Finite Group Databasebeta

← #49124291

Subgroups — #49124291

Order 36224.

576 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583587C211yesmaximal, characteristic
2C2×C4×Dic566C211yesmaximal, characteristic, complemented
3C2×#22583566C211yesmaximal, characteristic
4#31826369C211yesmaximal, characteristic, complemented
5#31826360C211yesmaximal, characteristic, complemented
6#31826359C211yesmaximal, characteristic
7#31826359C211yesmaximal, characteristic
8#31826358C211yesmaximal, characteristic
9#31826352C211yesmaximal, characteristic, complemented
10#31826350C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic283C211yesmaximal, characteristic, complemented
12C4×Dic283⋊C4C211yesmaximal, characteristic, complemented
13#31826316C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic283C211yesmaximal, characteristic
15#388×C283C211yesmaximal, characteristic
16C22×Dic566C2211yescharacteristic
17C23×Dic283C2211yescharacteristic
18#22583587C418yescomplemented
19C4×Dic566—24no—
20#22583566—24no—
21C2×C4⋊Dic283C2211yescharacteristic
22C2×C4⋊Dic283C2211yescharacteristic
23C2×Dic283⋊C4C2211yescharacteristic
24C2×Dic283⋊C4C2211yescharacteristic
25C2×Dic283⋊C4C2211yescharacteristic
26C2×Dic283⋊C4C2211yescharacteristic
27C2×Dic283⋊C4C2211yescharacteristic
28C2×Dic283⋊C4C2211yescharacteristic
29C2×C22⋊Dic283C2211yescharacteristic
30C2×C22⋊Dic283C2211yescharacteristic
31C2×C22⋊Dic283C2211yescharacteristic
32C2×C4×Dic283C2211yescharacteristic
33C2×C4×Dic283C2211yescharacteristic
34C2×C4×Dic283C2211yescharacteristic
35C2×C4×Dic283C2211yescharacteristic
36C2×C4×Dic283C2211yescharacteristic
37C2×C4×Dic283C2211yescharacteristic
38C2×C4×Dic283C2211yescharacteristic
39C283⋊#95C2211yescharacteristic
40C283⋊#95C2211yescharacteristic
41C283⋊#95C2211yescharacteristic
42C283⋊#95C2211yescharacteristic
43C283⋊#95C2211yescharacteristic
44C283⋊#95C2211yescharacteristic
45C283⋊#95C2211yescharacteristic
46C283⋊#95C2211yescharacteristic
47C2×C4⋊C4×C283C2211yescharacteristic
48C2×C22⋊C4×C283C2211yescharacteristic
49C2×C22⋊C4×C283C2211yescharacteristic
50C2×C22⋊C4×C283C2211yescharacteristic
51C2×C42×C283C2211yescharacteristic
52#95×C283C2211yescharacteristic
53#95×C283C2211yescharacteristic
54C2×Dic566—22no—
55C2×Dic566—22no—
56C2×Dic566C2×C414yes—
57C22×Dic283—22no—
58C22×Dic283—22no—
59C22×Dic283C2×C414yes—
60C22×Dic283C2311yescharacteristic
61C22×Dic283C2311yescharacteristic
62C22×Dic283C2311yescharacteristic
63C22×Dic283C2311yescharacteristic
64C22×Dic283C2311yescharacteristic
65C22×Dic283C2311yescharacteristic
66C22×Dic283C2311yescharacteristic
67C22×Dic283C2311yescharacteristic
68C4⋊Dic283—22no—
69C4⋊Dic283C2×C414yes—
70Dic283⋊C4—22no—
71Dic283⋊C4—22no—
72Dic283⋊C4—22no—
73Dic283⋊C4C2×C414yes—
74Dic283⋊C4C2×C414yes—
75Dic283⋊C4D414yes—
76C22⋊Dic283—22no—
77C22⋊Dic283—22no—
78C22⋊Dic283C2×C414yes—
79C4×Dic283—22no—
80C4×Dic283—22no—
81C4×Dic283—22no—
82C4×Dic283—22no—
83C4×Dic283—22no—
84C4×Dic283—22no—
85C4×Dic283—22no—
86C24×C283C2311yescharacteristic
87C22×C4×C283C2311yescharacteristic
88C22×C4×C283C2311yescharacteristic
