Finite Group Databasebeta

← #49134322

Subgroups — #49134322

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#31827201C211yesmaximal, characteristic, complemented
2#31827184C211yesmaximal, characteristic, complemented
3#31826856C211yesmaximal, characteristic, complemented
4#31826848C211yesmaximal, characteristic, complemented
5#31826685C211yesmaximal, characteristic, complemented
6#31826684C211yesmaximal, characteristic, complemented
7#31826678C211yesmaximal, characteristic, complemented
8#31826678C211yesmaximal, characteristic, complemented
9#31826668C211yesmaximal, characteristic, complemented
10#31826662C211yesmaximal, characteristic, complemented
11#31826593C211yesmaximal, characteristic, complemented
12#31826577C211yesmaximal, characteristic, complemented
13#31826573C211yesmaximal, characteristic, complemented
14SD16×Dic283C211yesmaximal, characteristic, complemented
15#496×C283C211yesmaximal, characteristic
16C2×Dic1132C2211yescharacteristic
17C2×#17985382C2211yescharacteristic
18C2×#17985380C2211yescharacteristic
19C2×#17985380C2211yescharacteristic
20C2×#17985380C2211yescharacteristic
21#22583608C2211yescharacteristic
22#22583603—21no—
23#22583603—21no—
24#22583597—21no—
25#22583597C2211yescharacteristic
26#22583597C2211yescharacteristic
27#22583597C2211yescharacteristic
28#22583591—21no—
29#22583591—21no—
30#22583587—21no—
31Q8×Dic283C2211yescharacteristic
32C4×Dic566C2211yescharacteristic
33C4×Dic566C2211yescharacteristic
34C4×Dic566C2211yescharacteristic
35D4×Dic283—21no—
36D4×Dic283C2211yescharacteristic
37D4×Dic283C2211yescharacteristic
38D4×Dic283C2211yescharacteristic
39#22583566—21no—
40#22583542C2211yescharacteristic
41C4⋊(C283⋊C8)C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43Dic283⋊C8C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583535C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49#22583531C2211yescharacteristic
50#22583520C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C2×SD16×C283C2211yescharacteristic
53C2×SD16×C283C2211yescharacteristic
54C2×D8×C283C2211yescharacteristic
55C2×D8×C283C2211yescharacteristic
56#127×C283C2211yescharacteristic
57#124×C283C2211yescharacteristic
58#97×C283C2211yescharacteristic
59C2×Dic566D411yescharacteristic
60C2×Dic566C2311yescharacteristic
61C2×Dic566C2311yescharacteristic
62C22×Dic283—21no—
63C22×Dic283—21no—
64C22×Dic283—21no—
65Dic1132—22no—
66#17985382—41no—
67#17985380—41no—
68#17985380—41no—
69#17985380—41no—
70C2×C283⋊C8C2311yescharacteristic
71C2×C283⋊C8C2311yescharacteristic
72C4⋊Dic283—21no—
73C4⋊Dic283D411yescharacteristic
74C4⋊Dic283C2311yescharacteristic
75C4⋊Dic283C2311yescharacteristic
76Dic283⋊C4—21no—
77Dic283⋊C4—21no—
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80Dic283⋊C4D411yescharacteristic
81Dic283⋊C4D411yescharacteristic
82C22⋊Dic283—21no—
83C22⋊Dic283—21no—
84C22⋊Dic283—21no—
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C22⋊Dic283—21no—
89C22⋊Dic283—21no—
90C22⋊Dic283—21no—
91C4×Dic283—21no—
92C4×Dic283—21no—
93C4×Dic283—21no—
94C4×Dic283C2311yescharacteristic
95C4×Dic283C2311yescharacteristic
96C2×Q8×C283C2311yescharacteristic
97C2×D4×C283—21no—
98C2×D4×C283—21no—
99C2×D4×C283C2311yescharacteristic
100C2×D4×C283C2311yescharacteristic
101C2×D4×C283C2311yescharacteristic
102SD16×C283—22no—
103SD16×C283—41no—
104D8×C283—22no—
105D8×C283—41no—
106C2×C8×C283C2311yescharacteristic
107C2×C8×C283C2311yescharacteristic
108C22⋊C4×C283—21no—
109C22⋊C4×C283—21no—
110C42×C283C2311yescharacteristic
111Dic566—21no—
112Dic566—21no—
113Dic566—21no—
114Dic566—21no—
115Dic566—41no—
116C2×Dic283—21no—
117C2×Dic283—21no—
118C2×Dic283—21no—
119C2×Dic283—21no—
120C2×Dic283—42no—
121C2×Dic283—81no—
122C2×Dic283—81no—
123C2×Dic283C2×D411yescharacteristic
124C2×Dic283C2×D411yescharacteristic
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C283⋊C8—21no—
128C283⋊C8—21no—
129C23×C283—21no—
130C23×C283—21no—
131C23×C283—21no—
132Q8×C283—21no—
133Q8×C283—21no—
134D4×C283—21no—
135D4×C283—21no—
136D4×C283—21no—
137D4×C283—21no—
138D4×C283—21no—
139D4×C283—21no—
140D4×C283—42no—
141D4×C283—81no—
142C2×C4×C283—21no—
143C2×C4×C283C2×D411yescharacteristic
144C2×C4×C283C2×D411yescharacteristic
145C2×C4×C283C2411yescharacteristic
146C8×C283—21no—
147C8×C283#4112yes—
148C22×C283—41no—
