Finite Group Databasebeta

← #35375258

Subgroups — #35375258

Order 21248.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918022C211yesmaximal, characteristic
2#24916557C211yesmaximal, characteristic, complemented
3#24916538C211yesmaximal, characteristic
4#24906343C212yesmaximal, complemented
5#24906322C212yesmaximal, complemented
6#24906308C212yesmaximal, complemented
7#24906301C212yesmaximal, complemented
8#24906226C211yesmaximal, characteristic, complemented
9#24905937C211yesmaximal, characteristic, complemented
10#24905933C211yesmaximal, characteristic
11C83×#1452C211yesmaximal, characteristic, complemented
12#19128258C2211yescharacteristic
13#19128251C2211yescharacteristic
14#19128099C411yescharacteristic
15#19128074C411yescharacteristic
16#19128052C411yescharacteristic
17#19128048C411yescharacteristic
18#19127930C412yes—
19#19127930C412yes—
20#19127831C2212yes—
21#19127830—22no—
22C8×Dic166C2212yes—
23C4×#13050579C2212yes—
24C4×#13050579C2212yes—
25#19127780C2212yes—
26#19127771—22no—
27C8×D332C2212yes—
28#19127759—22no—
29#19127758C2211yescharacteristic
30#19127758C2211yescharacteristic
31#19127666—22no—
32#19127454C2212yes—
33#19127452C2212yes—
34#19127433C2212yes—
35#19127433C2212yes—
36#19127430C2212yes—
37#19127430C2212yes—
38#431×C83C2211yescharacteristic
39#431×C83C2211yescharacteristic
40C2×C4×C8×C83C2211yescharacteristic
41#322×C83C2211yescharacteristic
42#322×C83C2211yescharacteristic
43#322×C83C2211yescharacteristic
44#322×C83C2211yescharacteristic
45C2×#13050590—21no—
46C2×#13050579C2×C412yes—
47C2×#13050579C2×C412yes—
48C2×#13050573—22no—
49C2×C8×D83—22no—
50#14581792C2×C411yescharacteristic
51#14581792C2×C411yescharacteristic
52#14581792C2×C411yescharacteristic
53#14581789C2×C411yescharacteristic
54#14581784—21no—
55#14581780—21no—
56#14581771—21no—
57#14581771C2×C412yes—
58#14581766—21no—
59#14581766C2×C412yes—
60C4×Dic166—21no—
61C4×Dic166C2312yes—
62C4×D332—21no—
63C4×D332C2312yes—
64#14581751—21no—
65#14581724C2×C412yes—
66#14581724C2×C412yes—
67C4⋊(C83⋊C8)C2312yes—
68C4⋊(C83⋊C8)C2312yes—
69Dic83⋊C8—22no—
70Dic83⋊C8—22no—
71#14581717C2×C412yes—
72#14581717C2×C412yes—
73#14581714C2×C412yes—
74#14581714C2×C412yes—
75C22⋊(C83⋊C8)—21no—
76C22⋊(C83⋊C8)—21no—
77(C2×D83)⋊C8—22no—
78(C2×D83)⋊C8—22no—
79#14581703—21no—
80#14581703—21no—
81#14581702—22no—
82C8×Dic83—22no—
83C22×C8×C83C2×C411yescharacteristic
84C22×C8×C83C2×C411yescharacteristic
85C2×C42×C83C2311yescharacteristic
86C4⋊C8×C83—21no—
87C4⋊C8×C83D412yes—
88C22⋊C8×C83—21no—
89C22⋊C8×C83D412yes—
90C4×C8×C83C2×C411yescharacteristic
91C4×C8×C83C2×C411yescharacteristic
92C4×C8×C83C2311yescharacteristic
93C4×C8×C83C2311yescharacteristic
94C4×C8×C83C2311yescharacteristic
95C4×C8×C83C2311yescharacteristic
96C4×C8×C83C2311yescharacteristic
97C4×C8×C83C2311yescharacteristic
98#13050590—82no—
99C2×Dic166—21no—
100C2×Dic166—21no—
101C2×Dic166C22×C412yes—
102C2×D332—21no—
103C2×D332—21no—
104C2×D332C22×C412yes—
105C2×C22⋊D83—22no—
106C2×C4×D83—22no—
107#13050579—42no—
108#13050579—42no—
109#13050573—82no—
110C2×C83⋊C8—22no—
111C2×C83⋊C8—22no—
112C8×D83—82no—
113C4⋊Dic83—21no—
114C4⋊Dic83—21no—
115C4⋊Dic83—21no—
116C4⋊Dic83—21no—
117C4⋊Dic83C22×C412yes—
118C4⋊Dic83C22×C412yes—
119Dic83⋊C4—22no—
120Dic83⋊C4—22no—
121C22⋊Dic83—22no—
122(C2×D83)⋊C4—22no—
