Finite Group Databasebeta

← #24912380

Subgroups — #24912380

Order 10624.

620 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581770C211yesmaximal, characteristic, complemented
2C2×#14581769C211yesmaximal, characteristic, complemented
3C2×#14581757C211yesmaximal, characteristic, complemented
4C2×(C22⋊D83)⋊C4C211yesmaximal, characteristic
5#19127624C211yesmaximal, characteristic, complemented
6#19127623C211yesmaximal, characteristic, complemented
7#19127623C211yesmaximal, characteristic, complemented
8#19127601C211yesmaximal, characteristic
9#19127601C211yesmaximal, characteristic
10#19127598C211yesmaximal, characteristic, complemented
11#19127596C211yesmaximal, characteristic, complemented
12#19127570C211yesmaximal, characteristic, complemented
13#19127560C211yesmaximal, characteristic, complemented
14#19127559C211yesmaximal, characteristic, complemented
15#396×C83C211yesmaximal, characteristic, complemented
16C22×Dic166C2211yescharacteristic
17C22×C22⋊D83C2211yescharacteristic
18C23×Dic83C2211yescharacteristic
19C22×C4×D83C2211yescharacteristic
20#14581770—24no—
21#14581769—24no—
22#14581757—24no—
23(C22⋊D83)⋊C4—24no—
24C2×C4⋊Dic83C2211yescharacteristic
25C2×C4⋊Dic83C2211yescharacteristic
26C2×Dic83⋊C4C2211yescharacteristic
27C2×Dic83⋊C4C2211yescharacteristic
28C2×Dic83⋊C4C2211yescharacteristic
29C2×Dic83⋊C4C2211yescharacteristic
30C2×Dic83⋊C4C2211yescharacteristic
31C2×Dic83⋊C4C2211yescharacteristic
32C2×Dic83⋊C4C2211yescharacteristic
33C2×Dic83⋊C4C2211yescharacteristic
34C2×Dic83⋊C4C2211yescharacteristic
35C2×C22⋊Dic83C2211yescharacteristic
36C2×C22⋊Dic83C2211yescharacteristic
37C2×(C2×D83)⋊C4C2211yescharacteristic
38C2×(C2×D83)⋊C4C2211yescharacteristic
39C2×(C2×D83)⋊C4C2211yescharacteristic
40C2×(C2×D83)⋊C4C2211yescharacteristic
41C2×(C2×D83)⋊C4C2211yescharacteristic
42C2×C4×Dic83C2211yescharacteristic
43C2×C4×Dic83C2211yescharacteristic
44C2×C4×Dic83C2211yescharacteristic
45C83⋊#95C2211yescharacteristic
46C83⋊#95C2211yescharacteristic
47C83⋊#95C2211yescharacteristic
48C2×C4⋊C4×C83C2211yescharacteristic
49C2×C4⋊C4×C83C2211yescharacteristic
50C2×C4⋊C4×C83C2211yescharacteristic
51C2×C22⋊C4×C83C2211yescharacteristic
52C2×C22⋊C4×C83C2211yescharacteristic
53C2×C22⋊C4×C83C2211yescharacteristic
54#95×C83C2211yescharacteristic
55C23×D83C2311yescharacteristic
56C2×Dic166—22no—
57C2×Dic166—22no—
58C2×Dic166D414yescomplemented
59C2×C22⋊D83—22no—
60C2×C22⋊D83—22no—
61C2×C22⋊D83Q814yescomplemented
62C22×Dic83—22no—
63C22×Dic83—22no—
64C22×Dic83—22no—
65C22×Dic83C2311yescharacteristic
66C22×Dic83C2311yescharacteristic
67C22×Dic83C2311yescharacteristic
68C22×Dic83C2311yescharacteristic
69C22×Dic83C2311yescharacteristic
70C22×Dic83C2311yescharacteristic
71C22×Dic83C2311yescharacteristic
72C2×C4×D83—22no—
73C2×C4×D83—22no—
74C2×C4×D83—22no—
75C4⋊Dic83—22no—
76C4⋊Dic83—41no—
77Dic83⋊C4—22no—
78Dic83⋊C4—22no—
79Dic83⋊C4—22no—
80Dic83⋊C4—22no—
81Dic83⋊C4—22no—
82Dic83⋊C4—22no—
83Dic83⋊C4—22no—
84Dic83⋊C4—22no—
85Dic83⋊C4—22no—
86C22⋊Dic83—22no—
87C22⋊Dic83—41no—
88(C2×D83)⋊C4—22no—
89(C2×D83)⋊C4—22no—
90(C2×D83)⋊C4—22no—
91(C2×D83)⋊C4—41no—
92(C2×D83)⋊C4—41no—
93C4×Dic83—22no—
94C4×Dic83—22no—
95C4×Dic83D414yes—
96C24×C83C2311yescharacteristic
97C22×C4×C83C2311yescharacteristic
