Finite Group Databasebeta

← C83×#9378 #35237202

Subgroups — C83×#9378 #35237202

Order 21248.

498 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3217C211yesmaximal, characteristic
2C83×#2992C211yesmaximal, characteristic, complemented
3C83×#1982C211yesmaximal, characteristic, complemented
4C83×#1555C211yesmaximal, characteristic, complemented
5C83×#1509C211yesmaximal, characteristic, complemented
6C83×#1290C211yesmaximal, characteristic, complemented
7C83×#1285C211yesmaximal, characteristic, complemented
8C2×#127×C83C2211yescharacteristic
9#529×C83C2211yescharacteristic
10#496×C83C412yescomplemented
11#490×C83C412yescomplemented
12#460×C83—21no—
13#459×C83—21no—
14#450×C83—21no—
15#447×C83—21no—
16#446×C83—21no—
17#435×C83—21no—
18#435×C83—21no—
19C8×D4×C83—21no—
20#433×C83—21no—
21#432×C83C2211yescharacteristic
22#432×C83C2211yescharacteristic
23#432×C83C2211yescharacteristic
24#424×C83—21no—
25C2×C4⋊C8×C83C2211yescharacteristic
26C2×#102×C83—21no—
27C2×#102×C83—21no—
28C4×C4⋊C4×C83C2211yescharacteristic
29#331×C83—21no—
30#331×C83—21no—
31#328×C83—21no—
32#327×C83—21no—
33C22×D4×C83—21no—
34C22×D4×C83—21no—
35C22×D4×C83—21no—
36C2×SD16×C83—22no—
37C2×D8×C83—22no—
38C2×M16×C83—21no—
39C22×C8×C83—21no—
40C22×C8×C83D411yescharacteristic
41C22×C8×C83D411yescharacteristic
42#127×C83—21no—
43#127×C83—21no—
44#127×C83—21no—
45#127×C83C2×C412yes—
46#126×C83—21no—
47#125×C83C2×C411yescharacteristic
48#124×C83C2×C411yescharacteristic
49#123×C83—21no—
50C22⋊Q8×C83D411yescharacteristic
51#121×C83D411yescharacteristic
52C4×Q8×C83D411yescharacteristic
53C4×D4×C83—21no—
54C4×D4×C83D411yescharacteristic
55#117×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C42×C83—21no—
58C2×C42×C83C2311yescharacteristic
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83D411yescharacteristic
66C4⋊C8×C83D411yescharacteristic
67#102×C83—22no—
68#102×C83—22no—
69#102×C83—42no—
70#102×C83—42no—
71C22⋊C8×C83—21no—
72C22⋊C8×C83—21no—
73C22⋊C8×C83—21no—
74C22⋊C8×C83—21no—
75C22⋊C8×C83—21no—
76#97×C83—21no—
77#97×C83—21no—
78#97×C83—21no—
79#97×C83C2×C412yes—
80C4×C8×C83—21no—
81#95×C83—21no—
82C24×C83—41no—
83C2×Q8×C83C22⋊C411yescharacteristic
84C2×D4×C83—21no—
85C2×D4×C83—22no—
86C2×D4×C83—22no—
87C2×D4×C83—22no—
88C2×D4×C83—41no—
89C2×D4×C83—41no—
90C2×D4×C83—41no—
91C2×D4×C83—41no—
92C2×D4×C83—41no—
93C2×D4×C83—41no—
94C2×D4×C83—41no—
95C2×D4×C83—81no—
96C2×D4×C83—81no—
97C2×D4×C83—82no—
98C2×D4×C83C22⋊C411yescharacteristic
99C22×C4×C83—21no—
100C22×C4×C83—41no—
101C22×C4×C83C2×D411yescharacteristic
102C22×C4×C83C2×D411yescharacteristic
103C22×C4×C83#4111yescharacteristic
104SD16×C83—82no—
105D8×C83—82no—
106M16×C83—81no—
107C2×C8×C83—22no—
108C2×C8×C83—21no—
109C2×C8×C83—21no—
110C2×C8×C83—21no—
111C2×C8×C83—21no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C2×C8×C83—81no—
116C4⋊C4×C83—21no—
117C4⋊C4×C83—21no—
118C4⋊C4×C83—21no—
119C4⋊C4×C83—21no—
120C4⋊C4×C83—22no—
121C4⋊C4×C83—41no—
122C4⋊C4×C83C22⋊C411yescharacteristic
123C4⋊C4×C83C22⋊C411yescharacteristic
124C22⋊C4×C83—21no—
125C22⋊C4×C83—21no—
126C22⋊C4×C83—21no—
