Finite Group Databasebeta

← #49134327

Subgroups — #49134327

Order 36224.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827200C211yesmaximal, characteristic, complemented
2#31826857C211yesmaximal, characteristic, complemented
3#31826855C211yesmaximal, characteristic
4#31826847C211yesmaximal, characteristic, complemented
5#31826687C211yesmaximal, characteristic, complemented
6#31826684C211yesmaximal, characteristic, complemented
7#31826680C211yesmaximal, characteristic, complemented
8#31826677C211yesmaximal, characteristic, complemented
9#31826667C211yesmaximal, characteristic, complemented
10#31826665C211yesmaximal, characteristic, complemented
11#31826596C211yesmaximal, characteristic, complemented
12#31826575C211yesmaximal, characteristic, complemented
13#31826570C211yesmaximal, characteristic, complemented
14C4×#17985379C211yesmaximal, characteristic, complemented
15#556×C283C211yesmaximal, characteristic, complemented
16C2×#17985382C2211yescharacteristic
17C2×#17985382C2211yescharacteristic
18C2×#17985380C2211yescharacteristic
19C2×#17985379C2211yescharacteristic
20C2×#17985379C2211yescharacteristic
21C2×#17985379C2211yescharacteristic
22C2×#17985379C2211yescharacteristic
23C2×#17985377C2211yescharacteristic
24C2×#17985377C2211yescharacteristic
25#22583607C2211yescharacteristic
26#22583599—21no—
27#22583598C2211yescharacteristic
28#22583598C2211yescharacteristic
29#22583595—21no—
30#22583595—21no—
31C4×Dic566C2211yescharacteristic
32C4×D1132C2211yescharacteristic
33C4×D1132C2211yescharacteristic
34#22583572—21no—
35#22583541C2211yescharacteristic
36C4⋊(C283⋊C8)C2211yescharacteristic
37C4⋊(C283⋊C8)C2211yescharacteristic
38C4⋊(C283⋊C8)C2211yescharacteristic
39#22583534C2211yescharacteristic
40#22583534C2211yescharacteristic
41#22583533C2211yescharacteristic
42#22583531C2211yescharacteristic
43#22583530C2211yescharacteristic
44#22583530C2211yescharacteristic
45#22583521C2211yescharacteristic
46#22583521C2211yescharacteristic
47C4×C283⋊C8C2211yescharacteristic
48C4⋊Q8×C283C2211yescharacteristic
49#126×C283—21no—
50#126×C283—21no—
51#124×C283C2211yescharacteristic
52#124×C283C2211yescharacteristic
53C22⋊Q8×C283—21no—
54C4×Q8×C283C2211yescharacteristic
55C4×Q8×C283C2211yescharacteristic
56C4×Q8×C283C2211yescharacteristic
57C4×D4×C283C2211yescharacteristic
58#117×C283—21no—
59C2×Dic566C2311yescharacteristic
60C2×D1132—21no—
61C2×D1132—21no—
62C2×D1132C2311yescharacteristic
63C2×D1132C2311yescharacteristic
64C2×C4×D283—21no—
65C2×C4×D283—21no—
66#17985382—22no—
67#17985382—41no—
68#17985380—41no—
69#17985379—22no—
70#17985379—41no—
71#17985379—41no—
72#17985379—41no—
73#17985377—22no—
74#17985377—41no—
75C2×C283⋊C8C2311yescharacteristic
76C2×C283⋊C8C2311yescharacteristic
77C2×C283⋊C8C2311yescharacteristic
78C2×C283⋊C8C2311yescharacteristic
79C4⋊Dic283C2311yescharacteristic
80Dic283⋊C4—21no—
81(C2×D283)⋊C4—21no—
82(C2×D283)⋊C4—21no—
83(C2×D283)⋊C4—21no—
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—21no—
86(C2×D283)⋊C4—21no—
87C4×Dic283—21no—
88C2×Q8×C283D411yescharacteristic