98C22×C4×C83C2311yescharacteristic
99C22×C4×C83C2311yescharacteristic
100C22×C4×C83C2311yescharacteristic
101C22×C4×C83C2311yescharacteristic
102C22×C4×C83C2311yescharacteristic
103C4⋊C4×C83—22no—
104C4⋊C4×C83—22no—
105C4⋊C4×C83—41no—
106C22⋊C4×C83—22no—
107C22⋊C4×C83—22no—
108C22⋊C4×C83—41no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83C2×Q812yes—
116C23×C83C2411yescharacteristic
117C22×D83—21no—
118C22×D83—21no—
119C22×D83—21no—
120C22×D83—21no—
121C22×D83—21no—
122C22×D83—21no—
123C22×D83C2×Q812yes—
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×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83C2×D412yes—
141C2×C4×C83C2×D412yes—
142C22⋊D83—28no—
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—21no—
155C2×Dic83—21no—
156C2×Dic83—21no—
157C2×Dic83—44no—
158C2×Dic83C2×D412yes—
159C2×Dic83C2×D412yes—
160C2×Dic83C2×D412yes—
161C2×Dic83C2×D412yes—
162C2×Dic83C2×Q812yes—
163C2×Dic83#4112yes—
164C2×Dic83#4112yes—
165C4×D83—44no—
166Dic166—28no—
167C22×C83—22no—
168C22×C83—22no—
169C22×C83—22no—
170C22×C83—41no—
171C22×C83—41no—
172C22×C83—41no—
173C22×C83—41no—
174C22×C83C22×D411yescharacteristic
175C22×C83C22×D411yescharacteristic
176C22×C83C22×Q811yescharacteristic
177C22×C83C2×#4111yescharacteristic
178C22×C83C2×#4111yescharacteristic
179C22×C83H(5,𝔽2)11yescharacteristic
180C22×C832−1+411yescharacteristic
181C2×D83—22no—
182C2×D83—22no—
183C2×D83—22no—
184C2×D83—41no—
185C2×D83—41no—
186C2×D83—41no—
187C2×D83—41no—
188C4×C83—22no—
189C4×C83—22no—
190C4×C83—41no—
191C4×C83—41no—
192C4×C83—41no—
193C4×C83—41no—
194Dic83—22no—
195Dic83—41no—
196Dic83—41no—
197Dic83—41no—
198Dic83—41no—
199Dic83#12114yes—
200Dic83C22⋊Q814yescomplemented
201C2×C83—42no—
202C2×C83C2×#12111yescharacteristic
203C2×C83C2×C22⋊Q811yescharacteristic
204C2×C83#53711yescharacteristic
205C2×C83D4×Q811yescharacteristic
206C2×C83#55011yescharacteristic
207C2×C83#55411yescharacteristic
208C2×C83#55611yescharacteristic
209D83—42no—
210#2538—831nomaximal, Sylow
211C83#253811yescharacteristic, complemented, Sylow
212C2×C22⋊Q8—831no—
213C2×C22⋊Q8—831no—
214C2×C4×Q8—831no—
215C2×C4×D4—831no—
216#396—831no—
217#396—831no—
218#396—831no—
219#396D8311yescharacteristic, complemented
220#390—831no—
221#390—831no—
222#389—831no—
223#389—831no—
224#384—831no—
225#380—831no—
226#380—831no—
227C22×Q8—831no—
228C22×D4—831no—
229C23×C4—831no—
230C23×C4—831no—
231C22⋊Q8—1664no—
232C22⋊Q8—1664no—
233C4×Q8—1664no—
234C4×D4—1664no—
235C2×C4⋊C4—831no—
236C2×C4⋊C4—831no—
237C2×C4⋊C4—831no—
238C2×C4⋊C4—831no—
239C2×C4⋊C4—831no—
240C2×C4⋊C4—831no—
241C2×C4⋊C4—831no—
242C2×C4⋊C4—831no—
243C2×C4⋊C4—831no—
244C2×C4⋊C4—831no—
245C2×C4⋊C4—831no—
246C2×C4⋊C4C2×D8311yescharacteristic
247C2×C4⋊C4C2×D8311yescharacteristic
248C2×C4⋊C4C2×D8311yescharacteristic
249C2×C22⋊C4—831no—
250C2×C22⋊C4—831no—
251C2×C22⋊C4—831no—
252C2×C22⋊C4—831no—
253C2×C22⋊C4—831no—
254C2×C22⋊C4—831no—
255C2×C22⋊C4—831no—
256C2×C22⋊C4C2×D8311yescharacteristic
257C2×C22⋊C4C2×D8311yescharacteristic
258C2×C22⋊C4C2×D8311yescharacteristic
259C2×C42—831no—
260C2×C42—831no—
261C2×C42—831no—
262#95—831no—
263#95—831no—
264#95—831no—
265#95C2×D8311yescharacteristic
266C24—831no—
267C24C22×D8311yescharacteristic
268C2×Q8—834no—
269C2×Q8—1662no—
