Finite Group Databasebeta

← #49133323

Subgroups — #49133323

Order 36224.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827183C211yesmaximal, characteristic, complemented
2#31827122C211yesmaximal, characteristic
3#31826823C211yesmaximal, characteristic, complemented
4#31826806C211yesmaximal, characteristic, complemented
5#31826787C211yesmaximal, characteristic, complemented
6#31826758C211yesmaximal, characteristic, complemented
7#31826710C211yesmaximal, characteristic, complemented
8#31826706C211yesmaximal, characteristic, complemented
9#31826686C211yesmaximal, characteristic, complemented
10#31826627C211yesmaximal, characteristic, complemented
11#31826622C211yesmaximal, characteristic, complemented
12#31826569C211yesmaximal, characteristic, complemented
13#31826536C211yesmaximal, characteristic, complemented
14#31826472C211yesmaximal, characteristic, complemented
15#416×C283C211yesmaximal, characteristic, complemented
16C2×Q8×D283C2211yescharacteristic
17C2×Dic1132C2211yescharacteristic
18C2×#17985382C2211yescharacteristic
19C2×#17985380C2211yescharacteristic
20C2×#17985375C2211yescharacteristic
21#22583609C2211yescharacteristic
22#22583604—21no—
23#22583603—21no—
24#22583597—21no—
25#22583597C2211yescharacteristic
26#22583596—21no—
27#22583590C2211yescharacteristic
28#22583589—21no—
29#22583589C2211yescharacteristic
30#22583589C2211yescharacteristic
31#22583581—21no—
32#22583581C2211yescharacteristic
33Q8×Dic283C2211yescharacteristic
34C4×Dic566—21no—
35#22583575C2211yescharacteristic
36C4×C22⋊D283—21no—
37C4×C22⋊D283C2211yescharacteristic
38#22583568C2211yescharacteristic
39#22583544C2211yescharacteristic
40#22583543C2211yescharacteristic
41#22583541C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583533C2211yescharacteristic
47#22583533C2211yescharacteristic
48#22583531C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51#22583520C2211yescharacteristic
52C2×#41×C283C2211yescharacteristic
53C22×C8×C283C2211yescharacteristic
54#117×C283C2211yescharacteristic
55#103×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#102×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59C2×Dic566—21no—
60C2×Dic566—21no—
61C2×Dic566C2311yescharacteristic
62C2×Dic566C2311yescharacteristic
63Q8×D283—22no—
64Q8×D283—41no—
65C2×C22⋊D283—21no—
66C2×C22⋊D283D411yescharacteristic
67C2×C4×D283—21no—
68C2×C4×D283C2311yescharacteristic
69Dic1132—41no—
70#17985382—22no—
71#17985380—41no—
72#17985375—41no—
73C2×C283⋊C8C2311yescharacteristic
74C2×C283⋊C8C2311yescharacteristic
75C4⋊Dic283—21no—
76C4⋊Dic283—21no—
77C4⋊Dic283C2311yescharacteristic
78C4⋊Dic283C2311yescharacteristic
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83Dic283⋊C4D411yescharacteristic
84C22⋊Dic283—21no—
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C22⋊Dic283—21no—
89C22⋊Dic283D411yescharacteristic
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4D411yescharacteristic
93C4×Dic283—21no—
94C4×Dic283—21no—
95C4×Dic283—21no—
96C4×Dic283C2311yescharacteristic
97#41×C283—22no—
98#41×C283—41no—
99C2×Q8×C283C2311yescharacteristic
100C2×D4×C283—21no—
101C2×D4×C283C2311yescharacteristic
102C22×C4×C283—21no—
103C22×C4×C283C2311yescharacteristic
104C2×C8×C283—41no—
105C2×C8×C283C2311yescharacteristic
106C2×C8×C283C2311yescharacteristic
107C4⋊C4×C283C2311yescharacteristic
108C4⋊C4×C283C2311yescharacteristic
109C22⋊C4×C283—21no—
110C42×C283—21no—
111C22×D283C2×D411yescharacteristic
112Dic566—21no—
113Dic566—21no—
114Dic566—21no—
115Dic566—21no—
116Dic566—42no—
117Dic566—42no—
118C22⋊D283—41no—
119C22⋊D283—42no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—21no—
124C2×Dic283C2×D411yescharacteristic
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C4×D283—21no—
128C4×D283—21no—
129C4×D283—42no—
130C283⋊C8—21no—
131C283⋊C8—21no—
132C23×C283—21no—
133C23×C283C2×D411yescharacteristic
134Q8×C283—21no—
135Q8×C283#4112yes—
136D4×C283—21no—
137D4×C283—21no—
138D4×C283—42no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—42no—
145C2×C4×C283C2×D411yescharacteristic
146C2×C4×C283C2411yescharacteristic
147C8×C283—21no—
148C8×C283—21no—
149C22×C283—21no—
150C22×C283—21no—
151C22×C283—21no—
152C22×C283—41no—
153C22×C283—41no—
154C22×C283—41no—
155C22×C283C22×D411yescharacteristic
156C2×D283—21no—
157C2×D283—21no—
158C2×D283—21no—
159C4×C283—21no—
160C4×C283—22no—
161C4×C283—41no—
162C4×C283—41no—
163C4×C283C2×#4111yescharacteristic
164C4×C283H(5,𝔽2)11yescharacteristic