127C22⋊C4×C83—21no—
128C22⋊C4×C83—21no—
129C42×C83—21no—
130C42×C83—81no—
131C42×C83C22×C411yescharacteristic
132C42×C83C2×D411yescharacteristic
133C42×C83C2×D411yescharacteristic
134C42×C83#4111yescharacteristic
135C23×C83—21no—
136C23×C83—42no—
137C23×C83—81no—
138C23×C83—81no—
139C23×C83—81no—
140C23×C83—81no—
141C23×C83—81no—
142C23×C83—81no—
143C23×C83#12111yescharacteristic
144Q8×C83—41no—
145D4×C83—41no—
146D4×C83—42no—
147D4×C83—42no—
148D4×C83—42no—
149D4×C83—42no—
150D4×C83—41no—
151D4×C83—41no—
152D4×C83—82no—
153D4×C83—82no—
154D4×C83—82no—
155D4×C83—81no—
156D4×C83—81no—
157C2×C4×C83—21no—
158C2×C4×C83—21no—
159C2×C4×C83—21no—
160C2×C4×C83—21no—
161C2×C4×C83—21no—
162C2×C4×C83—21no—
163C2×C4×C83—21no—
164C2×C4×C83—41no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—42no—
168C2×C4×C83—81no—
169C2×C4×C83—81no—
170C2×C4×C83C2×C22⋊C411yescharacteristic
171C2×C4×C83C4×D411yescharacteristic
172C2×C4×C83C4×D411yescharacteristic
173C2×C4×C83#12011yescharacteristic
174C2×C4×C83#12111yescharacteristic
175C2×C4×C83#12311yescharacteristic
176C8×C83—42no—
177C8×C83—41no—
178C8×C83—41no—
179C8×C83—41no—
180C8×C83—41no—
181C8×C83—81no—
182C22×C83—21no—
183C22×C83—21no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—82no—
188C22×C83—82no—
189C22×C83—82no—
190C22×C83—161no—
191C22×C83—161no—
192C22×C83—161no—
193C22×C83—161no—
194C22×C83#38611yescharacteristic
195C22×C83#47111yescharacteristic
196C22×C83#47211yescharacteristic
197C4×C83—21no—
198C4×C83—21no—
199C4×C83—21no—
200C4×C83—21no—
201C4×C83—21no—
202C4×C83—41no—
203C4×C83—41no—
204C4×C83—41no—
205C4×C83—82no—
206C4×C83#45311yescharacteristic
207C4×C83#45411yescharacteristic
208#9378C8311yesmaximal, characteristic, direct product factor, Sylow
209C2×C83—21no—
210C2×C83—21no—
211C2×C83—81no—
212C2×C83—162no—
213C2×C83#188311yescharacteristic
214C2×C83#190711yescharacteristic
215C2×C83#191011yescharacteristic
216#3217C2×C8311yescharacteristic
217#2992C2×C8311yescharacteristic, complemented
218#1982C2×C8311yescharacteristic, complemented
219#1555C2×C8311yescharacteristic, complemented
220#1509C2×C8311yescharacteristic, complemented
221#1290C2×C8311yescharacteristic, complemented
222#1285C2×C8311yescharacteristic, complemented
223C83#937811yescharacteristic, direct product factor, Sylow
224C2×#127C22×C8311yescharacteristic
225#529C22×C8311yescharacteristic
226#496C4×C8312yescomplemented
227#490C4×C8312yescomplemented
228#460—21no—
229#459—21no—
230#450—21no—
231#447—21no—
232#446—21no—
233#435—21no—
234#435—21no—
235C8×D4—21no—
236#433—21no—
237#432C22×C8311yescharacteristic
238#432C22×C8311yescharacteristic
239#432C22×C8311yescharacteristic
240#424—21no—
241C2×C4⋊C8C22×C8311yescharacteristic
242C2×#102—21no—
243C2×#102—21no—
244C4×C4⋊C4C22×C8311yescharacteristic
245#331—21no—
246#331—21no—
247#328—21no—
248#327—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C2×SD16—22no—
253C2×D8—22no—
254C2×M16—21no—
255C22×C8—21no—
256C22×C8D4×C8311yescharacteristic
257C22×C8D4×C8311yescharacteristic
258#127—21no—
259#127—21no—
260#127—21no—
