Finite Group Databasebeta

← C83×#9358 #35237182

Subgroups — C83×#9358 #35237182

Order 21248.

454 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3079C211yesmaximal, characteristic
2C83×#2991C211yesmaximal, characteristic
3C83×#1966C211yesmaximal, characteristic, complemented
4C83×#1549C211yesmaximal, characteristic, complemented
5C83×#1489C211yesmaximal, characteristic
6C83×#1309C211yesmaximal, characteristic, complemented
7C83×#1299C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C83C2211yescharacteristic
9C4×#41×C83C2211yescharacteristic
10#477×C83—21no—
11#476×C83—21no—
12#475×C83—21no—
13#474×C83—21no—
14C8×Q8×C83—21no—
15#436×C83—22no—
16#436×C83—21no—
17#431×C83—21no—
18#431×C83C2211yescharacteristic
19#428×C83C412yes—
20#423×C83—21no—
21#423×C83C2211yescharacteristic
22#423×C83C2211yescharacteristic
23#423×C83C2211yescharacteristic
24#417×C83C412yes—
25C2×#101×C83—22no—
26C4×M16×C83C2211yescharacteristic
27#332×C83—21no—
28#332×C83—21no—
29Q8⋊C8×C83—21no—
30D4⋊C8×C83—21no—
31C2×#41×C83C2×C411yescharacteristic
32C22×Q8×C83—21no—
33C2×M16×C83—22no—
34C2×M16×C83—21no—
35C2×M16×C83—21no—
36C2×M16×C83C2×C412yes—
37C2×M16×C83D412yes—
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83—21no—
40C4⋊Q8×C83—21no—
41C4⋊Q8×C83—21no—
42C4×Q8×C83D411yescharacteristic
43C4×Q8×C83D411yescharacteristic
44C4×D4×C83—21no—
45C4×D4×C83—21no—
46C4×D4×C83D411yescharacteristic
47C4×D4×C83D411yescharacteristic
48#117×C83—21no—
49#117×C83C2×C411yescharacteristic
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83C2×C412yes—
52C2×C42×C83—21no—
53C2×C42×C83C2311yescharacteristic
54#107×C83—22no—
55#106×C83—22no—
56C4⋊C8×C83—22no—
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83D411yescharacteristic
64C4⋊C8×C83D411yescharacteristic
65#103×C83—22no—
66#102×C83—22no—
67#101×C83—44no—
68C22⋊C8×C83—22no—
69C22⋊C8×C83—21no—
70#97×C83—21no—
71C4×C8×C83—22no—
72C4×C8×C83—21no—
73C4×C8×C83—21no—
74#41×C83—22no—
75#41×C83—41no—
76C2×Q8×C83—42no—
77C2×Q8×C83—42no—
78C2×Q8×C83—41no—
79C2×Q8×C83—41no—
80C2×Q8×C83C22⋊C411yescharacteristic
81C2×D4×C83—21no—
82C2×D4×C83C22⋊C411yescharacteristic
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—22no—
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83#4111yescharacteristic
90M16×C83—22no—
91M16×C83—41no—
92M16×C83—42no—
93M16×C83—42no—
94M16×C83—82no—
95M16×C83—81no—
96C2×C8×C83—22no—
97C2×C8×C83—21no—
98C2×C8×C83—22no—
99C2×C8×C83—21no—
100C2×C8×C83—41no—
101C2×C8×C83—41no—
102C4⋊C4×C83—21no—
103C4⋊C4×C83—21no—
104C4⋊C4×C83—22no—
105C4⋊C4×C83—22no—
106C4⋊C4×C83—41no—
107C4⋊C4×C83—41no—
108C4⋊C4×C83C22⋊C411yescharacteristic
109C4⋊C4×C83C22⋊C411yescharacteristic
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C22⋊C4×C83—21no—
113C42×C83—21no—
114C42×C83—21no—
115C42×C83—21no—
116C42×C83C2×D411yescharacteristic
117C42×C83C2×D411yescharacteristic
118C42×C83C2×D411yescharacteristic
119C42×C83#4111yescharacteristic
120C23×C83—21no—
121C23×C83C4×D411yescharacteristic
122Q8×C83—21no—
123Q8×C83—82no—
124Q8×C83—81no—
125Q8×C83—81no—
126Q8×C83#10112yescomplemented
127D4×C83—21no—
128D4×C83—21no—
129D4×C83—42no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—42no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—44no—
145C2×C4×C83—42no—
146C2×C4×C83C2×C22⋊C411yescharacteristic
147C2×C4×C83C4×D411yescharacteristic
148C2×C4×C83#12011yescharacteristic
149C2×C4×C83#12111yescharacteristic
150C2×C4×C83#12111yescharacteristic
151C2×C4×C83#12311yescharacteristic
152C8×C83—42no—
153C8×C83—42no—
154C8×C83—42no—
155C8×C83—41no—
156C8×C83—41no—
157C8×C83—81no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83#38611yescharacteristic
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—22no—
168C4×C83—21no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—82no—
172C4×C83C2×#10111yescharacteristic
173C4×C83#41311yescharacteristic
174C4×C83#45311yescharacteristic
175C4×C83#45411yescharacteristic
176C4×C83#47111yescharacteristic
177C4×C83#47311yescharacteristic
