Finite Group Databasebeta

← #24913952

Subgroups — #24913952

Order 10624.

536 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581770C211yesmaximal, characteristic, complemented
2C2×#14581769C211yesmaximal, characteristic, complemented
3#19127634C211yesmaximal, characteristic, complemented
4#19127623C211yesmaximal, characteristic, complemented
5#19127621C211yesmaximal, characteristic, complemented
6#19127614C211yesmaximal, characteristic, complemented
7#19127613C211yesmaximal, characteristic, complemented
8#19127606C211yesmaximal, characteristic, complemented
9#19127601C211yesmaximal, characteristic
10#19127583C211yesmaximal, characteristic, complemented
11#19127578C211yesmaximal, characteristic
12#19127577C211yesmaximal, characteristic
13#19127572C211yesmaximal, characteristic, complemented
14#19127559C211yesmaximal, characteristic, complemented
15#399×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×Dic166C2211yescharacteristic
18C22×C22⋊D83C2211yescharacteristic
19C23×Dic83C2211yescharacteristic
20C22×C4×D83C2211yescharacteristic
21#14581770—24no—
22#14581769—24no—
23C2×C4⋊Dic83C2211yescharacteristic
24C2×Dic83⋊C4C2211yescharacteristic
25C2×Dic83⋊C4C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×C22⋊Dic83C2211yescharacteristic
30C2×C22⋊Dic83C2211yescharacteristic
31C2×(C2×D83)⋊C4C2211yescharacteristic
32C2×(C2×D83)⋊C4C2211yescharacteristic
33C2×(C2×D83)⋊C4C2211yescharacteristic
34C2×(C2×D83)⋊C4C2211yescharacteristic
35C2×(C2×D83)⋊C4C2211yescharacteristic
36C2×C4×Dic83C2211yescharacteristic
37C83⋊#95C2211yescharacteristic
38C83⋊#95C2211yescharacteristic
39C83⋊#95C2211yescharacteristic
40C83⋊#95C2211yescharacteristic
41C83⋊#95C2211yescharacteristic
42C83⋊#95C2211yescharacteristic
43C83⋊#95C2211yescharacteristic
44C83⋊#95C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C2×C4⋊C4×C83C2211yescharacteristic
47C2×C4⋊C4×C83C2211yescharacteristic
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C22⋊C4×C83C2211yescharacteristic
50C2×C22⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52#95×C83C2211yescharacteristic
53C23×D83C2311yescharacteristic
54C2×Dic166—22no—
55C2×Dic166—22no—
56C2×Dic166—22no—
57C2×Dic166—41no—
58C2×Dic166—41no—
59C2×Dic166—41no—
60C2×C22⋊D83—41no—
61C2×C22⋊D83—41no—
62C2×C22⋊D83Q814yescomplemented
63C22×Dic83—22no—
64C22×Dic83—22no—
65C22×Dic83—41no—
66C22×Dic83C2311yescharacteristic
67C22×Dic83C2311yescharacteristic
68C22×Dic83C2311yescharacteristic
69C22×Dic83C2311yescharacteristic
70C22×Dic83C2311yescharacteristic
71C22×Dic83C2311yescharacteristic
72C22×Dic83C2311yescharacteristic
73C2×C4×D83—22no—
74C2×C4×D83—41no—
75C2×C4×D83—41no—
76C4⋊Dic83—22no—
77Dic83⋊C4—22no—
78Dic83⋊C4—22no—
79Dic83⋊C4—41no—
80Dic83⋊C4—41no—
81Dic83⋊C4D414yescomplemented
82C22⋊Dic83—22no—
83C22⋊Dic83—41no—
84(C2×D83)⋊C4—22no—
85(C2×D83)⋊C4—22no—
86(C2×D83)⋊C4—22no—
87(C2×D83)⋊C4—41no—
88(C2×D83)⋊C4—41no—
89C4×Dic83—41no—
90C24×C83C2311yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C22×C4×C83C2311yescharacteristic