123(C2×D83)⋊C4—22no—
124C4×Dic83—22no—
125C22×C4×C83C22×C411yescharacteristic
126C22×C4×C83C2×D411yescharacteristic
127C22×C4×C83#4111yescharacteristic
128C2×C8×C83—21no—
129C2×C8×C83—21no—
130C2×C8×C83—41no—
131C2×C8×C83—41no—
132C2×C8×C83C22×C411yescharacteristic
133C2×C8×C83C22×C411yescharacteristic
134C2×C8×C83C22×C411yescharacteristic
135C2×C8×C83C22×C411yescharacteristic
136C2×C8×C83C2×D412yes—
137C2×C8×C83C2×D412yes—
138C42×C83C22×C411yescharacteristic
139C42×C83C2×D411yescharacteristic
140C42×C83#4111yescharacteristic
141C42×C83C2411yescharacteristic
142C23×C83C4×D411yescharacteristic
143C22×D83—22no—
144C2×C4×C83—21no—
145C2×C4×C83—21no—
146C2×C4×C83—22no—
147C2×C4×C83—41no—
148C2×C4×C83C4×D411yescharacteristic
149C2×C4×C83C4×D411yescharacteristic
150C2×C4×C83C4×D411yescharacteristic
151C2×C4×C83C23×C411yescharacteristic
152C2×C4×C83C22×D411yescharacteristic
153C2×C4×C83C2×#4111yescharacteristic
154C22⋊D83—82no—
155C2×Dic83—22no—
156C2×Dic83—22no—
157C2×Dic83—22no—
158D332—22no—
159D332—22no—
160D332—81no—
161D332—81no—
162C4×D83—42no—
163C4×D83—42no—
164Dic166—22no—
165Dic166—22no—
166Dic166—81no—
167Dic166—81no—
168C8×C83—21no—
169C8×C83—21no—
170C8×C83—21no—
171C8×C83—21no—
172C8×C83—22no—
173C8×C83—22no—
174C8×C83—41no—
175C8×C83—41no—
176C83⋊C8—42no—
177C83⋊C8—42no—
178C22×C83—21no—
179C22×C83—21no—
180C22×C83—21no—
181C22×C83C2×C4×D411yescharacteristic
182C2×D83—42no—
183C2×D83—42no—
184C2×D83—42no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—21no—
189C4×C83#56811yescharacteristic
190C4×C83#56811yescharacteristic
191C4×C83#57712yes—
192Dic83—42no—
193Dic83—42no—
194Dic83—42no—
195#17990—831nomaximal, Sylow
196C2×C83—41no—
197C2×C83#293211yescharacteristic
198C2×C83#294911yescharacteristic
199C2×C83#295711yescharacteristic
200D83—82no—
201#3167—831no—
202#2985—831no—
203#2981—831no—
204#1588—832no—
205#1585—832no—
206#1583—832no—
207#1582—832no—
208#1568—831no—
209#1497—831no—
210#1496—831no—
211#1452D8311yescharacteristic, complemented
212C83#1799011yescharacteristic, complemented, Sylow
213#533—831no—
214#532—831no—
215#498—831no—
216#492—831no—
217#487—831no—
218#486—831no—
219#465—832no—
220#465—832no—
221#446—832no—
222#446—1662no—
223C8×Q8—832no—
224C4×SD16—832no—
225C4×SD16—832no—
226#436—832no—
227#435—1662no—
228C8×D4—832no—
229#433—1662no—
230#432—831no—
231#432—831no—
232#431C2×D8311yescharacteristic
233#431C2×D8311yescharacteristic
234#408—1662no—
235C2×C4×C8C2×D8311yescharacteristic
236#335—832no—
237#334—832no—
238Q8⋊C8—832no—
239Q8⋊C8—832no—
240D4⋊C8—832no—
241D4⋊C8—832no—
242#322C2×D8311yescharacteristic
243#322C2×D8311yescharacteristic
244#322C2×D8311yescharacteristic
245#322C2×D8311yescharacteristic
246C2×#41—1661no—
247C2×SD16—832no—
248C2×SD16—832no—
249C2×M16—1662no—
250C22×C8—1662no—
251C22×C8C4×D8311yescharacteristic
252C22×C8C4×D8311yescharacteristic
253C4⋊Q8—831no—
254C4⋊Q8—831no—
255C4⋊Q8—831no—
256#127—831no—
257#125—1661no—
258#124—1661no—
259C22⋊Q8—832no—
260C22⋊Q8—1661no—
261#121—832no—
262#121—1661no—
263C4×Q8—832no—
264C4×Q8—1661no—
265C4×D4—832no—
266C4×D4—1661no—
267#117—1661no—
268C2×C42C22×D8311yescharacteristic
269#106—832no—
270#106—832no—
271C4⋊C8—21no—
272C4⋊C8—832no—
273C4⋊C8—832no—
274C4⋊C8—1662no—
275C4⋊C8—1662no—
276C4⋊C8D33212yes—
277#103—832no—