89C22×C4×C283C2311yescharacteristic
90C22×C4×C283C2311yescharacteristic
91C22×C4×C283C2311yescharacteristic
92C22×C4×C283C2311yescharacteristic
93C4⋊C4×C283—22no—
94C22⋊C4×C283—22no—
95C22⋊C4×C283—22no—
96C22⋊C4×C283C2×C414yes—
97C42×C283—22no—
98Dic566—28no—
99C2×Dic283—21no—
100C2×Dic283—21no—
101C2×Dic283—21no—
102C2×Dic283—21no—
103C2×Dic283—21no—
104C2×Dic283—21no—
105C2×Dic283—21no—
106C2×Dic283—21no—
107C2×Dic283—21no—
108C2×Dic283—21no—
109C2×Dic283—28no—
110C2×Dic283—21no—
111C2×Dic283—21no—
112C2×Dic283—21no—
113C2×Dic283—21no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283C22×C412yes—
117C2×Dic283C22×C412yes—
118C2×Dic283C22×C412yes—
119C2×Dic283C22×C412yes—
120C2×Dic283C2×D412yes—
121C2×Dic283C2×D412yes—
122C2×Dic283#4112yes—
123C2×Dic283#4112yes—
124C23×C283—21no—
125C23×C283—21no—
126C23×C283—21no—
127C23×C283—21no—
128C23×C283—21no—
129C23×C283—21no—
130C23×C283C22×C412yes—
131C23×C283C2411yescharacteristic
132C2×C4×C283—21no—
133C2×C4×C283—21no—
134C2×C4×C283—21no—
135C2×C4×C283—21no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283C22×C412yes—
147C2×C4×C283C22×C412yes—
148C2×C4×C283C2×D412yes—
149C2×C4×C283#4112yes—
150C22×C283—22no—
151C22×C283—22no—
152C22×C283—22no—
153C22×C283—41no—
154C22×C283—41no—
155C22×C283—41no—
156C22×C283—41no—
157C22×C283C23×C411yescharacteristic
158C22×C283C22×D411yescharacteristic
159C22×C283C2×#4111yescharacteristic
160C22×C283C2×#4111yescharacteristic
161C22×C283C2×#4111yescharacteristic
162C22×C2832−1+411yescharacteristic
163C22×C2832−1+411yescharacteristic
164C4×C283—22no—
165C4×C283—22no—
166C4×C283—22no—
167C4×C283—22no—
168C4×C283—41no—
169C4×C283—41no—
170Dic283—22no—
171Dic283—22no—
172Dic283—22no—
173Dic283—22no—
174Dic283—22no—
175Dic283—41no—
176Dic283—41no—
177Dic283C4×D414yes—
178C2×C283—42no—
179C2×C283C2×C4×D411yescharacteristic
180C2×C283C4×#4111yescharacteristic
181C2×C283#51911yescharacteristic
182C2×C283#54711yescharacteristic
183C2×C283#54811yescharacteristic
184C2×C283#55211yescharacteristic
185C2×C283#55611yescharacteristic
186C283#236211yescharacteristic, complemented, Sylow
187#2362—2831nomaximal, Sylow
188C2×C22⋊Q8—2831no—
189C2×C4×Q8—2831no—
190C2×#117—2831no—
191#391—2831no—
192#389—2831no—
193#388—2831no—
194#388—2831no—
195#388—2831no—
196#388D28311yescharacteristic
197#387—2831no—
198#387—2831no—
199C4×C4⋊C4—2831no—
200C4×C4⋊C4—2831no—
201C4×C4⋊C4—2831no—
202C4×C22⋊C4—2831no—
203C22×Q8—2831no—
204C23×C4—2831no—
205C22⋊Q8—2838no—
206C4×Q8—5664no—
207#117—5664no—
208C2×C4⋊C4—2831no—
209C2×C4⋊C4—2831no—
210C2×C4⋊C4—2831no—
211C2×C4⋊C4—2831no—
212C2×C4⋊C4—2831no—
213C2×C4⋊C4—2831no—
214C2×C4⋊C4—2831no—
215C2×C4⋊C4—2831no—
216C2×C4⋊C4C2×D28311yescharacteristic
217C2×C22⋊C4—2831no—
218C2×C22⋊C4—2831no—
219C2×C22⋊C4—2831no—
220C2×C22⋊C4C2×D28311yescharacteristic
221C2×C22⋊C4C2×D28311yescharacteristic
222C2×C22⋊C4C2×D28311yescharacteristic
223C2×C42—2831no—
224C2×C42—2831no—
225C2×C42—2831no—
226C2×C42—2831no—
227C2×C42—2831no—
228C2×C42—2831no—
229C2×C42—2831no—
230C2×C42C2×D28311yescharacteristic
231#95—2831no—
232#95—2831no—
233#95—2831no—
