Finite Group Databasebeta

← #137×#87604 #27158492

Subgroups — #137×#87604 #27158492

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#87604C211yesmaximal, characteristic
2C2×Q8×#87604C211yesmaximal, characteristic, complemented
3#20261775C212yesmaximal, complemented
4#20261774C212yesmaximal, complemented
5Q16×#87604C212yesmaximal, complemented
6SD16×#87604C212yesmaximal, complemented
7#20261746C211yesmaximal, characteristic, complemented
8#20261728C211yesmaximal, characteristic, complemented
9#20261726C211yesmaximal, characteristic
10M16×#87604C211yesmaximal, characteristic, complemented
11#137×D97C211yesmaximal, characteristic
12D97⋊M16C2211yescharacteristic
13Q8×#87604—21no—
14Q8×#87604—21no—
15Q8×#87604C2212yes—
16Q8×#87604C2211yescharacteristic
17Q8×#87604C2212yes—
18Q8×#87604C2211yescharacteristic
19D4×#87604—21no—
20D4×#87604—21no—
21D4×#87604C2211yescharacteristic
22D4×#87604C2211yescharacteristic
23#15690889—21no—
24#15690889C2211yescharacteristic
25C2×C4⋊#87604—21no—
26C2×C4⋊#87604C2211yescharacteristic
27C2×C4×#87604—21no—
28C2×C4×#87604—21no—
29C2×C4×#87604C2211yescharacteristic
30#15690879C2212yes—
31#15690878C2212yes—
32#15690873C2212yes—
33#15690873C2212yes—
34#15690873C2212yes—
35#15690872C2212yes—
36#15690864C2212yes—
37C8×#87604C2212yes—
38#41×D97C2211yescharacteristic
39C2×Q8×D97C2211yescharacteristic
40#15690843C411yescharacteristic, complemented
41#15690842C412yescomplemented
42#15690841C411yescharacteristic, complemented
43#15690840C411yescharacteristic, complemented
44#15690839C412yescomplemented
45Q16×D97C2212yes—
46SD16×D97C2212yes—
47M16×D97C2211yescharacteristic
48#137×C97C411yescharacteristic, complemented
49C22×#87604—21no—
50C22×#87604—21no—
51C22×#87604—21no—
52C22×#87604D411yescharacteristic
53C22×#87604D411yescharacteristic
54#13095521—21no—
55#13095520—21no—
56#13095519C2312yes—
57C4⋊#87604—22no—
58C4⋊#87604—22no—
59C4⋊#87604—21no—
60C4⋊#87604—21no—
61C4⋊#87604—21no—
62C4⋊#87604C2312yes—
63C4⋊#87604C2311yescharacteristic
64C4⋊#87604C2311yescharacteristic
65C22⋊#87604—21no—
66C22⋊#87604—21no—
67C22⋊#87604—21no—
68C4×#87604—21no—
69C4×#87604—21no—
70C4×#87604—22no—
71C4×#87604—22no—
72C4×#87604—21no—
73C4×#87604D411yescharacteristic
74C4×#87604D411yescharacteristic
75C4×#87604C2311yescharacteristic
76C4×#87604C2311yescharacteristic
77#13095512C2×C411yescharacteristic
78#13095511—21no—
79#13095511C2×C411yescharacteristic
80#13095510—21no—
81#13095510C2×C411yescharacteristic
82C2×Dic194—21no—
83C2×Dic194C2×C411yescharacteristic, complemented
84Q8×D97—21no—
85Q8×D97—21no—
86Q8×D97C2×C412yescomplemented
87Q8×D97C2312yes—
88Q8×D97C2311yescharacteristic
89D4×D97—21no—
90D4×D97C2311yescharacteristic
91C2×C4×D97—21no—
92C2×C4×D97—21no—
93C2×C4×D97C2311yescharacteristic
94Dic388C2×C412yescomplemented
95#13095501C2×C412yescomplemented
96#13095501C2×C412yescomplemented
97#13095500C2×C412yescomplemented
98#13095499C2×C412yescomplemented
99#13095498C2×C412yescomplemented
100#13095495C2×C411yescharacteristic, complemented
101#13095494C2×C412yescomplemented
102C8×D97C2312yes—
103#41×C97C2×C411yescharacteristic
104C2×Q8×C97C2×C411yescharacteristic, complemented
105Q16×C97C2×C412yescomplemented
106SD16×C97C2×C412yescomplemented