96C22×C4×C83C2311yescharacteristic
97C4⋊C4×C83—22no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C22⋊C4×C83—22no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83C2×Q812yes—
110C23×C83C2411yescharacteristic
111C22×D83—21no—
112C22×D83—21no—
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—21no—
116C22×D83—21no—
117C22×D83C2×Q812yes—
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—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83C2×D412yes—
136C22⋊D83—44no—
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×Dic83—21no—
154C2×Dic83—44no—
155C2×Dic83C2×D412yes—
156C2×Dic83C2×D412yes—
157C2×Dic83C2×Q812yes—
158C2×Dic83#4112yes—
159C4×D83—44no—
160Dic166—44no—
161Dic166—44no—
162C22×C83—22no—
163C22×C83—22no—
164C22×C83—22no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83C22×D411yescharacteristic
170C22×C83C22×Q811yescharacteristic
171C22×C83C2×#4111yescharacteristic
172C22×C83H(5,𝔽2)11yescharacteristic
173C22×C83H(5,𝔽2)11yescharacteristic
174C22×C83H(5,𝔽2)11yescharacteristic
175C22×C832−1+411yescharacteristic
176C2×D83—22no—
177C2×D83—22no—
178C2×D83—22no—
179C2×D83—41no—
180C2×D83—41no—
181C2×D83—41no—
182C2×D83—41no—
183C4×C83—22no—
184C4×C83—41no—
185C4×C83—41no—
186C4×C83—41no—
187C4×C83—41no—
188C4×C83—41no—
189Dic83—22no—
190Dic83—22no—
191Dic83—22no—
192Dic83—22no—
193Dic83—41no—
194Dic83—41no—
195Dic83—41no—
196C2×C83—42no—
197C2×C83#53511yescharacteristic
198C2×C83#54211yescharacteristic
199C2×C83#54311yescharacteristic
200C2×C83#54611yescharacteristic
201C2×C83D4×Q811yescharacteristic
202C2×C83#55411yescharacteristic
203C2×C83#56211yescharacteristic
204D83—42no—
205#2692—831nomaximal, Sylow
206C83#269211yescharacteristic, complemented, Sylow
207C2×C22⋊Q8—831no—
208C2×C22⋊Q8—831no—
209#399D8311yescharacteristic, complemented
210#398—831no—
211#396—831no—
212#395—831no—
213#393—831no—
214#393—831no—
215#391—831no—
216#390—831no—
217#387—831no—
218#386—831no—
219#386—831no—
220#385—831no—
221#380—831no—
222C22×Q8—831no—
223C22×Q8—831no—
224C22×D4—831no—
225C23×C4—831no—
226C23×C4—831no—
227C22⋊Q8—1664no—
228C22⋊Q8—1664no—
229C2×C4⋊C4—831no—
230C2×C4⋊C4—831no—
231C2×C4⋊C4—831no—
232C2×C4⋊C4—831no—
233C2×C4⋊C4—831no—
234C2×C4⋊C4—831no—
235C2×C4⋊C4C2×D8311yescharacteristic
236C2×C4⋊C4C2×D8311yescharacteristic
237C2×C4⋊C4C2×D8311yescharacteristic
238C2×C22⋊C4—831no—
239C2×C22⋊C4—831no—
240C2×C22⋊C4—831no—
241C2×C22⋊C4—831no—
242C2×C22⋊C4—831no—
243C2×C22⋊C4—831no—
244C2×C22⋊C4—831no—
245C2×C22⋊C4C2×D8311yescharacteristic
246C2×C22⋊C4C2×D8311yescharacteristic
247C2×C22⋊C4C2×D8311yescharacteristic
248C2×C42—831no—
249#95—831no—
250#95—831no—
251#95—831no—
252#95—831no—
253#95—831no—
254#95—831no—
255#95—831no—
256#95—831no—
257#95—831no—
258#95C2×D8311yescharacteristic
259C24—831no—
260C24C22×D8311yescharacteristic
261C2×Q8—3321no—
262C2×Q8—3321no—
263C2×Q8—3321no—
264C2×Q8—1662no—
265C2×Q8—1662no—
266C2×Q8—1662no—
267C2×D4—3321no—
268C2×D4—3321no—
269C2×D4—834no—
270C22×C4—3321no—
