Finite Group Databasebeta

← #46821161

Subgroups — #46821161

Order 34432.

384 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#31771224C211yesmaximal, characteristic, complemented
2#31771087C211yesmaximal, characteristic, complemented
3#31770934C211yesmaximal, characteristic, complemented
4#31770922C211yesmaximal, characteristic, complemented
5#31770896C211yesmaximal, characteristic, complemented
6#31770890C211yesmaximal, characteristic, complemented
7#31770817C211yesmaximal, characteristic, complemented
8#31770766C211yesmaximal, characteristic, complemented
9#31770749C211yesmaximal, characteristic
10#31770714C211yesmaximal, characteristic, complemented
11#31770712C211yesmaximal, characteristic, complemented
12SD16×Dic269C211yesmaximal, characteristic, complemented
13#31770690C211yesmaximal, characteristic, complemented
14#31770665C211yesmaximal, characteristic, complemented
15#464×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864389C2211yescharacteristic
18C2×#16864387C2211yescharacteristic
19#22575453C2211yescharacteristic
20#22575446—21no—
21#22575439—21no—
22#22575437—21no—
23#22575430—21no—
24#22575429C2211yescharacteristic
25#22575424C2211yescharacteristic
26Q8×Dic269C2211yescharacteristic
27Q8×Dic269C2211yescharacteristic
28C4×D1076—21no—
29C4×D1076C2211yescharacteristic
30D4×Dic269C2211yescharacteristic
31#22575415—21no—
32#22575415C2211yescharacteristic
33#22575410—21no—
34#22575410C2211yescharacteristic
35C4⋊C4×D269—21no—
36C4⋊C4×D269C2211yescharacteristic
37#22575385C2211yescharacteristic
38#22575385C2211yescharacteristic
39#22575384C2211yescharacteristic
40#22575384C2211yescharacteristic
41C4⋊#14524130C2211yescharacteristic
42#22575377C2211yescharacteristic
43#22575377C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575375C2211yescharacteristic
46#22575375C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575365C2211yescharacteristic
49#22575365C2211yescharacteristic
50#22575363C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×Q16×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54#124×C269C2211yescharacteristic
55C4×Q8×C269C2211yescharacteristic
56C4⋊C8×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59#16864403—22no—
60#16864403—22no—
61C2×Dic538—21no—
62C2×D1076—21no—
63C2×D1076C2311yescharacteristic
64C2×C22⋊D269—21no—
65C22×Dic269—21no—
66C2×C4×D269—21no—
67C2×C4×D269—21no—
68C2×C4×D269—21no—
69C2×C4×D269—21no—
70C2×C4×D269C2311yescharacteristic
71#16864389—22no—
72#16864387—22no—
73C2×#14524130C2311yescharacteristic
74C2×#14524130C2311yescharacteristic
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77C4⋊Dic269—21no—
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80C4⋊Dic269C2311yescharacteristic
81#16864381—21no—
82#16864381—21no—
83#16864381D411yescharacteristic
84#16864381D411yescharacteristic
85C22⋊Dic269—21no—
86C22⋊Dic269—21no—
87#16864379—21no—
88#16864379—21no—
89#16864379—21no—
90#16864379D411yescharacteristic
91#16864379D411yescharacteristic
92C4×Dic269—21no—
93C4×Dic269—21no—
94C4×Dic269C2311yescharacteristic
95C2×Q8×C269C2311yescharacteristic
96C2×Q8×C269C2311yescharacteristic
97C2×D4×C269C2311yescharacteristic
98Q16×C269—22no—
99SD16×C269—22no—
100C2×C8×C269C2311yescharacteristic
101C2×C8×C269C2311yescharacteristic
102C4⋊C4×C269—21no—
103C4⋊C4×C269C2311yescharacteristic
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C42×C269—21no—
107C42×C269C2311yescharacteristic
108C22×D269—21no—
109C22×D269C2×D411yescharacteristic
110Dic538—81no—
111D1076—24no—
112D1076—21no—
113D1076#4112yes—
114C22⋊D269—81no—
115C2×Dic269—21no—
116C2×Dic269—21no—
117C2×Dic269—21no—
118C2×Dic269—81no—
119C2×Dic269C2×D411yescharacteristic
120C2×Dic269C2×D411yescharacteristic
121C2×Dic269C2×D411yescharacteristic
122C4×D269—24no—
123C4×D269—21no—
124C4×D269—21no—
125C4×D269—81no—
126C4×D269—81no—
127C4×D269—81no—
128#14524130—21no—
129#14524130—21no—
130C23×C269—21no—
131Q8×C269—21no—
132Q8×C269—21no—
133Q8×C269—21no—
134Q8×C269#4112yes—
135D4×C269—21no—
136D4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269C2×D411yescharacteristic
141C2×C4×C269C2×D411yescharacteristic
142C2×C4×C269C2411yescharacteristic
143C8×C269—21no—
144C8×C269—21no—
145C22×C269—41no—
146C22×C269—41no—
147C22×C269—41no—
148C22×C269C22×D411yescharacteristic
149C2×D269—21no—
150C2×D269—22no—
151C2×D269—21no—
152C2×D269—21no—
153C2×D269—41no—
154C2×D269—41no—
155C4×C269—22no—
156C4×C269—21no—
157C4×C269—21no—
158C4×C269—41no—
159C4×C269—41no—
