Finite Group Databasebeta

← #24914065

Subgroups — #24914065

Order 10624.

506 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
1C2×#14581779C211yesmaximal, characteristic
2C2×#14581771C211yesmaximal, characteristic, complemented
3#19127629C211yesmaximal, characteristic, complemented
4#19127626C211yesmaximal, characteristic, complemented
5#19127624C211yesmaximal, characteristic, complemented
6#19127622C211yesmaximal, characteristic, complemented
7#19127615C211yesmaximal, characteristic
8#19127613C211yesmaximal, characteristic, complemented
9#19127605C211yesmaximal, characteristic, complemented
10#19127605C211yesmaximal, characteristic, complemented
11#19127594C211yesmaximal, characteristic, complemented
12#19127589C211yesmaximal, characteristic, complemented
13#19127583C211yesmaximal, characteristic, complemented
14#19127564C211yesmaximal, characteristic, complemented
15#386×C83C211yesmaximal, characteristic
16C22×Dic166C2211yescharacteristic
17C22×Dic166C2211yescharacteristic
18#14581779—24no—
19#14581771—24no—
20C2×C4⋊Dic83C2211yescharacteristic
21C2×C4⋊Dic83C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×C22⋊Dic83C2211yescharacteristic
27C2×C22⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83C2211yescharacteristic
31C2×C22⋊Dic83C2211yescharacteristic
32C2×C22⋊Dic83C2211yescharacteristic
33C2×C22⋊Dic83C2211yescharacteristic
34C2×C4×Dic83C2211yescharacteristic
35C2×C4×Dic83C2211yescharacteristic
36C83⋊#95C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C22×D4×C83C2211yescharacteristic
47C23×C4×C83C2211yescharacteristic
48C2×C22⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51#95×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C2×Dic166—22no—
54C2×Dic166—22no—
55C2×Dic166—22no—
56C2×Dic166—22no—
57C2×Dic166—41no—
58C2×Dic166—41no—
59C22×Dic83C2311yescharacteristic
60C22×Dic83C2311yescharacteristic
61C22×Dic83C2311yescharacteristic
62C22×Dic83C2311yescharacteristic
63C22×Dic83C2311yescharacteristic
64C22×Dic83C2311yescharacteristic
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C4⋊Dic83—22no—
68C4⋊Dic83—41no—
69Dic83⋊C4—22no—
70Dic83⋊C4—22no—
71Dic83⋊C4—22no—
72Dic83⋊C4—41no—
73C22⋊Dic83—22no—
74C22⋊Dic83—22no—
75C22⋊Dic83—22no—
76C22⋊Dic83—22no—
77C22⋊Dic83—41no—
78C22⋊Dic83—41no—
79C22⋊Dic83—41no—
80C22⋊Dic83D414yes—
81C4×Dic83—22no—
82C4×Dic83—41no—
83C24×C83C2311yescharacteristic
84C24×C83C2311yescharacteristic
85C2×D4×C83—22no—
86C2×D4×C83—22no—
87C2×D4×C83—41no—
88C22×C4×C83—22no—
89C22×C4×C83—22no—
90C22×C4×C83—41no—
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C22⋊C4×C83—22no—
97C22⋊C4×C83—41no—
98C22⋊C4×C83—41no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83C2×D412yes—
113C23×C83C2411yescharacteristic
114D4×C83—44no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—44no—
129C2×C4×C83#4112yes—
130C2×C4×C83#4112yes—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—21no—
146C2×Dic83—21no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—21no—
151C2×Dic83—21no—
152C2×Dic83—21no—
153C2×Dic83C2×D412yes—
154C2×Dic83C2×D412yes—
155Dic166—44no—
156Dic166—44no—
157C22×C83—22no—
158C22×C83—22no—
159C22×C83—22no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83—41no—
171C22×C83C22×D411yescharacteristic
172C22×C83C2×#4111yescharacteristic
173C22×C83C2×#4111yescharacteristic
174C22×C83H(5,𝔽2)11yescharacteristic
175C22×C83H(5,𝔽2)11yescharacteristic
176C22×C832−1+411yescharacteristic
177C22×C832−1+411yescharacteristic
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183Dic83—22no—
184Dic83—22no—
185Dic83—41no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189Dic83—41no—
190Dic83—41no—
191C2×C83—42no—
192C2×C83—81no—
193C2×C83#53611yescharacteristic
194C2×C83#54211yescharacteristic
195C2×C83#54211yescharacteristic
196C2×C83#54611yescharacteristic
197C2×C83#54611yescharacteristic
198C2×C83#55611yescharacteristic
199C2×C83#56211yescharacteristic
200#2700—831nomaximal, Sylow