261#127C2×C4×C8312yes—
262#126—21no—
263#125C2×C4×C8311yescharacteristic
264#124C2×C4×C8311yescharacteristic
265#123—21no—
266C22⋊Q8D4×C8311yescharacteristic
267#121D4×C8311yescharacteristic
268C4×Q8D4×C8311yescharacteristic
269C4×D4—21no—
270C4×D4D4×C8311yescharacteristic
271#117—21no—
272C2×C4⋊C4—21no—
273C2×C42—21no—
274C2×C42C23×C8311yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8D4×C8311yescharacteristic
282C4⋊C8D4×C8311yescharacteristic
283#102—22no—
284#102—22no—
285#102—42no—
286#102—42no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292#97—21no—
293#97—21no—
294#97—21no—
295#97C2×C4×C8312yes—
296C4×C8—21no—
297#95—21no—
298C24—41no—
299C2×Q8C22⋊C4×C8311yescharacteristic
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—81no—
312C2×D4—81no—
313C2×D4—82no—
314C2×D4C22⋊C4×C8311yescharacteristic
315C22×C4—21no—
316C22×C4—41no—
317C22×C4C2×D4×C8311yescharacteristic
318C22×C4C2×D4×C8311yescharacteristic
319C22×C4#41×C8311yescharacteristic
320SD16—82no—
321D8—82no—
322M16—81no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—81no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4C22⋊C4×C8311yescharacteristic
339C4⋊C4C22⋊C4×C8311yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—81no—
347C42C22×C4×C8311yescharacteristic
348C42C2×D4×C8311yescharacteristic
349C42C2×D4×C8311yescharacteristic
350C42#41×C8311yescharacteristic
351C23—21no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23#121×C8311yescharacteristic
360Q8—41no—
361D4—41no—
362D4—42no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—41no—
367D4—41no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—81no—
372D4—81no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C2×C22⋊C4×C8311yescharacteristic
387C2×C4C4×D4×C8311yescharacteristic
388C2×C4C4×D4×C8311yescharacteristic
389C2×C4#120×C8311yescharacteristic
390C2×C4#121×C8311yescharacteristic
391C2×C4#123×C8311yescharacteristic
392C8—42no—
393C8—41no—
394C8—41no—
395C8—41no—
396C8—41no—
397C8—81no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22#386×C8311yescharacteristic
411C22#471×C8311yescharacteristic
412C22#472×C8311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4#453×C8311yescharacteristic
423C4#454×C8311yescharacteristic
424C2—21no—
425C2—21no—
426C2—81no—
427C2—162no—
428C2C83×#188311yescharacteristic
429C2C83×#190711yescharacteristic
430C2C83×#191011yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC83×#3217C2
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
2. C83×#2992C2semidirect
3. C83×#1982C2semidirect
4. C83×#1555C2semidirect
5. C83×#1509C2semidirect
6. C83×#1290C2semidirect
7. C83×#1285C2semidirect
10. #496×C83C4semidirect
11. #490×C83C4semidirect
208. #9378C83direct (N × H)
217. #2992C2×C83semidirect
218. #1982C2×C83semidirect
219. #1555C2×C83semidirect
220. #1509C2×C83semidirect
221. #1290C2×C83semidirect
222. #1285C2×C83semidirect
223. C83#9378direct (N × H)
226. #496C4×C83semidirect
227. #490C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32172
C83×#29922
C83×#19822
C83×#15552
C83×#15092
C83×#12902
C83×#12852
#937883