271C22×C4—3321no—
272C22×C4—3321no—
273C22×C4—831no—
274C22×C4—831no—
275C22×C4—831no—
276C22×C4—831no—
277C22×C4—831no—
278C22×C4—831no—
279C22×C4—831no—
280C22×C4—1662no—
281C22×C4—1662no—
282C22×C4—1662no—
283C22×C4C22×D8311yescharacteristic
284C22×C4C22×D8311yescharacteristic
285C22×C4C22×D8311yescharacteristic
286C22×C4C22×D8311yescharacteristic
287C22×C4C22×D8311yescharacteristic
288C22×C4C22×D8311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—3321no—
293C4⋊C4—3321no—
294C4⋊C4—834no—
295C4⋊C4—1662no—
296C4⋊C4—1662no—
297C4⋊C4—1662no—
298C22⋊C4—22no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—3321no—
302C22⋊C4—3321no—
303C22⋊C4—3321no—
304C22⋊C4—1662no—
305C22⋊C4—1662no—
306C22⋊C4—1662no—
307C22⋊C4—1662no—
308C42—3321no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—832no—
316C23—1661no—
317C23—1661no—
318C23—1661no—
319C23—1661no—
320C23—1661no—
321C23—1661no—
322C23Q8×D8312yes—
323C23C23×D8311yescharacteristic
324Q8—3324no—
325Q8—3324no—
326D4—3324no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3324no—
345C2×C4—3324no—
346C2×C4—832no—
347C2×C4—832no—
348C2×C4—832no—
349C2×C4—832no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4—1661no—
354C2×C4—1661no—
355C2×C4—1661no—
356C2×C4—1661no—
357C2×C4—1661no—
358C2×C4—1661no—
359C2×C4—1661no—
360C2×C4—1661no—
361C2×C4—1661no—
362C2×C4—1661no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4D4×D8312yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—3321no—
376C22—3321no—
377C22—3321no—
378C22—3321no—
379C22—1662no—
380C22—1662no—
381C22—1662no—
382C22C2×D4×D8311yescharacteristic
383C22C2×Q8×D8311yescharacteristic
384C22#41×D8311yescharacteristic
385C22#1458183711yescharacteristic
386C22#1458183711yescharacteristic
387C22#1458183811yescharacteristic
388C22#1458183911yescharacteristic
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3321no—
396C4—3321no—
397C4—3321no—
398C4—1662no—
399C4—1662no—
400C4—1662no—
401C4—1662no—
402C2—42no—
403C2—3322no—
404C2#1912826411yescharacteristic
405C2#1912831911yescharacteristic
406C2#1912832611yescharacteristic
407C2#1912834311yescharacteristic
408C2#1912836411yescharacteristic
409C2#1912840511yescharacteristic
410C2#1912844811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#399×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#399D83
hypocenterC83#2692
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14581770C2semidirect
2. C2×#14581769C2semidirect
3. #19127634C2semidirect
4. #19127623C2semidirect
5. #19127621C2semidirect
6. #19127614C2semidirect
7. #19127613C2semidirect
8. #19127606C2semidirect
10. #19127583C2semidirect
13. #19127572C2semidirect
14. #19127559C2semidirect
15. #399×C83C2semidirect
62. C2×C22⋊D83Q8semidirect
81. Dic83⋊C4D4semidirect
206. C83#2692semidirect
209. #399D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
C2×#145817692
#191276342
#191276232
#191276212
#191276142
#191276132
#191276062
#191276012
#191275832
#191275782
#191275772
#191275722
#191275592
#399×C832
#269283