Finite Group Databasebeta

← #46822791

Subgroups — #46822791

Order 34432.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771307C211yesmaximal, characteristic, complemented
2#31771266C211yesmaximal, characteristic, complemented
3#31770961C211yesmaximal, characteristic, complemented
4#31770944C211yesmaximal, characteristic, complemented
5#31770924C211yesmaximal, characteristic, complemented
6#31770910C211yesmaximal, characteristic, complemented
7#31770839C211yesmaximal, characteristic, complemented
8#31770831C211yesmaximal, characteristic, complemented
9#31770822C211yesmaximal, characteristic, complemented
10#31770749C211yesmaximal, characteristic, complemented
11SD16×Dic269C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13#31770600C211yesmaximal, characteristic, complemented
14C2×#22575377C211yesmaximal, characteristic, complemented
15#448×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864387C2211yescharacteristic
18C22×#14524130C2211yescharacteristic
19C2×C8×D269C2211yescharacteristic
20#22575446—21no—
21#22575444C2211yescharacteristic
22#22575442—21no—
23#22575442C2211yescharacteristic
24#22575437—21no—
25#22575429—21no—
26#22575429C2211yescharacteristic
27#22575427—21no—
28#22575427C2211yescharacteristic
29#22575425—21no—
30#22575424C2211yescharacteristic
31Q8×Dic269C2211yescharacteristic
32D4×Dic269—21no—
33D4×Dic269C2211yescharacteristic
34C4×C22⋊D269—21no—
35C4×C22⋊D269C2211yescharacteristic
36#22575411C2211yescharacteristic
37C2×C4⋊Dic269C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575385C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575377C2211yescharacteristic
43#22575377C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575375C2211yescharacteristic
47#22575375C2211yescharacteristic
48#22575374C2211yescharacteristic
49#22575373C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C22×Q8×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54C2×SD16×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59#16864403—22no—
60#16864403—41no—
61C2×D1076—21no—
62C2×D1076C2311yescharacteristic
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65C2×C22⋊D269D411yescharacteristic
66C22×Dic269—21no—
67C22×Dic269—21no—
68C2×C4×D269—21no—
69C2×C4×D269C2311yescharacteristic
70#16864387—41no—
71C2×#14524130—41no—
72C2×#14524130C2311yescharacteristic
73C2×#14524130C2311yescharacteristic
74C8×D269—41no—
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77C4⋊Dic269—21no—
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80C4⋊Dic269C2311yescharacteristic
81#16864381—21no—
82#16864381—21no—
83#16864381D411yescharacteristic
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269D411yescharacteristic
87#16864379—21no—
88#16864379—21no—
89#16864379—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379D411yescharacteristic
93C4×Dic269—21no—
94C4×Dic269—21no—
95C4×Dic269C2311yescharacteristic
96C2×Q8×C269—21no—
97C2×Q8×C269—22no—
98C2×Q8×C269—41no—
99C2×Q8×C269C2311yescharacteristic
100C2×Q8×C269C2311yescharacteristic
101C2×D4×C269—21no—
102C2×D4×C269C2311yescharacteristic
103C22×C4×C269—21no—
104C22×C4×C269C2311yescharacteristic
105SD16×C269—22no—
106SD16×C269—41no—
107C2×C8×C269C2311yescharacteristic
108C2×C8×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22×D269—21no—
112C22×D269C2×D411yescharacteristic
113D1076—21no—
114D1076—21no—
115D1076—42no—
116C22⋊D269—22no—
117C22⋊D269—42no—
118C22⋊D269—81no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—81no—
123C2×Dic269—81no—
124C2×Dic269C2×D411yescharacteristic
125C2×Dic269C2×D411yescharacteristic
126C2×Dic269C2×D411yescharacteristic
127C4×D269—21no—
128C4×D269—21no—
129C4×D269—42no—
130#14524130—21no—
131#14524130—21no—
132C23×C269—21no—
133C23×C269C2×D411yescharacteristic
134Q8×C269—21no—
135Q8×C269—21no—
136Q8×C269—21no—
137Q8×C269—42no—
138Q8×C269#4112yes—
139D4×C269—21no—
140D4×C269—21no—
141D4×C269—42no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—42no—
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2411yescharacteristic
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—41no—
155C22×C269—41no—
156C22×C269—41no—
157C22×C269#13512yes—
158C22×C269C22×D411yescharacteristic
159C2×D269—21no—
160C2×D269—21no—
161C2×D269—21no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—41no—
165C4×C269—21no—
166C4×C269—22no—
167C4×C269—41no—
168C4×C269—41no—
169C4×C269C2×#4111yescharacteristic
170C4×C269H(5,𝔽2)11yescharacteristic