178#9358C8311yesmaximal, characteristic, direct product factor, Sylow
179C2×C83—41no—
180C2×C83—81no—
181C2×C83#188311yescharacteristic
182C2×C83#190111yescharacteristic
183C2×C83#190811yescharacteristic
184#3079C2×C8311yescharacteristic
185#2991C2×C8311yescharacteristic
186#1966C2×C8311yescharacteristic, complemented
187#1549C2×C8311yescharacteristic, complemented
188#1489C2×C8311yescharacteristic
189#1309C2×C8311yescharacteristic, complemented
190#1299C2×C8311yescharacteristic, complemented
191C83#935811yescharacteristic, direct product factor, Sylow
192C2×C4⋊Q8C22×C8311yescharacteristic
193C4×#41C22×C8311yescharacteristic
194#477—21no—
195#476—21no—
196#475—21no—
197#474—21no—
198C8×Q8—21no—
199#436—22no—
200#436—21no—
201#431—21no—
202#431C22×C8311yescharacteristic
203#428C4×C8312yes—
204#423—21no—
205#423C22×C8311yescharacteristic
206#423C22×C8311yescharacteristic
207#423C22×C8311yescharacteristic
208#417C4×C8312yes—
209C2×#101—22no—
210C4×M16C22×C8311yescharacteristic
211#332—21no—
212#332—21no—
213Q8⋊C8—21no—
214D4⋊C8—21no—
215C2×#41C2×C4×C8311yescharacteristic
216C22×Q8—21no—
217C2×M16—22no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16C2×C4×C8312yes—
221C2×M16D4×C8312yes—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4×Q8D4×C8311yescharacteristic
227C4×Q8D4×C8311yescharacteristic
228C4×D4—21no—
229C4×D4—21no—
230C4×D4D4×C8311yescharacteristic
231C4×D4D4×C8311yescharacteristic
232#117—21no—
233#117C2×C4×C8311yescharacteristic
234C2×C4⋊C4—21no—
235C2×C4⋊C4C2×C4×C8312yes—
236C2×C42—21no—
237C2×C42C23×C8311yescharacteristic
238#107—22no—
239#106—22no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C8311yescharacteristic
248C4⋊C8D4×C8311yescharacteristic
249#103—22no—
250#102—22no—
251#101—44no—
252C22⋊C8—22no—
253C22⋊C8—21no—
254#97—21no—
255C4×C8—22no—
256C4×C8—21no—
257C4×C8—21no—
258#41—22no—
259#41—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8C22⋊C4×C8311yescharacteristic
265C2×D4—21no—
266C2×D4C22⋊C4×C8311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4C22×C4×C8311yescharacteristic
272C22×C4C2×D4×C8311yescharacteristic
273C22×C4#41×C8311yescharacteristic
274M16—22no—
275M16—41no—
276M16—42no—
277M16—42no—
278M16—82no—
279M16—81no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4C22⋊C4×C8311yescharacteristic
293C4⋊C4C22⋊C4×C8311yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42C2×D4×C8311yescharacteristic
301C42C2×D4×C8311yescharacteristic
302C42C2×D4×C8311yescharacteristic
303C42#41×C8311yescharacteristic
304C23—21no—
305C23C4×D4×C8311yescharacteristic
306Q8—21no—
307Q8—82no—
308Q8—81no—
309Q8—81no—
310Q8#101×C8312yescomplemented
311D4—21no—
312D4—21no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4C2×C22⋊C4×C8311yescharacteristic
331C2×C4C4×D4×C8311yescharacteristic
332C2×C4#120×C8311yescharacteristic
333C2×C4#121×C8311yescharacteristic
334C2×C4#121×C8311yescharacteristic
335C2×C4#123×C8311yescharacteristic
336C8—42no—
337C8—42no—
338C8—42no—
339C8—41no—
340C8—41no—
341C8—81no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22#386×C8311yescharacteristic
349C4—21no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—82no—
356C4C2×#101×C8311yescharacteristic
357C4#413×C8311yescharacteristic
358C4#453×C8311yescharacteristic
359C4#454×C8311yescharacteristic
360C4#471×C8311yescharacteristic
361C4#473×C8311yescharacteristic
362C2—41no—
363C2—81no—
364C2C83×#188311yescharacteristic
365C2C83×#190111yescharacteristic
366C2C83×#190811yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#41×C83C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1966C2semidirect
4. C83×#1549C2semidirect
6. C83×#1309C2semidirect
7. C83×#1299C2semidirect
126. Q8×C83#101semidirect
178. #9358C83direct (N × H)
186. #1966C2×C83semidirect
187. #1549C2×C83semidirect
189. #1309C2×C83semidirect
190. #1299C2×C83semidirect
191. C83#9358direct (N × H)
310. Q8#101×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#30792
C83×#29912
C83×#19662
C83×#15492
C83×#14892
C83×#13092
C83×#12992
#935883