Finite Group Databasebeta

← #49131546

Subgroups — #49131546

Order 36224.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827068C211yesmaximal, characteristic
2#31826943C211yesmaximal, characteristic, complemented
3#31826790C211yesmaximal, characteristic, complemented
4#31826786C211yesmaximal, characteristic, complemented
5#31826774C211yesmaximal, characteristic, complemented
6#31826770C211yesmaximal, characteristic, complemented
7#31826663C211yesmaximal, characteristic, complemented
8#31826663C211yesmaximal, characteristic, complemented
9#31826604C211yesmaximal, characteristic, complemented
10#31826568C211yesmaximal, characteristic, complemented
11#31826566C211yesmaximal, characteristic, complemented
12SD16×Dic283C211yesmaximal, characteristic, complemented
13#31826550C211yesmaximal, characteristic, complemented
14#31826523C211yesmaximal, characteristic, complemented
15#448×C283C211yesmaximal, characteristic, complemented
16C2×#17985379C2211yescharacteristic
17C2×#17985379C2211yescharacteristic
18#22583606C2211yescharacteristic
19#22583601C2211yescharacteristic
20#22583594—21no—
21#22583592—21no—
22#22583584C2211yescharacteristic
23#22583582—21no—
24#22583581—21no—
25Q8×Dic283—21no—
26Q8×Dic283—21no—
27Q8×Dic283C2211yescharacteristic
28Q8×Dic283C2211yescharacteristic
29Q8×Dic283C2211yescharacteristic
30Q8×Dic283C2211yescharacteristic
31D4×Dic283C2211yescharacteristic
32#22583572C2211yescharacteristic
33#22583569—21no—
34#22583569—21no—
35#22583569C2211yescharacteristic
36#22583569C2211yescharacteristic
37#22583544C2211yescharacteristic
38#22583543C2211yescharacteristic
39#22583542C2211yescharacteristic
40#22583541C2211yescharacteristic
41Dic283⋊C8C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583534C2211yescharacteristic
44#22583534C2211yescharacteristic
45#22583532C2211yescharacteristic
46#22583532C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50#22583520C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C22×Q8×C283C2211yescharacteristic
53C2×SD16×C283C2211yescharacteristic
54C2×SD16×C283C2211yescharacteristic
55#121×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58C22⋊C8×C283C2211yescharacteristic
59C2×D1132C2311yescharacteristic
60C2×C22⋊D283—21no—
61C2×C22⋊D283—21no—
62C2×C22⋊D283—21no—
63C22×Dic283—21no—
64C2×C4×D283—21no—
65#17985379—22no—
66#17985379—22no—
67C2×C283⋊C8C2311yescharacteristic
68C2×C283⋊C8C2311yescharacteristic
69C4⋊Dic283—21no—
70C4⋊Dic283—21no—
71C4⋊Dic283—21no—
72C4⋊Dic283—21no—
73C4⋊Dic283—21no—
74C4⋊Dic283D411yescharacteristic
75C4⋊Dic283D411yescharacteristic
76C4⋊Dic283C2311yescharacteristic
77C4⋊Dic283C2311yescharacteristic
78C4⋊Dic283C2311yescharacteristic
79Dic283⋊C4—21no—
80C22⋊Dic283—21no—
81C22⋊Dic283—21no—
82C22⋊Dic283D411yescharacteristic
83C22⋊Dic283D411yescharacteristic
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—21no—
86C4×Dic283—21no—
87C4×Dic283—21no—
88C4×Dic283—21no—
89C4×Dic283—21no—
90C4×Dic283—21no—
91C4×Dic283C2311yescharacteristic
92C4×Dic283C2311yescharacteristic
93C2×Q8×C283—22no—
94C2×Q8×C283—22no—
95C2×Q8×C283—21no—
96C2×Q8×C283C2311yescharacteristic
97C2×Q8×C283C2311yescharacteristic
98C2×D4×C283—21no—
99C2×D4×C283C2311yescharacteristic
100C22×C4×C283—21no—
101C22×C4×C283C2311yescharacteristic
102SD16×C283—22no—
103SD16×C283—22no—
104C2×C8×C283C2311yescharacteristic
105C2×C8×C283C2311yescharacteristic
106C4⋊C4×C283C2311yescharacteristic
107C22⋊C4×C283—21no—
108C22×D283—21no—
109D1132—21no—
110D1132—21no—
111C22⋊D283—81no—
112C22⋊D283—81no—
113C22⋊D283—81no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—81no—
118C2×Dic283C2×D411yescharacteristic
119C2×Dic283C2×D411yescharacteristic
120C2×Dic283C2×D411yescharacteristic
121C2×Dic283C2×D411yescharacteristic
122C4×D283—81no—
123C283⋊C8—21no—
124C283⋊C8—21no—
125C23×C283—21no—
126C23×C283C2×D411yescharacteristic
127Q8×C283—24no—
128Q8×C283—21no—
129Q8×C283—21no—
130Q8×C283#4112yes—
131Q8×C283#4112yes—
132D4×C283—21no—
133D4×C283—21no—
134D4×C283—81no—
135C2×C4×C283—24no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283C2×D411yescharacteristic
142C2×C4×C283C2411yescharacteristic
143C8×C283—21no—
144C8×C283—21no—
145C22×C283—21no—
146C22×C283—21no—
147C22×C283—21no—
148C22×C283—41no—
149C22×C283—41no—
150C22×C283—41no—
151C22×C283C22×D411yescharacteristic
152C2×D283—41no—
153C2×D283—41no—
154C2×D283—41no—
155C4×C283—22no—
156C4×C283—22no—
157C4×C283—21no—
