Finite Group Databasebeta

← #27158494

Subgroups — #27158494

Order 12416.

470 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261804C211yesmaximal, characteristic
2#20261803C211yesmaximal, characteristic, complemented
3#20261775C212yesmaximal, complemented
4#20261774C212yesmaximal, complemented
5Q16×#87604C212yesmaximal, complemented
6SD16×#87604C212yesmaximal, complemented
7#20261746C211yesmaximal, characteristic, complemented
8#20261725C211yesmaximal, characteristic
9C2×#15690873C211yesmaximal, characteristic, complemented
10#20261701C211yesmaximal, characteristic, complemented
11#137×D97C211yesmaximal, characteristic
12C2×#13095519C2211yescharacteristic
13Q8×#87604—21no—
14Q8×#87604—21no—
15Q8×#87604C2212yes—
16Q8×#87604C2212yes—
17Q8×#87604C2212yes—
18D4×#87604—22no—
19D4×#87604C2212yes—
20#15690889—21no—
21#15690889—22no—
22#15690889C2211yescharacteristic
23#15690889C2211yescharacteristic
24C2×C4⋊#87604—21no—
25C2×C4⋊#87604C2211yescharacteristic
26#15690879C2212yes—
27#15690878C2212yes—
28#15690873C2212yes—
29#15690873C2212yes—
30#15690873C2211yescharacteristic
31#15690873C2211yescharacteristic
32#15690872C2211yescharacteristic
33#15690872C2211yescharacteristic
34#15690864C2212yes—
35C8×#87604C2212yes—
36#41×D97C2211yescharacteristic
37C2×Q8×D97C2211yescharacteristic
38#15690843C411yescharacteristic, complemented
39#15690842C412yescomplemented
40#15690841C411yescharacteristic, complemented
41#15690840C411yescharacteristic, complemented
42#15690839C412yescomplemented
43Q16×D97C2212yes—
44SD16×D97C2212yes—
45M16×D97C2211yescharacteristic
46#137×C97C411yescharacteristic, complemented
47C22×#87604—21no—
48C22×#87604—22no—
49#13095519—21no—
50#13095519C2311yescharacteristic
51#13095519C2311yescharacteristic
52C2×#11715897—21no—
53C4⋊#87604—22no—
54C4⋊#87604—22no—
55C4⋊#87604—22no—
56C4⋊#87604—21no—
57C4⋊#87604D411yescharacteristic
58C4⋊#87604D411yescharacteristic
59C4⋊#87604C2311yescharacteristic
60C4⋊#87604C2311yescharacteristic
61C4⋊#87604C2312yes—
62C22⋊#87604—21no—
63C22⋊#87604—22no—
64C22⋊#87604D412yes—
65C4×#87604—22no—
66C4×#87604—22no—
67C4×#87604—22no—
68C4×#87604—21no—
69C4×#87604C2312yes—
70#13095512C2×C411yescharacteristic
71#13095511—21no—
72#13095511C2×C411yescharacteristic
73#13095510—21no—
74#13095510C2×C411yescharacteristic
75C2×Dic194—21no—
76C2×Dic194C2×C411yescharacteristic, complemented
77Q8×D97—21no—
78Q8×D97—21no—
79Q8×D97C2×C412yescomplemented
80Q8×D97C2312yes—
81Q8×D97C2311yescharacteristic
82D4×D97—21no—
83D4×D97C2311yescharacteristic
84C2×C4×D97—21no—
85C2×C4×D97—21no—
86C2×C4×D97C2311yescharacteristic
87Dic388C2×C412yescomplemented
88#13095501C2×C412yescomplemented
89#13095501C2×C412yescomplemented
90#13095500C2×C412yescomplemented
91#13095499C2×C412yescomplemented
92#13095498C2×C412yescomplemented
93#13095495C2×C411yescharacteristic, complemented
94#13095494C2×C412yescomplemented
95C8×D97C2312yes—
96#41×C97C2×C411yescharacteristic
97C2×Q8×C97C2×C411yescharacteristic, complemented
98Q16×C97C2×C412yescomplemented
99SD16×C97C2×C412yescomplemented