89C2×Q8×C283C2311yescharacteristic
90C2×Q8×C283C2311yescharacteristic
91C2×D4×C283C2311yescharacteristic
92C22×C4×C283—21no—
93C4⋊C4×C283—21no—
94C4⋊C4×C283—21no—
95C4⋊C4×C283—21no—
96C4⋊C4×C283—21no—
97C4⋊C4×C283D411yescharacteristic
98C4⋊C4×C283D411yescharacteristic
99C4⋊C4×C283D411yescharacteristic
100C4⋊C4×C283C2311yescharacteristic
101C22⋊C4×C283—21no—
102C22⋊C4×C283—21no—
103C22⋊C4×C283—21no—
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C42×C283—21no—
107C42×C283—21no—
108C42×C283C2311yescharacteristic
109C42×C283C2311yescharacteristic
110C42×C283C2311yescharacteristic
111C22×D283—21no—
112C22×D283—21no—
113Dic566—21no—
114Dic566—21no—
115D1132—21no—
116D1132—21no—
117D1132—21no—
118D1132—21no—
119D1132—42no—
120D1132—81no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C4×D283—81no—
124C4×D283—81no—
125C283⋊C8—21no—
126C283⋊C8—21no—
127C283⋊C8—21no—
128C283⋊C8#4112yes—
129C23×C283—21no—
130Q8×C283—21no—
131Q8×C283—21no—
132Q8×C283—21no—
133Q8×C283—21no—
134Q8×C283—41no—
135D4×C283—21no—
136D4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—42no—
141C2×C4×C283C2×D411yescharacteristic
142C2×C4×C283C2×D411yescharacteristic
143C2×C4×C283C2×D411yescharacteristic
144C2×C4×C283C2×D411yescharacteristic
145C2×C4×C283C2×D411yescharacteristic
146C2×C4×C283C2×D411yescharacteristic
147C2×C4×C283C2411yescharacteristic
148C22×C283—41no—
149C22×C283—41no—
150C22×C283—41no—
151C22×C283C22×D411yescharacteristic
152C2×D283—41no—
153C2×D283—41no—
154C2×D283—41no—
155C2×D283—41no—
156C2×D283—41no—
157C2×D283—41no—
158C4×C283—21no—
159C4×C283—21no—
160C4×C283—21no—
161C4×C283—21no—
162C4×C283—21no—
163C4×C283—21no—
164C4×C283—41no—
165C4×C283—41no—
166C4×C283—41no—
167C4×C283C2×#4111yescharacteristic
168C4×C283H(5,𝔽2)11yescharacteristic
169Dic283—41no—
170Dic283—41no—
171C2×C283—81no—
172C2×C283#55011yescharacteristic
173C2×C283#57511yescharacteristic
174C2×C283#57711yescharacteristic
175D283—81no—
176D283—81no—
177C283#340111yescharacteristic, complemented, Sylow
178#3401—2831nomaximal, Sylow
179#559—2831no—
180#556D28311yescharacteristic, complemented
181#497—2831no—
182#496—2831no—
183#495—2831no—
184#464—2831no—
185#464—2831no—
186#463—2831no—
187#463—2831no—
188#461—2831no—
189#460—2831no—
190#446—2831no—
191#442—2831no—
192#441—2831no—
193C4×SD16—2831no—
194C2×Q16—2831no—
195C2×Q16—2831no—
196C2×SD16—2831no—
197C2×SD16—2831no—
198C2×SD16—2831no—
199C2×SD16—2831no—
200C2×SD16—2831no—
201C2×D8—2831no—
202C2×D8—2831no—
203C4⋊Q8C2×D28311yescharacteristic
204#127—2831no—
205#126—21no—
206#126—21no—
207#124—2831no—
208#124—2831no—
209#124—5661no—
210#124C2×D28311yescharacteristic
211#124C2×D28311yescharacteristic
212#123—5661no—
213#123—5661no—
214C22⋊Q8—21no—
215C4×Q8—2831no—
216C4×Q8C2×D28311yescharacteristic
217C4×Q8C2×D28311yescharacteristic
218C4×Q8C2×D28311yescharacteristic
219C4×D4—2831no—
220C4×D4—2831no—
221C4×D4—5661no—
222C4×D4C2×D28311yescharacteristic
223#117—21no—
224#106—2831no—
225C4⋊C8—2831no—
226C4⋊C8—2831no—
227C4⋊C8—2831no—
228#103—2831no—
229#103—2831no—
230#103—2831no—