158C4×C283—41no—
159C4×C283C2×#4111yescharacteristic
160C4×C283C2×#4111yescharacteristic
161Dic283—21no—
162Dic283—21no—
163Dic283—21no—
164Dic283—21no—
165Dic283—41no—
166Dic283—41no—
167Dic283—41no—
168C2×C283—41no—
169C2×C283—81no—
170C2×C283#52511yescharacteristic
171C2×C283#57711yescharacteristic
172C2×C283#57711yescharacteristic
173D283—81no—
174C283#312611yescharacteristic, complemented, Sylow
175#3126—2831nomaximal, Sylow
176#540—2831no—
177#519—2831no—
178#482—2831no—
179#482—2831no—
180#479—2831no—
181#479—2831no—
182#460—2831no—
183#460—2831no—
184#448—2831no—
185#448D28311yescharacteristic, complemented
186#440—2831no—
187#440—2831no—
188C4×SD16—2831no—
189C4×SD16—2831no—
190#433—2831no—
191C22×Q8C2×D28311yescharacteristic
192C2×SD16—2831no—
193C2×SD16—2831no—
194C2×SD16C2×D28311yescharacteristic
195C2×SD16C2×D28311yescharacteristic
196#127—2831no—
197#125—2831no—
198#123—5661no—
199#123—5661no—
200#121—2831no—
201#121—5661no—
202#121—5661no—
203#121C2×D28311yescharacteristic
204C4×Q8—2831no—
205C4×Q8—2831no—
206C4×Q8—2831no—
207C4×Q8—2831no—
208C4×Q8—5661no—
209C4×Q8—5661no—
210C4×D4—2831no—
211C4×D4—2831no—
212#117—2831no—
213#117—2831no—
214#117—5661no—
215#117—5661no—
216#107—2831no—
217#107—2831no—
218#106—2831no—
219#106—2831no—
220C4⋊C8—2831no—
221C4⋊C8—2831no—
222#103—2831no—
223#103—2831no—
224#103C2×D28311yescharacteristic
225#103C2×D28311yescharacteristic
226#102—2831no—
227#102—2831no—
228#102—2831no—
229#102—2831no—
230C22⋊C8—2831no—
231C22⋊C8C2×D28311yescharacteristic
232#97—2831no—
233C4×C8—2831no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D28311yescharacteristic
238C2×Q8C22×D28311yescharacteristic
239C2×D4—21no—
240C2×D4—2831no—
241C2×D4—5661no—
242C2×D4—5661no—
243C2×D4—5661no—
244C2×D4C22×D28311yescharacteristic
245C22×C4—21no—
246C22×C4—5661no—
247C22×C4—5661no—
248C22×C4C22×D28311yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—5662no—
252SD16—5662no—
253C2×C8—2831no—
254C2×C8—2831no—
255C2×C8C22×D28311yescharacteristic
256C2×C8C22×D28311yescharacteristic
257C4⋊C4—2831no—
258C4⋊C4—2831no—
259C4⋊C4—2831no—
260C4⋊C4—2831no—
261C4⋊C4—2831no—
262C4⋊C4—5661no—
263C4⋊C4—5661no—
264C4⋊C4—5661no—
265C4⋊C4—5661no—
266C4⋊C4—5661no—
267C4⋊C4—5661no—
268C4⋊C4C22×D28311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—2831no—
271C22⋊C4—2831no—
272C22⋊C4—5661no—
273C22⋊C4—5661no—
274C22⋊C4—5661no—
275C22⋊C4—5661no—
276C42—2831no—
277C42—2831no—
278C42—5661no—
279C42—5661no—
280C42—5661no—
281C42—5661no—
282C42—5661no—
283C23—21no—
284C23—5661no—
285C23D4×D28311yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1798539112yes—
290Q8#1798539112yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—5661no—
295D4—5661no—
296D4—22641no—
297D4—22641no—
298D4—22641no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—2831no—
306C2×C4—2831no—
307C2×C4—2831no—
308C2×C4—2831no—
309C2×C4—5661no—
310C2×C4—5661no—
311C2×C4—5661no—
312C2×C4—22641no—
313C2×C4—22641no—
314C2×C4D4×D28311yescharacteristic
315C2×C4C23×D28311yescharacteristic
316C8—21no—
317C8—21no—
318C8—5661no—
319C8—5661no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—11321no—
327C22—11321no—
328C22—11321no—
329C22C2×D4×D28311yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—5661no—
335C4—5661no—
336C4—5661no—
337C4—5661no—
338C4—11321no—
339C4—11321no—
340C4—11321no—
341C4C2×#1798539111yescharacteristic
342C4C2×#1798539111yescharacteristic
343C2—41no—
344C2—81no—
345C2—22641no—
346C2#3182697111yescharacteristic
347C2#3182727111yescharacteristic
348C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#448×C283C2
socleC22×C283C22×D4
prime radical#31827068C2
hypercenter#448D283
hypocenterC283#3126
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #31826943C2semidirect
3. #31826790C2semidirect
4. #31826786C2semidirect
5. #31826774C2semidirect
6. #31826770C2semidirect
7. #31826663C2semidirect
8. #31826663C2semidirect
9. #31826604C2semidirect
10. #31826568C2semidirect
11. #31826566C2semidirect
12. SD16×Dic283C2semidirect
13. #31826550C2semidirect
14. #31826523C2semidirect
15. #448×C283C2semidirect
174. C283#3126semidirect
185. #448D283semidirect

Maximal subgroups

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

subgroupindex
#318270682
#318269432
#318267902
#318267862
#318267742
#318267702
#318266632
#318266042
#318265682
#318265662
SD16×Dic2832
#318265502
#318265232
#448×C2832
#3126283