100M16×C97C2×C411yescharacteristic, complemented
101C22×D97—21no—
102C22×D97C2×D411yescharacteristic
103C2×#87604—21no—
104C2×#87604—21no—
105C2×#87604—22no—
106C2×#87604—82no—
107C2×#87604—81no—
108C2×#87604C2×D412yes—
109C2×#87604C2×D412yes—
110Q8×C97—21no—
111Q8×C97C22×C412yes—
112Q8×C97C22×C411yescharacteristic
113D4×C97—21no—
114D4×C97C22×C411yescharacteristic
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97C22×C411yescharacteristic
118C22⋊D97—21no—
119C22⋊D97—21no—
120C22⋊D97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97#4111yescharacteristic
124D388—21no—
125D388C22×C411yescharacteristic
126C4×D97—22no—
127C4×D97—21no—
128C4×D97—22no—
129C4×D97—21no—
130C4×D97—21no—
131C4×D97C22×C411yescharacteristic
132C4×D97C2×D411yescharacteristic
133C4×D97#4111yescharacteristic
134C4×D97C2411yescharacteristic
135Dic194—21no—
136Dic194—22no—
137Dic194—22no—
138Dic194—21no—
139Dic194C22×C412yes—
140Dic194C22×C411yescharacteristic
141#11715897—21no—
142#11715897—21no—
143C8×C97C22×C412yes—
144#11715895C22×C412yes—
145C22×C97—21no—
146C22×C97C4×D411yescharacteristic
147C2×D97—21no—
148C2×D97—21no—
149C2×D97—21no—
150C2×D97—41no—
151C2×D97—41no—
152C2×D97C4×D411yescharacteristic
153C2×D97C22×D411yescharacteristic
154#87604—22no—
155#87604—22no—
156#87604—41no—
157#87604—42no—
158#87604—41no—
159C4×C97—22no—
160C4×C97—21no—
161C4×C97C4×D411yescharacteristic
162C4×C97C23×C411yescharacteristic
163Dic97—22no—
164Dic97—21no—
165Dic97C4×D411yescharacteristic
166Dic97C2×#4111yescharacteristic
167C2×C97—21no—
168C2×C97—41no—
169C2×C97C2×C4×D411yescharacteristic
170D97—21no—
171D97—41no—
172D97#57711yescharacteristic
173D97#57811yescharacteristic
174#2947—971nomaximal, Sylow
175C97#294711yescharacteristic, complemented, Sylow
176C2×#137—971no—
177#520—971no—
178#519—971no—
179#441—972no—
180#440—972no—
181C4×Q16—972no—
182C4×SD16—972no—
183#428—971no—
184#416—971no—
185C2×#103—971no—
186#405—971no—
187C2×#41—971no—
188C22×Q8—971no—
189#137—971no—
190#137—971no—
191#137—972no—
192#137—972no—
193#137—971no—
194#137#8760411yescharacteristic, complemented
195C2×Q16—972no—
196C2×SD16—972no—
197C2×M16—971no—
198C22×C8—971no—
199C4×Q8—972no—
200C4×Q8—972no—
201C4×Q8—972no—
202C4×Q8—1941no—
203C4×Q8—1941no—
204C4×D4—972no—
205C4×D4—1942no—
206#117—971no—
207#117—971no—
208#117—1942no—
209#117—1941no—
210C2×C4⋊C4—971no—
211C2×C4⋊C4—1941no—
212#107—972no—
213#106—972no—
214#103—972no—
215#103—972no—
216#103—971no—
217#103—971no—
218#102—971no—
219#102—971no—
220#97—972no—
221C4×C8—972no—
222#41—971no—
223#41—971no—
224#41—971no—
225#41—1941no—
226#41—1941no—
227#41C2×#8760411yescharacteristic
228C2×Q8—972no—
229C2×Q8—972no—
230C2×Q8—971no—
231C2×Q8—971no—
232C2×Q8—1941no—
233C2×Q8—1941no—
234C2×Q8—1941no—
235C2×Q8C2×#8760411yescharacteristic, complemented
236C2×D4—971no—
237C2×D4—1941no—
238C22×C4—971no—
239C22×C4—1941no—
240C22×C4—1941no—
241C22×C4—1942no—
242C22×C4—1941no—
243Q16—972no—
244Q16—972no—
245Q16—972no—
246Q16C2×#8760412yescomplemented
247SD16—972no—
248SD16—972no—
249SD16—972no—
250SD16C2×#8760412yescomplemented
