Finite Group Databasebeta

← #46822766

Subgroups — #46822766

Order 34432.

386 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#31771264C211yesmaximal, characteristic
2#31770962C211yesmaximal, characteristic, complemented
3#31770946C211yesmaximal, characteristic, complemented
4#31770918C211yesmaximal, characteristic, complemented
5#31770909C211yesmaximal, characteristic, complemented
6#31770844C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770827C211yesmaximal, characteristic, complemented
9#31770765C211yesmaximal, characteristic, complemented
10#31770760C211yesmaximal, characteristic, complemented
11C4×#16864392C211yesmaximal, characteristic, complemented
12D4×#14524130C211yesmaximal, characteristic, complemented
13#31770599C211yesmaximal, characteristic, complemented
14C2×#22575376C211yesmaximal, characteristic, complemented
15#552×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×Dic538C2211yescharacteristic
18C2×#16864392C2211yescharacteristic
19C2×#16864392C2211yescharacteristic
20C2×#16864389C2211yescharacteristic
21C22×#14524130C2211yescharacteristic
22C22×#14524130C2211yescharacteristic
23#22575441C2211yescharacteristic
24#22575439—21no—
25#22575432C2211yescharacteristic
26#22575431—21no—
27#22575427C2211yescharacteristic
28#22575425—21no—
29C4×Dic538C2211yescharacteristic
30#22575413—21no—
31#22575386C2211yescharacteristic
32#22575386C2211yescharacteristic
33#22575384C2211yescharacteristic
34C4⋊#14524130C2211yescharacteristic
35#22575377C2211yescharacteristic
36#22575376C2211yescharacteristic
37#22575376C2211yescharacteristic
38#22575376C2211yescharacteristic
39#22575376C2211yescharacteristic
40#22575376C2211yescharacteristic
41#22575376C2211yescharacteristic
42#22575373C2211yescharacteristic
43#22575373C2211yescharacteristic
44#22575373C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic
47C4×#14524130C2211yescharacteristic
48#126×C269—21no—
49#125×C269—21no—
50#125×C269C2211yescharacteristic
51#123×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic
53C22⋊Q8×C269C2211yescharacteristic
54C4×Q8×C269C2211yescharacteristic
55C4×D4×C269C2211yescharacteristic
56#117×C269—21no—
57#117×C269C2211yescharacteristic
58C2×C4⋊C4×C269C2211yescharacteristic
59#16864402—22no—
60#16864402—41no—
61C2×Dic538—22no—
62C2×Dic538—21no—
63C2×Dic538—41no—
64C2×Dic538C2311yescharacteristic
65C2×Dic538C2311yescharacteristic
66C2×D1076C2311yescharacteristic
67C2×C22⋊D269—21no—
68C2×C22⋊D269—21no—
69C22×Dic269—21no—
70C2×C4×D269—21no—
71#16864392—22no—
72#16864392—41no—
73#16864389—41no—
74C2×#14524130—41no—
75C2×#14524130—41no—
76C2×#14524130C2311yescharacteristic
77C2×#14524130C2311yescharacteristic
78C2×#14524130C2311yescharacteristic
79C2×#14524130C2311yescharacteristic
80C4⋊Dic269C2311yescharacteristic
81#16864381—21no—
82#16864381—21no—
83C22⋊Dic269—21no—
84#16864379—21no—
85#16864379—21no—
86#16864379—21no—
87C4×Dic269—21no—
88C2×Q8×C269C2311yescharacteristic
89C2×D4×C269D411yescharacteristic
90C22×C4×C269—21no—
91C22×C4×C269C2311yescharacteristic
92C22×C4×C269C2311yescharacteristic
93C4⋊C4×C269—21no—
94C4⋊C4×C269—21no—
95C4⋊C4×C269—21no—
96C4⋊C4×C269—21no—
97C4⋊C4×C269—21no—
98C4⋊C4×C269—21no—
99C4⋊C4×C269D411yescharacteristic
100C4⋊C4×C269C2311yescharacteristic
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269D411yescharacteristic
107C22⋊C4×C269D411yescharacteristic
108C42×C269—21no—
109C42×C269—21no—
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112Dic538—21no—
113Dic538—21no—
114Dic538—21no—
115Dic538—42no—
116Dic538#4112yes—
117D1076—21no—
118D1076—21no—
119C22⋊D269—42no—
120C22⋊D269—42no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—42no—
125C4×D269—42no—
126#14524130—21no—
127#14524130—21no—
128#14524130—21no—
129#14524130—21no—
130C23×C269C2×D411yescharacteristic
131C23×C269C2×D411yescharacteristic
132Q8×C269—21no—
133Q8×C269—21no—
134D4×C269—22no—
135C2×C4×C269—21no—
136C2×C4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—42no—
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2411yescharacteristic
149C22×C269—21no—
150C22×C269—21no—
151C22×C269—21no—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269#13512yes—
155C22×C269C22×D411yescharacteristic
156C2×D269—41no—
157C2×D269—41no—
158C2×D269—41no—
159C4×C269—21no—
160C4×C269—21no—
161C4×C269—21no—
162C4×C269—21no—
163C4×C269—41no—
164C4×C269—41no—
165C4×C269—41no—
166C4×C269—41no—
167C4×C269C2×#4111yescharacteristic
