Finite Group Databasebeta

← C83×#9297 #35237121

Subgroups — C83×#9297 #35237121

Order 21248.

454 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3209C211yesmaximal, characteristic
2C83×#2331C211yesmaximal, characteristic
3C83×#1995C211yesmaximal, characteristic
4C83×#1548C211yesmaximal, characteristic
5C83×#1518C211yesmaximal, characteristic
6C83×#1286C211yesmaximal, characteristic
7C83×#1274C211yesmaximal, characteristic
8#533×C83C2211yescharacteristic
9C2×C4⋊Q8×C83C2211yescharacteristic
10#494×C83C412yescomplemented
11#487×C83C412yescomplemented
12#462×C83—22no—
13#451×C83—22no—
14#431×C83C2211yescharacteristic
15#431×C83C2211yescharacteristic
16#423×C83C2211yescharacteristic
17#391×C83—22no—
18#389×C83—21no—
19#387×C83—22no—
20#382×C83C2211yescharacteristic
21C4×C4⋊C4×C83—21no—
22#376×C83C2211yescharacteristic
23#339×C83—21no—
24#339×C83—21no—
25#333×C83—22no—
26#328×C83—22no—
27C22×Q8×C83D412yes—
28C2×Q16×C83—24no—
29C2×M16×C83—21no—
30C4⋊Q8×C83—21no—
31C4⋊Q8×C83—21no—
32C4⋊Q8×C83C2×C411yescharacteristic
33C4⋊Q8×C83C2×C411yescharacteristic
34C4⋊Q8×C83C2×C412yes—
35C4⋊Q8×C83D412yes—
36#125×C83—21no—
37C22⋊Q8×C83—21no—
38C22⋊Q8×C83—21no—
39C4×Q8×C83—21no—
40C4×Q8×C83—21no—
41#117×C83—21no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83—22no—
46C2×C42×C83—22no—
47C2×C42×C83D411yescharacteristic
48C2×C42×C83D411yescharacteristic
49C2×C42×C83C2311yescharacteristic
50C4⋊C8×C83—21no—
51C4⋊C8×C83—21no—
52C4⋊C8×C83—21no—
53C4⋊C8×C83—21no—
54#103×C83—24no—
55C22⋊C8×C83—21no—
56C22⋊C8×C83—21no—
57C4×C8×C83—21no—
58C4×C8×C83C2×C412yes—
59#95×C83—22no—
60#95×C83—22no—
61#95×C83—21no—
62#95×C83—21no—
63#95×C83—21no—
64#95×C83D412yes—
65C2×Q8×C83—22no—
66C2×Q8×C83—21no—
67C2×Q8×C83—21no—
68C2×Q8×C83—82no—
69C2×Q8×C83C22⋊C414yes—
70C22×C4×C83—21no—
71C22×C4×C83—22no—
72C22×C4×C83—21no—
73C22×C4×C83—22no—
74C22×C4×C83C2×D411yescharacteristic
75C22×C4×C83C2×D411yescharacteristic
76C22×C4×C83C2×D411yescharacteristic
77Q16×C83—84no—
78M16×C83—41no—
79M16×C83—41no—
80C2×C8×C83—22no—
81C2×C8×C83—41no—
82C2×C8×C83—41no—
83C4⋊C4×C83—22no—
84C4⋊C4×C83—22no—
85C4⋊C4×C83—22no—
86C4⋊C4×C83—22no—
87C4⋊C4×C83—21no—
88C4⋊C4×C83—21no—
89C4⋊C4×C83—21no—
90C4⋊C4×C83—21no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—82no—
96C22⋊C4×C83—41no—
97C42×C83—21no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—41no—
101C42×C83—82no—
102C42×C83Q1612yescomplemented
103C42×C83C22×C411yescharacteristic
104C42×C83C2×D411yescharacteristic
105C42×C83#4111yescharacteristic
106C42×C83#4111yescharacteristic
107C23×C83#12011yescharacteristic
108Q8×C83—44no—
109Q8×C83—42no—
110Q8×C83—82no—
111Q8×C83—81no—
112C2×C4×C83—24no—
113C2×C4×C83—22no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—42no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—42no—
128C2×C4×C83C2×C22⋊C411yescharacteristic
129C2×C4×C83C4×D411yescharacteristic
130C2×C4×C83C4×D411yescharacteristic
131C2×C4×C83#12111yescharacteristic
132C2×C4×C83#12111yescharacteristic
133C2×C4×C83#12311yescharacteristic
134C2×C4×C83C2×Q1611yescharacteristic
135C2×C4×C83#13511yescharacteristic
136C2×C4×C83#13711yescharacteristic
137C2×C4×C83#13711yescharacteristic
138C8×C83—42no—
139C8×C83—81no—
140C8×C83—81no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83#38611yescharacteristic
144C22×C83#45111yescharacteristic
145C22×C83#45211yescharacteristic
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—21no—
150C4×C83—42no—
151C4×C83—44no—
152C4×C83—82no—
153C4×C83—81no—
154C4×C83—81no—
155C4×C83—81no—
156C4×C83C4×Q1611yescharacteristic
157C4×C83#44111yescharacteristic
158C4×C83#46211yescharacteristic
159C4×C83#46411yescharacteristic
160#9297C8311yesmaximal, characteristic, direct product factor, Sylow