107M16×C97C2×C411yescharacteristic, complemented
108C22×D97—21no—
109C22×D97C2×D411yescharacteristic
110C2×#87604—22no—
111C2×#87604—21no—
112C2×#87604—21no—
113C2×#87604—22no—
114C2×#87604—41no—
115C2×#87604—42no—
116C2×#87604—42no—
117C2×#87604—42no—
118C2×#87604C2×D411yescharacteristic
119C2×#87604C2×D411yescharacteristic
120C2×#87604C2×D411yescharacteristic
121C2×#87604C2×D411yescharacteristic
122Q8×C97—21no—
123Q8×C97C22×C412yes—
124Q8×C97C22×C411yescharacteristic
125D4×C97—21no—
126D4×C97C22×C411yescharacteristic
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97C22×C411yescharacteristic
130C22⋊D97—21no—
131C22⋊D97—21no—
132C22⋊D97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97#4111yescharacteristic
136D388—21no—
137D388C22×C411yescharacteristic
138C4×D97—22no—
139C4×D97—21no—
140C4×D97—22no—
141C4×D97—21no—
142C4×D97—21no—
143C4×D97C22×C411yescharacteristic
144C4×D97C2×D411yescharacteristic
145C4×D97#4111yescharacteristic
146C4×D97C2411yescharacteristic
147Dic194—21no—
148Dic194—22no—
149Dic194—22no—
150Dic194—21no—
151Dic194C22×C412yes—
152Dic194C22×C411yescharacteristic
153#11715897—22no—
154C8×C97C22×C412yes—
155#11715895C22×C412yes—
156C22×C97—21no—
157C22×C97C4×D411yescharacteristic
158C2×D97—21no—
159C2×D97—21no—
160C2×D97—21no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97C4×D411yescharacteristic
164C2×D97C22×D411yescharacteristic
165#87604—21no—
166#87604—21no—
167#87604—21no—
168#87604—41no—
169#87604—41no—
170#87604—42no—
171#87604#13712yesdirect product factor
172C4×C97—22no—
173C4×C97—21no—
174C4×C97C4×D411yescharacteristic
175C4×C97C23×C411yescharacteristic
176Dic97—22no—
177Dic97—21no—
178Dic97C4×D411yescharacteristic
179Dic97C2×#4111yescharacteristic
180C2×C97—21no—
181C2×C97—41no—
182C2×C97C2×C4×D411yescharacteristic
183D97—21no—
184D97—41no—
185D97C2×#13711yescharacteristic
186D97#57511yescharacteristic
187C4×#137—971nomaximal, Sylow
188C97C4×#13711yescharacteristic, complemented, Sylow
189C2×#137—971no—
190C4×#41—971no—
191C2×C4×Q8—971no—
192#441—972no—
193#440—972no—
194C4×Q16—972no—
195C4×SD16—972no—
196#428—971no—
197#419—971no—
198#417—971no—
199C4×M16—971no—
200C2×#41—971no—
201C22×Q8—971no—
202#137—971no—
203#137—971no—
204#137—972no—
205#137—972no—
206#137—971no—
207#137#8760411yescharacteristic, direct product factor
208C2×Q16—972no—
209C2×SD16—972no—
210C2×M16—971no—
211C2×M16—971no—
212C4×Q8—972no—
213C4×Q8—972no—
214C4×Q8—971no—
215C4×Q8—971no—
216C4×Q8—1941no—
217C4×Q8—1941no—
218C4×D4—971no—
219C4×D4—971no—
220C4×D4—1941no—
221C4×D4—1941no—
222#117—971no—
223#117—1941no—
224C2×C4⋊C4—971no—
225C2×C4⋊C4—1941no—
226C2×C42—971no—
227C2×C42—1941no—
228C2×C42—1941no—
229#107—972no—
230#106—972no—
231#103—972no—
232#103—972no—
233#103—972no—
234#102—972no—
235#97—972no—
236C4×C8—972no—
237#41—971no—
238#41—971no—
239#41—971no—
240#41—1941no—
241#41—1941no—
242#41C2×#8760411yescharacteristic
243C2×Q8—972no—
244C2×Q8—972no—
245C2×Q8—971no—
246C2×Q8—971no—
247C2×Q8—1941no—
248C2×Q8—1941no—
249C2×Q8—1941no—
250C2×Q8C2×#8760411yescharacteristic, complemented
251C2×D4—971no—
252C2×D4—1941no—
253C22×C4—971no—
254C22×C4—971no—
255C22×C4—971no—
256C22×C4—1941no—
257C22×C4—1941no—
258C22×C4—1941no—
259C22×C4—1941no—
260C22×C4—1941no—