171Dic269—21no—
172Dic269—21no—
173Dic269—21no—
174Dic269—41no—
175Dic269—41no—
176Dic269—41no—
177C2×C269—22no—
178C2×C269—81no—
179C2×C269#54611yescharacteristic
180C2×C269C2×#13511yescharacteristic
181C2×C269#57711yescharacteristic
182D269—41no—
183D269—81no—
184C269#330111yescharacteristic, complemented, Sylow
185#3301—2691nomaximal, Sylow
186#553—2691no—
187#548—2691no—
188#488—2691no—
189#485—2691no—
190#481—2691no—
191#479—2691no—
192#466—2691no—
193#465—2691no—
194#463—2691no—
195#449—2691no—
196#448D26911yescharacteristic, complemented
197C4×SD16—2691no—
198C8×D4—2691no—
199#416—2691no—
200C2×#103—2691no—
201C2×#41—2691no—
202C22×Q8C2×D26911yescharacteristic
203C2×SD16C2×D26911yescharacteristic
204C2×SD16C2×D26911yescharacteristic
205C2×D8—2691no—
206C22×C8—2691no—
207C22×C8—2691no—
208#126—5381no—
209#125—2691no—
210#124—2691no—
211#124—5381no—
212#123—5381no—
213C22⋊Q8—2691no—
214C22⋊Q8—5381no—
215#121—2691no—
216#121—2691no—
217#121—5381no—
218#121—5381no—
219#121C2×D26911yescharacteristic
220C4×Q8—2691no—
221C4×D4—2691no—
222C4×D4—2691no—
223C4×D4—5381no—
224C4×D4—5381no—
225#117—2691no—
226C2×C4⋊C4—2691no—
227#107—2691no—
228#106—2691no—
229#106—2691no—
230C4⋊C8—2691no—
231#103—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103C2×D26911yescharacteristic
236#103C2×D26911yescharacteristic
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102—2691no—
241C22⋊C8—2691no—
242C22⋊C8C2×D26911yescharacteristic
243C4×C8—2691no—
244#41—5382no—
245#41—10761no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D26911yescharacteristic
250C2×Q8C22×D26911yescharacteristic
251C2×D4—21no—
252C2×D4—2691no—
253C2×D4—2691no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4—5381no—
257C2×D4C22×D26911yescharacteristic
258C22×C4—21no—
259C22×C4—2691no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4—5381no—
263C22×C4C22×D26911yescharacteristic
264SD16—22no—
265SD16—41no—
266D8—10761no—
267C2×C8—2691no—
268C2×C8—2691no—
269C2×C8—10761no—
270C2×C8—10761no—
271C2×C8C22×D26911yescharacteristic
272C2×C8C22×D26911yescharacteristic
273C4⋊C4—2691no—
274C4⋊C4—2691no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—5381no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4C22×D26911yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2691no—
285C22⋊C4—2691no—
286C22⋊C4—5381no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C42—2691no—
294C42—5381no—
295C42—5381no—
296C23—21no—
297C23—2691no—
298C23—5381no—
299C23D4×D26911yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1686440112yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—5382no—
309D4—5381no—
310D4—5381no—
311D4—10762no—
312D4—10762no—
313D4—21521no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—2691no—
321C2×C4—2691no—
322C2×C4—2691no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—10762no—
329C2×C4—21521no—
330C2×C4—21521no—
331C2×C4D4×D26911yescharacteristic
332C2×C4C23×D26911yescharacteristic
333C8—21no—
334C8—21no—
335C8—5381no—
336C8—5381no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—5381no—
343C22—5381no—
344C22—5381no—
345C22—10761no—
346C22—10761no—
347C22—10761no—
348C22#2257547412yes—
349C22C2×D4×D26911yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—5381no—
355C4—5381no—
356C4—5381no—
357C4—10761no—
358C4—10761no—
359C4—10761no—
360C4C2×#1686440111yescharacteristic
361C4#2257549811yescharacteristic
362C2—22no—
363C2—81no—
364C2—10761no—
365C2—21521no—
366C2#3177124211yescharacteristic
367C2C2×#2257547411yescharacteristic
368C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#448×C269C2
socleC22×C269C22×D4
hypercenter#448D269
hypocenterC269#3301
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #31771307C2semidirect
2. #31771266C2semidirect
3. #31770961C2semidirect
4. #31770944C2semidirect
5. #31770924C2semidirect
6. #31770910C2semidirect
7. #31770839C2semidirect
8. #31770831C2semidirect
9. #31770822C2semidirect
10. #31770749C2semidirect
11. SD16×Dic269C2semidirect
12. #31770668C2semidirect
13. #31770600C2semidirect
14. C2×#22575377C2semidirect
15. #448×C269C2semidirect
184. C269#3301semidirect
196. #448D269semidirect

Maximal subgroups

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

subgroupindex
#317713072
#317712662
#317709612
#317709442
#317709242
#317709102
#317708392
#317708312
#317708222
#317707492
SD16×Dic2692
#317706682
#317706002
C2×#225753772
#448×C2692
#3301269