168C4×C269H(5,𝔽2)11yescharacteristic
169Dic269—22no—
170Dic269—41no—
171Dic269—41no—
172C2×C269—22no—
173C2×C269—41no—
174C2×C269#54611yescharacteristic
175C2×C269C2×#13511yescharacteristic
176C2×C269#57811yescharacteristic
177D269—81no—
178C269#330011yescharacteristic, complemented, Sylow
179#3300—2691nomaximal, Sylow
180#552D26911yescharacteristic, complemented
181#548—2691no—
182#488—2691no—
183#485—2691no—
184#480—2691no—
185#479—2691no—
186#467—2691no—
187#465—2691no—
188#464—2691no—
189#452—2691no—
190#451—2691no—
191C4×Q16—2691no—
192C8×D4—2691no—
193#416—2691no—
194C2×#103—2691no—
195C2×#41—2691no—
196C22×Q8—2691no—
197C2×Q16—2691no—
198C2×Q16—2691no—
199C2×SD16—2691no—
200C22×C8—2691no—
201C22×C8—2691no—
202#126—21no—
203#125—21no—
204#125C2×D26911yescharacteristic
205#124—2691no—
206#124—5381no—
207#123—21no—
208C22⋊Q8—2691no—
209C22⋊Q8—5381no—
210C22⋊Q8C2×D26911yescharacteristic
211C22⋊Q8C2×D26911yescharacteristic
212#121—2691no—
213#121—5381no—
214C4×Q8—2691no—
215C4×Q8C2×D26911yescharacteristic
216C4×D4—5381no—
217C4×D4C2×D26911yescharacteristic
218#117—21no—
219#117C2×D26911yescharacteristic
220C2×C4⋊C4C2×D26911yescharacteristic
221#107—2691no—
222#107—2691no—
223#106—2691no—
224C4⋊C8—2691no—
225#103—2691no—
226#103—2691no—
227#103—2691no—
228#103—2691no—
229#103—2691no—
230#103—2691no—
231#103—2691no—
232#102—2691no—
233#102—2691no—
234#102—2691no—
235C22⋊C8—2691no—
236C22⋊C8—2691no—
237C4×C8—2691no—
238#41—5382no—
239#41—10761no—
240C2×Q8—2691no—
241C2×Q8—2691no—
242C2×Q8—5382no—
243C2×Q8—5381no—
244C2×Q8—10761no—
245C2×Q8C22×D26911yescharacteristic
246C2×D4—2691no—
247C2×D4—5381no—
248C2×D4—5381no—
249C2×D4C22⋊D26911yescharacteristic
250C22×C4—21no—
251C22×C4—5381no—
252C22×C4—5381no—
253C22×C4C22×D26911yescharacteristic
254C22×C4C22×D26911yescharacteristic
255Q16—5382no—
256Q16—10761no—
257SD16—10761no—
258C2×C8—2691no—
259C2×C8—2691no—
260C2×C8—2691no—
261C2×C8—2691no—
262C2×C8—10761no—
263C2×C8—10761no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—2691no—
271C4⋊C4—5381no—
272C4⋊C4—5381no—
273C4⋊C4C22⋊D26911yescharacteristic
274C4⋊C4C22×D26911yescharacteristic
275C4⋊C4C22×D26911yescharacteristic
276C4⋊C4C22×D26911yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—5381no—
281C22⋊C4—5381no—
282C22⋊C4—5381no—
283C22⋊C4—5381no—
284C22⋊C4C22⋊D26911yescharacteristic
285C22⋊C4C22⋊D26911yescharacteristic
286C42—21no—
287C42—21no—
288C42—5381no—
289C42C22×D26911yescharacteristic
290C23—5381no—
291C23C2×C22⋊D26911yescharacteristic
292C23C2×C22⋊D26911yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2692no—
296Q8—5381no—
297Q8—5381no—
298Q8—5381no—
299Q8—10762no—
300D4—22no—
301D4—5381no—
302D4—5381no—
303D4—10762no—
304D4—10762no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—5381no—
315C2×C4—5381no—
316C2×C4—5381no—
317C2×C4—10762no—
318C2×C4—10762no—
319C2×C4C2×C22⋊D26911yescharacteristic
320C2×C4C2×C22⋊D26911yescharacteristic
321C2×C4C2×C22⋊D26911yescharacteristic
322C2×C4C2×C22⋊D26911yescharacteristic
323C2×C4C23×D26911yescharacteristic
324C8—5381no—
325C8—5381no—
326C8—5381no—
327C8—5381no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—10761no—
334C22—10761no—
335C22—10761no—
336C22#2257547312yes—
337C22C22×C22⋊D26911yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—5382no—
347C4—10761no—
348C4—10761no—
349C4#41×D26911yescharacteristic
350C4#2257549911yescharacteristic
351C2—22no—
352C2—41no—
353C2—21521no—
354C2#3177124311yescharacteristic
355C2C2×#2257547311yescharacteristic
356C2#3177142011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#552×C269C2
socleC22×C269C22×D4
prime radical#31771264C2
hypercenter#552D269
hypocenterC269#3300
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31770962C2semidirect
3. #31770946C2semidirect
4. #31770918C2semidirect
5. #31770909C2semidirect
6. #31770844C2semidirect
7. #31770836C2semidirect
8. #31770827C2semidirect
9. #31770765C2semidirect
10. #31770760C2semidirect
11. C4×#16864392C2semidirect
12. D4×#14524130C2semidirect
13. #31770599C2semidirect
14. C2×#22575376C2semidirect
15. #552×C269C2semidirect
178. C269#3300semidirect
180. #552D269semidirect

Maximal subgroups

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

subgroupindex
#317712642
#317709622
#317709462
#317709182
#317709092
#317708442
#317708362
#317708272
#317707652
#317707602
C4×#168643922
D4×#145241302
#317705992
C2×#225753762
#552×C2692
#3300269