161C2×C83—21no—
162C2×C83—21no—
163C2×C83#187811yescharacteristic
164C2×C83#190511yescharacteristic
165C2×C83#191111yescharacteristic
166#3209C2×C8311yescharacteristic
167#2331C2×C8311yescharacteristic
168#1995C2×C8311yescharacteristic
169#1548C2×C8311yescharacteristic
170#1518C2×C8311yescharacteristic
171#1286C2×C8311yescharacteristic
172#1274C2×C8311yescharacteristic
173C83#929711yescharacteristic, direct product factor, Sylow
174#533C22×C8311yescharacteristic
175C2×C4⋊Q8C22×C8311yescharacteristic
176#494C4×C8312yescomplemented
177#487C4×C8312yescomplemented
178#462—22no—
179#451—22no—
180#431C22×C8311yescharacteristic
181#431C22×C8311yescharacteristic
182#423C22×C8311yescharacteristic
183#391—22no—
184#389—21no—
185#387—22no—
186#382C22×C8311yescharacteristic
187C4×C4⋊C4—21no—
188#376C22×C8311yescharacteristic
189#339—21no—
190#339—21no—
191#333—22no—
192#328—22no—
193C22×Q8D4×C8312yes—
194C2×Q16—24no—
195C2×M16—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C2×C4×C8311yescharacteristic
199C4⋊Q8C2×C4×C8311yescharacteristic
200C4⋊Q8C2×C4×C8312yes—
201C4⋊Q8D4×C8312yes—
202#125—21no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207#117—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C42—22no—
213C2×C42D4×C8311yescharacteristic
214C2×C42D4×C8311yescharacteristic
215C2×C42C23×C8311yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220#103—24no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C4×C8—21no—
224C4×C8C2×C4×C8312yes—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95D4×C8312yes—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—82no—
235C2×Q8C22⋊C4×C8314yes—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4C2×D4×C8311yescharacteristic
241C22×C4C2×D4×C8311yescharacteristic
242C22×C4C2×D4×C8311yescharacteristic
243Q16—84no—
244M16—41no—
245M16—41no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—41no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—82no—
262C22⋊C4—41no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—41no—
267C42—82no—
268C42Q16×C8312yescomplemented
269C42C22×C4×C8311yescharacteristic
270C42C2×D4×C8311yescharacteristic
271C42#41×C8311yescharacteristic
272C42#41×C8311yescharacteristic
273C23#120×C8311yescharacteristic
274Q8—44no—
275Q8—42no—
276Q8—82no—
277Q8—81no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4C2×C22⋊C4×C8311yescharacteristic
295C2×C4C4×D4×C8311yescharacteristic
296C2×C4C4×D4×C8311yescharacteristic
297C2×C4#121×C8311yescharacteristic
298C2×C4#121×C8311yescharacteristic
299C2×C4#123×C8311yescharacteristic
300C2×C4C2×Q16×C8311yescharacteristic
301C2×C4#135×C8311yescharacteristic
302C2×C4#137×C8311yescharacteristic
303C2×C4#137×C8311yescharacteristic
304C8—42no—
305C8—81no—
306C8—81no—
307C22—21no—
308C22—21no—
309C22#386×C8311yescharacteristic
310C22#451×C8311yescharacteristic
311C22#452×C8311yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—44no—
318C4—82no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4C4×Q16×C8311yescharacteristic
323C4#441×C8311yescharacteristic
324C4#462×C8311yescharacteristic
325C4#464×C8311yescharacteristic
326C2—21no—
327C2—21no—
328C2C83×#187811yescharacteristic
329C2C83×#190511yescharacteristic
330C2C83×#191111yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC23×C83#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
10. #494×C83C4semidirect
11. #487×C83C4semidirect
102. C42×C83Q16semidirect
160. #9297C83direct (N × H)
173. C83#9297direct (N × H)
176. #494C4×C83semidirect
177. #487C4×C83semidirect
268. C42Q16×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32092
C83×#23312
C83×#19952
C83×#15482
C83×#15182
C83×#12862
C83×#12742
#929783