261Q16—972no—
262Q16—972no—
263Q16—972no—
264Q16C2×#8760412yescomplemented
265SD16—972no—
266SD16—972no—
267SD16—972no—
268SD16C2×#8760412yescomplemented
269M16—971no—
270M16—972no—
271M16—1941no—
272M16—1941no—
273M16C2×#8760411yescharacteristic, complemented
274C2×C8—972no—
275C2×C8—972no—
276C4⋊C4—972no—
277C4⋊C4—971no—
278C4⋊C4—971no—
279C4⋊C4—1941no—
280C4⋊C4—1942no—
281C4⋊C4—1942no—
282C4⋊C4—1941no—
283C4⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C22⋊C4—1941no—
287C42—971no—
288C42—971no—
289C42—971no—
290C42—971no—
291C42—1941no—
292C42—1941no—
293C42—1941no—
294C42—1942no—
295C42—1942no—
296C23—971no—
297C23—1941no—
298Q8—21no—
299Q8—972no—
300Q8—971no—
301Q8—1942no—
302Q8—1942no—
303Q8—1941no—
304Q8—1941no—
305Q8C22×#8760412yes—
306Q8C22×#8760411yescharacteristic
307D4—21no—
308D4—971no—
309D4—1941no—
310D4—1941no—
311D4—1941no—
312D4—1941no—
313D4C22×#8760411yescharacteristic
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—971no—
317C2×C4—971no—
318C2×C4—971no—
319C2×C4—971no—
320C2×C4—971no—
321C2×C4—971no—
322C2×C4—971no—
323C2×C4—971no—
324C2×C4—971no—
325C2×C4—3882no—
326C2×C4—3882no—
327C2×C4—3882no—
328C2×C4—3881no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1942no—
332C2×C4—1942no—
333C2×C4—1941no—
334C2×C4—1942no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1942no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4C22×#8760411yescharacteristic
341C8—972no—
342C8—1942no—
343C8C22×#8760412yes—
344C22—21no—
345C22—971no—
346C22—971no—
347C22—3881no—
348C22—3881no—
349C22—1941no—
350C22—1941no—
351C22—1941no—
352C22D4×#8760411yescharacteristic
353C4—21no—
354C4—22no—
355C4—972no—
356C4—971no—
357C4—971no—
358C4—3881no—
359C4—3882no—
360C4—3881no—
361C4—1941no—
362C4—1941no—
363C4—1941no—
364C4—1941no—
365C4—1942no—
366C4D4×#8760411yescharacteristic
367C4C23×#8760411yescharacteristic
368C2—21no—
369C2—41no—
370C2—971no—
371C2—971no—
372C2—3881no—
373C2—1941no—
374C2C2×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
hypocenterC97C4×#137
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. C2×Q8×#87604C2semidirect
3. #20261775C2semidirect
4. #20261774C2semidirect
5. Q16×#87604C2semidirect
6. SD16×#87604C2semidirect
7. #20261746C2semidirect
8. #20261728C2semidirect
10. M16×#87604C2semidirect
40. #15690843C4semidirect
41. #15690842C4semidirect
42. #15690841C4semidirect
43. #15690840C4semidirect
44. #15690839C4semidirect
48. #137×C97C4semidirect
83. C2×Dic194C2×C4semidirect
86. Q8×D97C2×C4semidirect
94. Dic388C2×C4semidirect
95. #13095501C2×C4semidirect
96. #13095501C2×C4semidirect
97. #13095500C2×C4semidirect
98. #13095499C2×C4semidirect
99. #13095498C2×C4semidirect
100. #13095495C2×C4semidirect
101. #13095494C2×C4semidirect
104. C2×Q8×C97C2×C4semidirect
105. Q16×C97C2×C4semidirect
106. SD16×C97C2×C4semidirect
107. M16×C97C2×C4semidirect
171. #87604#137direct (N × H)
188. C97C4×#137semidirect
207. #137#87604direct (N × H)
250. C2×Q8C2×#87604semidirect
264. Q16C2×#87604semidirect
268. SD16C2×#87604semidirect
273. M16C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#41×#876042
C2×Q8×#876042
#202617752
#202617742
Q16×#876042
SD16×#876042
#202617462
#202617282
#202617262
M16×#876042
#137×D972
C4×#13797