270C2×Q8—1662no—
271C2×D4—834no—
272C2×D4—1662no—
273C2×D4—1662no—
274C22×C4—831no—
275C22×C4—831no—
276C22×C4—831no—
277C22×C4—831no—
278C22×C4—831no—
279C22×C4—831no—
280C22×C4—831no—
281C22×C4—1662no—
282C22×C4—1662no—
283C22×C4—1662no—
284C22×C4—1662no—
285C22×C4—1662no—
286C22×C4—1662no—
287C22×C4C22×D8311yescharacteristic
288C22×C4C22×D8311yescharacteristic
289C22×C4C22×D8311yescharacteristic
290C22×C4C22×D8311yescharacteristic
291C22×C4C22×D8311yescharacteristic
292C22×C4C22×D8311yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—3321no—
297C4⋊C4—1662no—
298C4⋊C4—1662no—
299C4⋊C4—1662no—
300C4⋊C4—1662no—
301C4⋊C4—1662no—
302C4⋊C4—1662no—
303C4⋊C4—1662no—
304C4⋊C4—1662no—
305C4⋊C4—1662no—
306C4⋊C4—1662no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—3321no—
311C22⋊C4—3321no—
312C22⋊C4—3321no—
313C22⋊C4—1662no—
314C22⋊C4—1662no—
315C22⋊C4—1662no—
316C22⋊C4—1662no—
317C42—834no—
318C42—1662no—
319C42—1662no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—832no—
327C23—1661no—
328C23—1661no—
329C23—1661no—
330C23—1661no—
331C23—1661no—
332C23—1661no—
333C23Q8×D8312yes—
334C23C23×D8311yescharacteristic
335Q8—1668no—
336D4—1668no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—3324no—
354C2×C4—3324no—
355C2×C4—832no—
356C2×C4—832no—
357C2×C4—832no—
358C2×C4—832no—
359C2×C4—832no—
360C2×C4—832no—
361C2×C4—832no—
362C2×C4—1661no—
363C2×C4—1661no—
364C2×C4—1661no—
365C2×C4—1661no—
366C2×C4—1661no—
367C2×C4—1661no—
368C2×C4—1661no—
369C2×C4—1661no—
370C2×C4—1661no—
371C2×C4—1661no—
372C2×C4—1661no—
373C2×C4—1661no—
374C2×C4—1661no—
375C2×C4—1661no—
376C2×C4D4×D8312yes—
377C2×C4D4×D8312yes—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—3321no—
386C22—3321no—
387C22—3321no—
388C22—3321no—
389C22—1662no—
390C22—1662no—
391C22—1662no—
392C22C2×D4×D8311yescharacteristic
393C22C2×D4×D8311yescharacteristic
394C22C2×Q8×D8311yescharacteristic
395C22#41×D8311yescharacteristic
396C22#41×D8311yescharacteristic
397C22#1458183711yescharacteristic
398C22#1458183911yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—3321no—
406C4—3321no—
407C4—3321no—
408C4—3321no—
409C4—834no—
410C4—834no—
411C4—1662no—
412C2—42no—
413C2—3322no—
414C2#121×D8311yescharacteristic
415C2C22⋊Q8×D8311yescharacteristic
416C2#1912827811yescharacteristic
417C2#1912836411yescharacteristic
418C2#1912836911yescharacteristic
419C2#1912840511yescharacteristic
420C2#1912841811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#396×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#396D83
hypocenterC83#2538
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#14581770C2semidirect
2. C2×#14581769C2semidirect
3. C2×#14581757C2semidirect
5. #19127624C2semidirect
6. #19127623C2semidirect
7. #19127623C2semidirect
10. #19127598C2semidirect
11. #19127596C2semidirect
12. #19127570C2semidirect
13. #19127560C2semidirect
14. #19127559C2semidirect
15. #396×C83C2semidirect
58. C2×Dic166D4semidirect
61. C2×C22⋊D83Q8semidirect
200. Dic83C22⋊Q8semidirect
211. C83#2538semidirect
219. #396D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
C2×#145817692
C2×#145817572
C2×(C22⋊D83)⋊C42
#191276242
#191276232
#191276012
#191275982
#191275962
#191275702
#191275602
#191275592
#396×C832
#253883