165Dic283—21no—
166Dic283—21no—
167Dic283—21no—
168Dic283—41no—
169Dic283—41no—
170Dic283—41no—
171Dic283—41no—
172C2×C283—41no—
173C2×C283—81no—
174C2×C283#54611yescharacteristic
175C2×C283#57511yescharacteristic
176C2×C283#57811yescharacteristic
177D283—41no—
178C283#331311yescharacteristic, complemented, Sylow
179#3313—2831nomaximal, Sylow
180#556—2831no—
181#548—2831no—
182#489—2831no—
183#485—2831no—
184#482—2831no—
185#477—2831no—
186#468—2831no—
187#467—2831no—
188#464—2831no—
189#452—2831no—
190#451—2831no—
191#441—2831no—
192#435—2831no—
193#419—2831no—
194#416D28311yescharacteristic, complemented
195C2×#41C2×D28311yescharacteristic
196C22×Q8—2831no—
197C2×Q16—2831no—
198C2×Q16—2831no—
199C2×SD16—2831no—
200C2×M16—2831no—
201C22×C8C2×D28311yescharacteristic
202C4⋊Q8—2831no—
203#126—5661no—
204#126—5661no—
205#124—2831no—
206#124—5661no—
207#124—5661no—
208C22⋊Q8—2831no—
209C22⋊Q8—2831no—
210C22⋊Q8—2831no—
211C22⋊Q8—5661no—
212#121—2831no—
213#121—5661no—
214C4×Q8—2831no—
215C4×Q8—2831no—
216C4×Q8—5661no—
217C4×D4—2831no—
218C4×D4—5661no—
219#117—2831no—
220#117C2×D28311yescharacteristic
221#107—2831no—
222#107—2831no—
223#106—2831no—
224C4⋊C8—2831no—
225#103—2831no—
226#103—2831no—
227#103—2831no—
228#103—2831no—
229#103—2831no—
230#103C2×D28311yescharacteristic
231#103C2×D28311yescharacteristic
232#102—2831no—
233#102C2×D28311yescharacteristic
234#102C2×D28311yescharacteristic
235C22⋊C8—2831no—
236C22⋊C8—2831no—
237#97—2831no—
238#41—22no—
239#41—41no—
240C2×Q8—2831no—
241C2×Q8—2831no—
242C2×Q8—5661no—
243C2×Q8—5661no—
244C2×Q8—5662no—
245C2×Q8—11321no—
246C2×Q8C22×D28311yescharacteristic
247C2×D4—21no—
248C2×D4—2831no—
249C2×D4—5661no—
250C2×D4C22×D28311yescharacteristic
251C22×C4—21no—
252C22×C4—2831no—
253C22×C4—5661no—
254C22×C4C22×D28311yescharacteristic
255Q16—5662no—
256Q16—11321no—
257SD16—11321no—
258M16—11321no—
259C2×C8—41no—
260C2×C8—2831no—
261C2×C8—2831no—
262C2×C8C22×D28311yescharacteristic
263C2×C8C22×D28311yescharacteristic
264C4⋊C4—2831no—
265C4⋊C4—2831no—
266C4⋊C4—2831no—
267C4⋊C4—5661no—
268C4⋊C4—5661no—
269C4⋊C4—5661no—
270C4⋊C4—5661no—
271C4⋊C4—5661no—
272C4⋊C4—5661no—
273C4⋊C4C22×D28311yescharacteristic
274C4⋊C4C22×D28311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—2831no—
277C22⋊C4—2831no—
278C22⋊C4—5661no—
279C22⋊C4—5661no—
280C22⋊C4—5661no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C22⋊C4—5661no—
284C22⋊C4—5661no—
285C42—21no—
286C42—2831no—
287C42—5661no—
288C42—5661no—
289C42—5661no—
290C23—21no—
291C23—2831no—
292C23D4×D28311yescharacteristic
293Q8—21no—
294Q8—5661no—
295Q8—5661no—
296Q8—5661no—
297Q8—5661no—
298Q8—11322no—
299Q8—11322no—
300Q8#1798539212yes—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—11321no—
305D4—11322no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—2831no—
313C2×C4—2831no—
314C2×C4—2831no—
315C2×C4—5661no—
316C2×C4—5661no—
317C2×C4—5661no—
318C2×C4—5661no—
319C2×C4—5661no—
320C2×C4—5661no—
321C2×C4—11322no—
322C2×C4D4×D28311yescharacteristic
323C2×C4C23×D28311yescharacteristic
324C8—21no—
325C8—21no—
326C8—5661no—
327C8—5661no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—5661no—
335C22—5661no—
336C22—5661no—
337C22C2×D4×D28311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—5661no—
343C4—5661no—
344C4—5661no—
345C4—11321no—
346C4—11321no—
347C4—11321no—
348C4—11321no—
349C4C2×#1798539211yescharacteristic
350C4#2258365511yescharacteristic
351C2—41no—
352C2—81no—
353C2—11321no—
354C2#3182710511yescharacteristic
355C2#3182726211yescharacteristic
356C2#3182727911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#416×C283C2
socleC22×C283C22×D4
prime radical#31827122C2
hypercenter#416D283
hypocenterC283#3313
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #31827183C2semidirect
3. #31826823C2semidirect
4. #31826806C2semidirect
5. #31826787C2semidirect
6. #31826758C2semidirect
7. #31826710C2semidirect
8. #31826706C2semidirect
9. #31826686C2semidirect
10. #31826627C2semidirect
11. #31826622C2semidirect
12. #31826569C2semidirect
13. #31826536C2semidirect
14. #31826472C2semidirect
15. #416×C283C2semidirect
178. C283#3313semidirect
194. #416D283semidirect

Maximal subgroups

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

subgroupindex
#318271832
#318271222
#318268232
#318268062
#318267872
#318267582
#318267102
#318267062
#318266862
#318266272
#318266222
#318265692
#318265362
#318264722
#416×C2832
#3313283