278#103—832no—
279#102—832no—
280#102—832no—
281C22⋊C8—21no—
282C22⋊C8—1661no—
283C22⋊C8—1661no—
284C22⋊C8—1662no—
285C22⋊C8—1662no—
286C22⋊C8D33212yes—
287#97—1662no—
288#97—1661no—
289#97—1661no—
290C4×C8—1662no—
291C4×C8C4×D8311yescharacteristic
292C4×C8C4×D8311yescharacteristic
293C4×C8C22×D8311yescharacteristic
294C4×C8C22×D8311yescharacteristic
295C4×C8C22×D8311yescharacteristic
296C4×C8C22×D8311yescharacteristic
297C4×C8C22×D8311yescharacteristic
298C4×C8C22×D8311yescharacteristic
299#41—6642no—
300C2×Q8—832no—
301C2×Q8—1661no—
302C2×Q8—1661no—
303C2×D4—832no—
304C2×D4—1662no—
305C2×D4—1661no—
306C2×D4—1661no—
307C22×C4—1662no—
308C22×C4C2×C4×D8311yescharacteristic
309C22×C4C2×D33211yescharacteristic
310C22×C4#1305059011yescharacteristic
311SD16—3322no—
312SD16—3322no—
313M16—6642no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—6642no—
319C2×C8—1662no—
320C2×C8—1662no—
321C2×C8C2×C4×D8311yescharacteristic
322C2×C8C2×C4×D8311yescharacteristic
323C2×C8C2×C4×D8311yescharacteristic
324C2×C8C2×C4×D8311yescharacteristic
325C2×C8C2×D33212yes—
326C2×C8C2×D33212yes—
327C4⋊C4—832no—
328C4⋊C4—832no—
329C4⋊C4—1662no—
330C4⋊C4—1661no—
331C4⋊C4—1661no—
332C4⋊C4—1661no—
333C4⋊C4—1662no—
334C4⋊C4—1661no—
335C22⋊C4—1662no—
336C22⋊C4—1662no—
337C22⋊C4—1662no—
338C42—1662no—
339C42C2×C4×D8311yescharacteristic
340C42C2×D33211yescharacteristic
341C42#1305059011yescharacteristic
342C42C23×D8311yescharacteristic
343C23—1662no—
344C23C4×D33211yescharacteristic
345Q8—6641no—
346Q8—6641no—
347Q8—1662no—
348Q8—1662no—
349D4—6641no—
350D4—6641no—
351D4—6642no—
352D4—1662no—
353D4—1662no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—3322no—
359C2×C4—3322no—
360C2×C4—1662no—
361C2×C4—1662no—
362C2×C4—1662no—
363C2×C4C4×D33211yescharacteristic
364C2×C4C4×D33211yescharacteristic
365C2×C4C4×D33211yescharacteristic
366C2×C4C22×C4×D8311yescharacteristic
367C2×C4C22×D33211yescharacteristic
368C2×C4C2×#1305059011yescharacteristic
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—22no—
374C8—22no—
375C8—41no—
376C8—41no—
377C8—3322no—
378C8—3322no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—3322no—
383C22—3322no—
384C22—3322no—
385C22C2×C4×D33211yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—3322no—
391C4—3322no—
392C4—3322no—
393C4#1912846711yescharacteristic
394C4#1912846711yescharacteristic
395C4#1912851312yes—
396C2—41no—
397C2—6642no—
398C2#2491618211yescharacteristic
399C2#2491632411yescharacteristic
400C2#2491638111yescharacteristic
subgroupG/H
centerC2×C4C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1452C2
socleC22×C83C2×C4×D4
prime radical#19128251C22
cosocleC42×C83C24
hypercenter#1452D83
hypocenterC83#17990
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1452

Complements

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

NH ≅ G/Ntype
2. #24916557C2semidirect
4. #24906343C2semidirect
5. #24906322C2semidirect
6. #24906308C2semidirect
7. #24906301C2semidirect
8. #24906226C2semidirect
9. #24905937C2semidirect
11. C83×#1452C2semidirect
211. #1452D83semidirect
212. C83#17990semidirect

Maximal subgroups

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

subgroupindex
#249180222
#249165572
#249165382
#249063432
#249063222
#249063082
#249063012
#249062262
#249059372
#249059332
C83×#14522
#1799083