234#95—2831no—
235#95—2831no—
236#95—2831no—
237#95—2831no—
238#95—2831no—
239#95C2×D28311yescharacteristic
240#95C2×D28311yescharacteristic
241C24C22×D28311yescharacteristic
242C2×Q8—2834no—
243C2×Q8—5662no—
244C2×Q8—5662no—
245C22×C4—2831no—
246C22×C4—2834no—
247C22×C4—2831no—
248C22×C4—2831no—
249C22×C4—2831no—
250C22×C4—2831no—
251C22×C4—2831no—
252C22×C4—2831no—
253C22×C4—2831no—
254C22×C4—5662no—
255C22×C4—5662no—
256C22×C4C22×D28311yescharacteristic
257C22×C4C22×D28311yescharacteristic
258C22×C4C22×D28311yescharacteristic
259C22×C4C22×D28311yescharacteristic
260C22×C4C22×D28311yescharacteristic
261C22×C4C22×D28311yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—2834no—
264C4⋊C4—2834no—
265C4⋊C4—2834no—
266C4⋊C4—2834no—
267C4⋊C4—5662no—
268C4⋊C4—5662no—
269C4⋊C4—5662no—
270C4⋊C4—5662no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—2834no—
274C22⋊C4—5662no—
275C22⋊C4—5662no—
276C22⋊C4C4×D28314yes—
277C42—22no—
278C42—5662no—
279C42—5662no—
280C42—5662no—
281C42—5662no—
282C42—5662no—
283C42—5662no—
284C42—5662no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23C2×C4×D28312yes—
292C23C23×D28311yescharacteristic
293Q8—5668no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—2832no—
309C2×C4—2832no—
310C2×C4—2832no—
311C2×C4—2832no—
312C2×C4—2832no—
313C2×C4—2832no—
314C2×C4—2832no—
315C2×C4—2832no—
316C2×C4—5668no—
317C2×C4—5661no—
318C2×C4—5661no—
319C2×C4—5661no—
320C2×C4—5661no—
321C2×C4—5661no—
322C2×C4—5661no—
323C2×C4—5661no—
324C2×C4—5661no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—5661no—
332C2×C4—5661no—
333C2×C4C2×C4×D28312yes—
334C2×C4C2×C4×D28312yes—
335C2×C4D4×D28312yes—
336C2×C4#1798539112yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C22×C4×D28311yescharacteristic
345C22C2×D4×D28311yescharacteristic
346C22#41×D28311yescharacteristic
347C22#41×D28311yescharacteristic
348C22C2×#1798539111yescharacteristic
349C22#2258365711yescharacteristic
350C22#2258365711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—2834no—
358C4—5662no—
359C4—5662no—
360C4—5662no—
361C4—5662no—
362C4—5662no—
363C4—11321no—
364C4—11321no—
365C2—42no—
366C2C4×D4×D28311yescharacteristic
367C2C4×#1798539111yescharacteristic
368C2#3182694211yescharacteristic
369C2#3182711311yescharacteristic
370C2#3182712111yescharacteristic
371C2#3182715111yescharacteristic
372C2#3182718411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#388×C283C2
socle, cosocleC23×C283C24
prime radicalC24×C283C23
hypercenter#388D283
hypocenterC283#2362
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic566C2semidirect
4. #31826369C2semidirect
5. #31826360C2semidirect
9. #31826352C2semidirect
10. #31826350C2semidirect
11. C4⋊C4×Dic283C2semidirect
12. C4×Dic283⋊C4C2semidirect
13. #31826316C2semidirect
18. #22583587C4semidirect
186. C283#2362semidirect

Maximal subgroups

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

subgroupindex
C2×#225835872
C2×C4×Dic5662
C2×#225835662
#318263692
#318263602
#318263592
#318263582
#318263522
#318263502
C4⋊C4×Dic2832
C4×Dic283⋊C42
#318263162
C22⋊C4×Dic2832
#388×C2832
#2362283