201C83#270011yescharacteristic, complemented, Sylow
202C2×#124—831no—
203C2×C22⋊Q8—831no—
204#397—831no—
205#397—831no—
206#396—831no—
207#395—831no—
208#394—831no—
209#393—831no—
210#391—831no—
211#391—831no—
212#388—831no—
213#388—831no—
214#387—831no—
215#386D8311yescharacteristic
216#381—831no—
217C22×Q8—831no—
218C22×Q8—831no—
219C22×D4C2×D8311yescharacteristic
220C23×C4C2×D8311yescharacteristic
221#124—1664no—
222C22⋊Q8—1664no—
223C2×C4⋊C4—831no—
224C2×C4⋊C4—831no—
225C2×C4⋊C4—831no—
226C2×C4⋊C4—831no—
227C2×C4⋊C4—831no—
228C2×C4⋊C4—831no—
229C2×C22⋊C4—831no—
230C2×C22⋊C4—831no—
231C2×C22⋊C4—831no—
232C2×C22⋊C4—831no—
233C2×C22⋊C4—831no—
234C2×C22⋊C4—831no—
235C2×C22⋊C4—831no—
236C2×C22⋊C4—831no—
237C2×C22⋊C4C2×D8311yescharacteristic
238C2×C22⋊C4C2×D8311yescharacteristic
239C2×C22⋊C4C2×D8311yescharacteristic
240C2×C42—831no—
241C2×C42—831no—
242#95—831no—
243#95—831no—
244#95—831no—
245#95—831no—
246#95—831no—
247#95—831no—
248#95—831no—
249#95—831no—
250#95—831no—
251#95—831no—
252#95C2×D8311yescharacteristic
253#95C2×D8311yescharacteristic
254C24C22×D8311yescharacteristic
255C24C22×D8311yescharacteristic
256C2×Q8—3321no—
257C2×Q8—3321no—
258C2×Q8—1662no—
259C2×Q8—1662no—
260C2×Q8—1662no—
261C2×Q8—1662no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—41no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—831no—
269C22×C4—831no—
270C22×C4—831no—
271C22×C4—831no—
272C22×C4—831no—
273C22×C4—831no—
274C22×C4—831no—
275C22×C4—831no—
276C22×C4C22×D8311yescharacteristic
277C22×C4C22×D8311yescharacteristic
278C22×C4C22×D8311yescharacteristic
279C22×C4C22×D8311yescharacteristic
280C22×C4C22×D8311yescharacteristic
281C4⋊C4—3321no—
282C4⋊C4—3321no—
283C4⋊C4—1662no—
284C4⋊C4—1662no—
285C4⋊C4—1662no—
286C4⋊C4—1662no—
287C22⋊C4—22no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—3321no—
291C22⋊C4—3321no—
292C22⋊C4—3321no—
293C22⋊C4—834no—
294C22⋊C4—1662no—
295C22⋊C4—1662no—
296C22⋊C4—1662no—
297C22⋊C4—1662no—
298C42—3321no—
299C42—1662no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23D4×D8312yes—
314C23C23×D8311yescharacteristic
315Q8—3324no—
316Q8—3324no—
317D4—44no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—832no—
333C2×C4—832no—
334C2×C4—1661no—
335C2×C4—1661no—
336C2×C4—1661no—
337C2×C4—1661no—
338C2×C4—1661no—
339C2×C4—1661no—
340C2×C4—1661no—
341C2×C4—1661no—
342C2×C4—1661no—
343C2×C4—1661no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4—1661no—
354C2×C4—1661no—
355C2×C4—1661no—
356C2×C4#1305058912yes—
357C2×C4#1305059012yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22C2×D4×D8311yescharacteristic
373C22C2×#1305058911yescharacteristic
374C22C2×#1305059011yescharacteristic
375C22#1458183711yescharacteristic
376C22#1458183711yescharacteristic
377C22#1458183911yescharacteristic
378C22#1458183911yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3321no—
385C4—3321no—
386C4—3321no—
387C4—3321no—
388C4—3321no—
389C4—3321no—
390C4—1662no—
391C4—1662no—
392C2—42no—
393C2—81no—
394C2#1912827311yescharacteristic
395C2#1912831611yescharacteristic
396C2#1912831811yescharacteristic
397C2#1912833811yescharacteristic
398C2#1912834111yescharacteristic
399C2#1912842111yescharacteristic
400C2#1912844611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC22×D4×C83C22
hypercenter#386D83
hypocenterC83#2700
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#14581771C2semidirect
3. #19127629C2semidirect
4. #19127626C2semidirect
5. #19127624C2semidirect
6. #19127622C2semidirect
8. #19127613C2semidirect
9. #19127605C2semidirect
10. #19127605C2semidirect
11. #19127594C2semidirect
12. #19127589C2semidirect
13. #19127583C2semidirect
14. #19127564C2semidirect
201. C83#2700semidirect

Maximal subgroups

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

subgroupindex
C2×#145817792
C2×#145817712
#191276292
#191276262
#191276242
#191276222
#191276152
#191276132
#191276052
#191275942
#191275892
#191275832
#191275642
#386×C832
#270083