149C22×C283—41no—
150C22×C283—41no—
151C22×C283—41no—
152C22×C283—41no—
153C22×C283—41no—
154C22×C283—41no—
155C22×C283—41no—
156C22×C283—41no—
157C22×C283C22×D411yescharacteristic
158C4×C283—21no—
159C4×C283—21no—
160C4×C283—41no—
161C4×C283C2×#4111yescharacteristic
162C4×C283H(5,𝔽2)11yescharacteristic
163Dic283—21no—
164Dic283—21no—
165Dic283—21no—
166Dic283—21no—
167Dic283—41no—
168Dic283—41no—
169Dic283—41no—
170Dic283—41no—
171C2×C283—81no—
172C2×C283—81no—
173C2×C283—81no—
174C2×C283#55011yescharacteristic
175C2×C283#57511yescharacteristic
176C2×C283#57711yescharacteristic
177C283#340111yescharacteristic, complemented, Sylow
178#3401—2831nomaximal, Sylow
179#559—2831no—
180#556—2831no—
181#497—2831no—
182#496D28311yescharacteristic
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×SD16C2×D28311yescharacteristic
200C2×SD16C2×D28311yescharacteristic
201C2×D8C2×D28311yescharacteristic
202C2×D8C2×D28311yescharacteristic
203C4⋊Q8—2831no—
204#127C2×D28311yescharacteristic
205#126—5661no—
206#126—5661no—
207#124—2831no—
208#124—2831no—
209#124—2831no—
210#124—5661no—
211#124C2×D28311yescharacteristic
212#123—5661no—
213#123—5661no—
214C22⋊Q8—5661no—
215C4×Q8—2831no—
216C4×Q8—2831no—
217C4×Q8—2831no—
218C4×Q8—2831no—
219C4×D4—2831no—
220C4×D4—2831no—
221C4×D4—2831no—
222C4×D4—5661no—
223#117—5661no—
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#97C2×D28311yescharacteristic
236C4×C8—2831no—
237C2×Q8—2831no—
238C2×Q8—2831no—
239C2×Q8—2831no—
240C2×Q8C22×D28311yescharacteristic
241C2×D4—21no—
242C2×D4—21no—
243C2×D4C22×D28311yescharacteristic
244C2×D4C22×D28311yescharacteristic
245C2×D4C22×D28311yescharacteristic
246C22×C4—5661no—
247C22×C4—5661no—
248C22×C4—5661no—
249Q16—5662no—
250Q16—11321no—
251SD16—22no—
252SD16—41no—
253SD16—11321no—
254SD16—11321no—
255SD16—11321no—
256D8—22no—
257D8—41no—
258C2×C8—2831no—
259C2×C8—2831no—
260C2×C8C22×D28311yescharacteristic
261C2×C8C22×D28311yescharacteristic
262C4⋊C4—2831no—
263C4⋊C4—2831no—
264C4⋊C4—2831no—
265C4⋊C4—2831no—
266C4⋊C4—2831no—
267C4⋊C4—5661no—
268C4⋊C4—5661no—
269C4⋊C4—5661no—
270C4⋊C4—5661no—
271C4⋊C4—5661no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—5661no—
275C22⋊C4—5661no—
276C22⋊C4—5661no—
277C22⋊C4—5661no—
278C22⋊C4—5661no—
279C22⋊C4—5661no—
280C22⋊C4—5661no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C42—2831no—
284C42—2831no—
285C42—5661no—
286C42—5661no—
287C42—5661no—
288C42C22×D28311yescharacteristic
289C23—21no—
290C23—21no—
291C23—21no—
292Q8—21no—
293Q8—21no—
294Q8—5661no—
295Q8—5661no—
296Q8—5661no—
297Q8—5661no—
298Q8—11321no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—42no—
306D4—81no—
307C2×C4—21no—
308C2×C4—2831no—
309C2×C4—2831no—
310C2×C4—2831no—
311C2×C4—2831no—
312C2×C4—5661no—
313C2×C4—5661no—
314C2×C4—5661no—
315C2×C4—5661no—
316C2×C4—11322no—
317C2×C4—22641no—
318C2×C4—22641no—
319C2×C4D4×D28311yescharacteristic
320C2×C4D4×D28311yescharacteristic
321C2×C4C23×D28311yescharacteristic
322C8—21no—
323C8—5661no—
324C8—5661no—
325C8#1798539112yes—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22C2×D4×D28311yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—5661no—
340C4—5661no—
341C4—5661no—
342C4—5661no—
343C4—11321no—
344C4—11321no—
345C4—11321no—
346C4—11321no—
347C4C2×#1798539111yescharacteristic
348C4#2258365511yescharacteristic
349C2—81no—
350C2—81no—
351C2—81no—
352C2#3182713511yescharacteristic
353C2#3182726111yescharacteristic
354C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting, prime radical#496×C283C2
socleC22×C283C22×D4
hypercenter#496D283
hypocenterC283#3401
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #31827201C2semidirect
2. #31827184C2semidirect
3. #31826856C2semidirect
4. #31826848C2semidirect
5. #31826685C2semidirect
6. #31826684C2semidirect
7. #31826678C2semidirect
8. #31826678C2semidirect
9. #31826668C2semidirect
10. #31826662C2semidirect
11. #31826593C2semidirect
12. #31826577C2semidirect
13. #31826573C2semidirect
14. SD16×Dic283C2semidirect
177. C283#3401semidirect

Maximal subgroups

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

subgroupindex
#318272012
#318271842
#318268562
#318268482
#318266852
#318266842
#318266782
#318266682
#318266622
#318265932
#318265772
#318265732
SD16×Dic2832
#496×C2832
#3401283