—
244#124—976no—
245#124—973no—
246#123—976no—
247#123—973no—
248#123—973no—
249C22⋊Q8—973no—
250C22⋊Q8—973no—
251C22⋊Q8—973no—
252C22⋊Q8—973no—
253C22⋊Q8—976no—
254C22⋊Q8—976no—
255C22⋊Q8—973no—
256C22⋊Q8—973no—
257C22⋊Q8C2×D9713yes—
258C22⋊Q8C2×D9713yescomplemented
259#121—973no—
260#121—973no—
261#121—973no—
262#121—973no—
263#120—971no—
264#120—973no—
265C4×Q8—971no—
266C4×Q8—973no—
267C4×Q8—973no—
268C4×Q8C2×D9711yescharacteristic, complemented
269C4×D4—973no—
270C4×D4—973no—
271C4×D4—976no—
272C4×D4—976no—
273C4×D4—973no—
274C4×D4C2×D9713yes—
275#117—973no—
276#117—973no—
277C24—971no—
278#41—19412no—
279#41—19412no—
280#41—19412no—
281#41—1944no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—978no—
285C2×Q8—973no—
286C2×Q8—976no—
287C2×Q8—973no—
288C2×Q8—973no—
289C2×Q8—19412no—
290C2×Q8—19412no—
291C2×Q8—19412no—
292C2×Q8—19412no—
293C2×Q8—1944no—
294C2×Q8C22×D9713yes—
295C2×Q8C22×D9713yes—
296C2×Q8C22×D9711yescharacteristic
297C2×D4—971no—
298C2×D4—973no—
299C2×D4—971no—
300C2×D4—973no—
301C2×D4—973no—
302C2×D4—973no—
303C2×D4—973no—
304C2×D4C22×D9711yescharacteristic
305C22×C4—973no—
306C22×C4—973no—
307C22×C4—973no—
308C22×C4—973no—
309C22×C4—976no—
310C22×C4—973no—
311C22×C4—973no—
312C22×C4—19412no—
313C22×C4C22×D9713yes—
314C22×C4C22×D9713yes—
315C4⋊C4—973no—
316C4⋊C4—973no—
317C4⋊C4—976no—
318C4⋊C4—973no—
319C4⋊C4—973no—
320C4⋊C4—973no—
321C4⋊C4—973no—
322C4⋊C4C22×D9713yes—
323C4⋊C4C22×D9716yes—
324C4⋊C4C22×D9713yes—
325C22⋊C4—971no—
326C22⋊C4—973no—
327C22⋊C4—973no—
328C22⋊C4—971no—
329C22⋊C4—973no—
330C22⋊C4—971no—
331C22⋊C4—973no—
332C22⋊C4—976no—
333C22⋊C4—976no—
334C22⋊C4—973no—
335C22⋊C4C22×D9713yes—
336C22⋊C4C22×D9713yes—
337C42—976no—
338C42—973no—
339C42C22×D9713yes—
340C23—971no—
341C23—971no—
342C23—973no—
343C23—1942no—
344C23—1942no—
345C23—1942no—
346C23C23×D9711yescharacteristic
347C23C23×D9711yescharacteristic
348Q8—26no—
349Q8—26no—
350Q8—9712no—
351Q8—19412no—
352Q8—1946no—
353Q8—1946no—
354Q8D4×D9714yes—
355D4—22no—
356D4—1946no—
357D4—1946no—
358D4—1946no—
359D4—1942no—
360D4—1942no—
361D4—1946no—
362D4—1946no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—973no—
366C2×C4—9712no—
367C2×C4—971no—
368C2×C4—973no—
369C2×C4—973no—
370C2×C4—971no—
371C2×C4—1946no—
372C2×C4—19412no—
373C2×C4—1946no—
374C2×C4—1946no—
375C2×C4—1946no—
376C2×C4—1946no—
377C2×C4C23×D9713yes—
378C2×C4C23×D9711yescharacteristic
379C2×C4C23×D9713yes—
380C2×C4C23×D9713yes—
381C2×C4C23×D9713yes—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—972no—
388C22—972no—
389C22—1948no—
390C22—1943no—
391C22—1943no—
392C22—1943no—
393C22—1941no—
394C22—1941no—
395C22—1941no—
396C22—1941no—
397C22#1569085812yescomplemented
398C22C24×D9711yescharacteristic
399C4—23no—
400C4—23no—
401C4—23no—
402C4—21no—
403C4—976no—
404C4—1941no—
405C4—1943no—
406C4—1941no—
407C4—1943no—
408C4C2×D4×D9716yes—
409C2—22no—
410C2—41no—
411C2—3883no—
412C2—1942no—
413C2—1942no—
414C2C22×D4×D9711yescharacteristic
415C22−1+4×D9711yescharacteristic
416C2C2×#1569085811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
FittingD4×Q8×C97C2
hypercenterD4×Q8D97
hypocenterC97#3486
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15690858C2semidirect
2. C22×Q8×D97C2semidirect
3. #20261400C2semidirect
4. Q8×C22⋊D97C2semidirect
5. #20261398C2semidirect
6. #20261397C2semidirect
7. #20261396C2semidirect
8. #20261395C2semidirect
9. #20261394C2semidirect
10. #20261310C2semidirect
11. #20261308C2semidirect
12. #20261248C2semidirect
13. #20261244C2semidirect
14. #20261211C2semidirect
15. D4×Q8×C97C2semidirect
18. C2×#13095512C22semidirect
20. C2×#13095511C22semidirect
27. C2×Q8×D97C22semidirect
29. #15690809C22semidirect
30. #15690808C22semidirect
31. #15690799C22semidirect
32. #15690798C22semidirect
33. #15690796C22semidirect
37. #15690790C22semidirect
38. #15690789C22semidirect
40. #15690788C22semidirect
41. #15690788C22semidirect
44. #15690786C22semidirect
51. Q8×Dic97C22semidirect
52. C4×Dic194C22semidirect
53. #15690775C22semidirect
62. C22×Q8×C97C22semidirect
63. C4⋊Q8×C97C22semidirect
65. C22⋊Q8×C97C22semidirect
66. C4×Q8×C97C22semidirect
82. Q8×D97D4semidirect
178. C22×C972−1+4semidirect
189. C2×D972−1+4semidirect
209. C97#3486semidirect
215. D4×Q8D97semidirect
238. C22×Q8C2×D97semidirect
242. C4⋊Q8C2×D97semidirect
258. C22⋊Q8C2×D97semidirect
268. C4×Q8C2×D97semidirect
397. C22#15690858semidirect

Maximal subgroups

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

subgroupindex
C2×#156908582
C22×Q8×D972
#202614002
Q8×C22⋊D972
#202613982
#202613972
#202613962
#202613952
#202613942
#202613102
#202613082
#202612482
#202612442
#202612112
D4×Q8×C972
#348697