160C4×C269C2×#4111yescharacteristic
161C4×C269C2×#4111yescharacteristic
162Dic269—21no—
163Dic269—21no—
164Dic269—21no—
165Dic269—41no—
166Dic269—41no—
167Dic269—41no—
168C2×C269—81no—
169C2×C269#52511yescharacteristic
170C2×C269#57711yescharacteristic
171C2×C269#57711yescharacteristic
172D269—41no—
173D269—42no—
174C269#312511yescharacteristic, complemented, Sylow
175#3125—2691nomaximal, Sylow
176#542—2691no—
177#520—2691no—
178#483—2691no—
179#481—2691no—
180#476—2691no—
181#475—2691no—
182#464D26911yescharacteristic, complemented
183#463—2691no—
184#452—2691no—
185#449—2691no—
186#442—2691no—
187#441—2691no—
188C4×SD16—2691no—
189C4×SD16—2691no—
190#433—2691no—
191C2×#41—2691no—
192C2×Q16C2×D26911yescharacteristic
193C2×SD16—2691no—
194C2×SD16C2×D26911yescharacteristic
195C2×D8—2691no—
196C4⋊Q8—2691no—
197#126—5381no—
198#124—5381no—
199#124C2×D26911yescharacteristic
200#123—5381no—
201C22⋊Q8—2691no—
202C22⋊Q8—5381no—
203#121—2691no—
204C4×Q8—2691no—
205C4×Q8—2691no—
206C4×Q8C2×D26911yescharacteristic
207C4×D4—2691no—
208C4×D4—2691no—
209C4×D4—2691no—
210C4×D4—5381no—
211C4×D4—5381no—
212#117—2691no—
213#117—5381no—
214C2×C4⋊C4—2691no—
215C2×C4⋊C4—5381no—
216#106—2691no—
217#106—2691no—
218#106—2691no—
219#106—2691no—
220C4⋊C8—2691no—
221C4⋊C8C2×D26911yescharacteristic
222#103—2691no—
223#103—2691no—
224#103—2691no—
225#103C2×D26911yescharacteristic
226#102—2691no—
227#102—2691no—
228#102—2691no—
229#102C2×D26911yescharacteristic
230C22⋊C8—2691no—
231C22⋊C8—2691no—
232#97—2691no—
233C4×C8—2691no—
234#41—5382no—
235#41—5382no—
236C2×Q8—5381no—
237C2×Q8C22×D26911yescharacteristic
238C2×Q8C22×D26911yescharacteristic
239C2×D4—2691no—
240C2×D4—5381no—
241C2×D4—5381no—
242C2×D4C22×D26911yescharacteristic
243C22×C4—2691no—
244C22×C4—5381no—
245C22×C4—5381no—
246C22×C4—5381no—
247C22×C4—5381no—
248C22×C4—5381no—
249Q16—22no—
250SD16—22no—
251SD16—5382no—
252D8—5382no—
253C2×C8—2691no—
254C2×C8—2691no—
255C2×C8C22×D26911yescharacteristic
256C2×C8C22×D26911yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—2691no—
259C4⋊C4—2691no—
260C4⋊C4—2691no—
261C4⋊C4—2691no—
262C4⋊C4—2691no—
263C4⋊C4—5381no—
264C4⋊C4—5381no—
265C4⋊C4—5381no—
266C4⋊C4—5381no—
267C4⋊C4—5381no—
268C4⋊C4C22×D26911yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—2691no—
272C22⋊C4—2691no—
273C22⋊C4—5381no—
274C22⋊C4—5381no—
275C22⋊C4—5381no—
276C22⋊C4—5381no—
277C22⋊C4—5381no—
278C42—21no—
279C42—2691no—
280C42—5381no—
281C42—5381no—
282C42C22×D26911yescharacteristic
283C23—21no—
284C23—2691no—
285C23—5381no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21521no—
290Q8#1686440112yes—
291D4—21no—
292D4—21no—
293D4—2692no—
294D4—5384no—
295D4—5381no—
296D4—21521no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—2691no—
301C2×C4—2691no—
302C2×C4—2691no—
303C2×C4—5384no—
304C2×C4—5381no—
305C2×C4—5381no—
306C2×C4—5381no—
307C2×C4—5381no—
308C2×C4—5381no—
309C2×C4—21521no—
310C2×C4—21521no—
311C2×C4—21521no—
312C2×C4—21521no—
313C2×C4D4×D26911yescharacteristic
314C2×C4D4×D26911yescharacteristic
315C2×C4C23×D26911yescharacteristic
316C8—21no—
317C8—21no—
318C8—5381no—
319C8—5381no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—5381no—
324C22—5381no—
325C22—5382no—
326C22—5381no—
327C22—10761no—
328C22—10761no—
329C22C2×D4×D26911yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—5381no—
336C4—5381no—
337C4—5381no—
338C4—10761no—
339C4—10761no—
340C4—10761no—
341C4#41×D26911yescharacteristic
342C4C2×#1686440111yescharacteristic
343C2—81no—
344C2—10761no—
345C2—10762no—
346C2#3177111511yescharacteristic
347C2#3177141211yescharacteristic
348C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#464×C269C2
socleC22×C269C22×D4
prime radical#31770749C2
hypercenter#464D269
hypocenterC269#3125
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #31771224C2semidirect
2. #31771087C2semidirect
3. #31770934C2semidirect
4. #31770922C2semidirect
5. #31770896C2semidirect
6. #31770890C2semidirect
7. #31770817C2semidirect
8. #31770766C2semidirect
10. #31770714C2semidirect
11. #31770712C2semidirect
12. SD16×Dic269C2semidirect
13. #31770690C2semidirect
14. #31770665C2semidirect
15. #464×C269C2semidirect
174. C269#3125semidirect
182. #464D269semidirect

Maximal subgroups

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

subgroupindex
#317712242
#317710872
#317709342
#317709222
#317708962
#317708902
#317708172
#317707662
#317707492
#317707142
#317707122
SD16×Dic2692
#317706902
#317706652
#464×C2692
#3125269