231#102—2831no—
232#102—2831no—
233#102—2831no—
234#97—2831no—
235#97—2831no—
236C4×C8—2831no—
237C2×Q8—2831no—
238C2×Q8C22⋊D28311yescharacteristic
239C2×Q8C22×D28311yescharacteristic
240C2×Q8C22×D28311yescharacteristic
241C2×D4—2831no—
242C2×D4—2831no—
243C2×D4—5661no—
244C2×D4—5661no—
245C2×D4C22×D28311yescharacteristic
246C22×C4—21no—
247C22×C4—5661no—
248C22×C4—5661no—
249Q16—5662no—
250Q16—11321no—
251SD16—5662no—
252SD16—11321no—
253SD16—11321no—
254SD16—11321no—
255SD16—11321no—
256D8—5662no—
257D8—11321no—
258C2×C8—2831no—
259C2×C8—2831no—
260C2×C8—2831no—
261C2×C8—2831no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—2831no—
267C4⋊C4—5661no—
268C4⋊C4C22⋊D28311yescharacteristic
269C4⋊C4C22⋊D28311yescharacteristic
270C4⋊C4C22⋊D28311yescharacteristic
271C4⋊C4C22×D28311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—5661no—
278C22⋊C4—5661no—
279C22⋊C4—5661no—
280C22⋊C4—5661no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C42—21no—
284C42—21no—
285C42—5661no—
286C42C22×D28311yescharacteristic
287C42C22×D28311yescharacteristic
288C42C22×D28311yescharacteristic
289C23—21no—
290C23—5661no—
291C23—5661no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—5661no—
298Q8—5661no—
299D4—21no—
300D4—21no—
301D4—5661no—
302D4—5661no—
303D4—5661no—
304D4—5661no—
305D4—11322no—
306D4—22641no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—5661no—
312C2×C4—5661no—
313C2×C4—22641no—
314C2×C4—22641no—
315C2×C4C2×C22⋊D28311yescharacteristic
316C2×C4C2×C22⋊D28311yescharacteristic
317C2×C4C2×C22⋊D28311yescharacteristic
318C2×C4C2×C22⋊D28311yescharacteristic
319C2×C4C2×C22⋊D28311yescharacteristic
320C2×C4C2×C22⋊D28311yescharacteristic
321C2×C4C23×D28311yescharacteristic
322C8—2832no—
323C8—5661no—
324C8—5661no—
325C8—5661no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—11321no—
330C22—11321no—
331C22—11321no—
332C22—11321no—
333C22—11321no—
334C22—11321no—
335C22C22×C22⋊D28311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—11321no—
346C4—11321no—
347C4#41×D28311yescharacteristic
348C4#2258365611yescharacteristic
349C2—81no—
350C2—22641no—
351C2—22641no—
352C2#3182713411yescharacteristic
353C2#3182726311yescharacteristic
354C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#556×C283C2
socleC22×C283C22×D4
prime radical#31826855C2
hypercenter#556D283
hypocenterC283#3401
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #31827200C2semidirect
2. #31826857C2semidirect
4. #31826847C2semidirect
5. #31826687C2semidirect
6. #31826684C2semidirect
7. #31826680C2semidirect
8. #31826677C2semidirect
9. #31826667C2semidirect
10. #31826665C2semidirect
11. #31826596C2semidirect
12. #31826575C2semidirect
13. #31826570C2semidirect
14. C4×#17985379C2semidirect
15. #556×C283C2semidirect
177. C283#3401semidirect
180. #556D283semidirect

Maximal subgroups

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

subgroupindex
#318272002
#318268572
#318268552
#318268472
#318266872
#318266842
#318266802
#318266772
#318266672
#318266652
#318265962
#318265752
#318265702
C4×#179853792
#556×C2832
#3401283