251M16—971no—
252M16—972no—
253M16C2×#8760411yescharacteristic, complemented
254C2×C8—972no—
255C2×C8—971no—
256C2×C8—971no—
257C2×C8—1941no—
258C2×C8—1941no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4—971no—
263C4⋊C4—972no—
264C4⋊C4—1941no—
265C4⋊C4—1942no—
266C4⋊C4—1942no—
267C4⋊C4—1942no—
268C22⋊C4—972no—
269C22⋊C4—1942no—
270C22⋊C4—1941no—
271C42—972no—
272C42—1941no—
273C42—1942no—
274C42—1942no—
275C42—1942no—
276C23—971no—
277C23—1941no—
278Q8—21no—
279Q8—972no—
280Q8—971no—
281Q8—1942no—
282Q8—1942no—
283Q8—1941no—
284Q8—1941no—
285Q8C22×#8760412yes—
286Q8C22×#8760411yescharacteristic
287D4—21no—
288D4—971no—
289D4—1941no—
290D4—1941no—
291D4—1941no—
292D4—1941no—
293D4C22×#8760411yescharacteristic
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—7761no—
297C2×C4—7762no—
298C2×C4—971no—
299C2×C4—971no—
300C2×C4—971no—
301C2×C4—971no—
302C2×C4—971no—
303C2×C4—972no—
304C2×C4—972no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1942no—
308C2×C4—1942no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4—1941no—
312C2×C4—1942no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4C22×#8760411yescharacteristic
316C8—972no—
317C8—1941no—
318C8—1941no—
319C8C22×#8760412yes—
320C22—21no—
321C22—971no—
322C22—971no—
323C22—3881no—
324C22—3881no—
325C22—1941no—
326C22—1941no—
327C22—1941no—
328C22D4×#8760411yescharacteristic
329C4—21no—
330C4—22no—
331C4—971no—
332C4—971no—
333C4—3882no—
334C4—3881no—
335C4—3881no—
336C4—1942no—
337C4—1942no—
338C4—1941no—
339C4—1942no—
340C4D4×#8760411yescharacteristic
341C4C23×#8760411yescharacteristic
342C2—21no—
343C2—41no—
344C2—971no—
345C2—971no—
346C2—3881no—
347C2—1941no—
348C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC2C2×D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
Fitting#137×C97C4
socleC2×C97C2×C4×D4
prime radical#41×D97C22
cosocleC4×D97C24
hypercenter#137#87604
hypocenterC97#2947
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #20261803C2semidirect
3. #20261775C2semidirect
4. #20261774C2semidirect
5. Q16×#87604C2semidirect
6. SD16×#87604C2semidirect
7. #20261746C2semidirect
9. C2×#15690873C2semidirect
10. #20261701C2semidirect
38. #15690843C4semidirect
39. #15690842C4semidirect
40. #15690841C4semidirect
41. #15690840C4semidirect
42. #15690839C4semidirect
46. #137×C97C4semidirect
76. C2×Dic194C2×C4semidirect
79. Q8×D97C2×C4semidirect
87. Dic388C2×C4semidirect
88. #13095501C2×C4semidirect
89. #13095501C2×C4semidirect
90. #13095500C2×C4semidirect
91. #13095499C2×C4semidirect
92. #13095498C2×C4semidirect
93. #13095495C2×C4semidirect
94. #13095494C2×C4semidirect
97. C2×Q8×C97C2×C4semidirect
98. Q16×C97C2×C4semidirect
99. SD16×C97C2×C4semidirect
100. M16×C97C2×C4semidirect
175. C97#2947semidirect
194. #137#87604semidirect
235. C2×Q8C2×#87604semidirect
246. Q16C2×#87604semidirect
250. SD16C2×#87604semidirect
253. M16C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618042
#202618032
#202617752
#202617742
Q16×#876042
SD16×#876042
#202617462
#202617252
C2×#156908732
#202617012
#137×D972
#294797