Finite Group Databasebeta

← #46822206

Subgroups — #46822206

Order 34432.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771150C211yesmaximal, characteristic, complemented
2#31771141C211yesmaximal, characteristic, complemented
3#31770904C211yesmaximal, characteristic, complemented
4#31770903C211yesmaximal, characteristic, complemented
5#31770898C211yesmaximal, characteristic, complemented
6#31770894C211yesmaximal, characteristic, complemented
7#31770841C211yesmaximal, characteristic, complemented
8#31770839C211yesmaximal, characteristic, complemented
9#31770836C211yesmaximal, characteristic, complemented
10#31770823C211yesmaximal, characteristic, complemented
11#31770822C211yesmaximal, characteristic, complemented
12#31770803C211yesmaximal, characteristic, complemented
13#31770799C211yesmaximal, characteristic, complemented
14C2×#22575382C211yesmaximal, characteristic, complemented
15#465×C269C211yesmaximal, characteristic, complemented
16C2×#16864401—21no—
17C2×#16864389C2211yescharacteristic
18C2×D2152C2211yescharacteristic
19C2×#16864387C2211yescharacteristic
20C22×#14524130C2211yescharacteristic
21#22575452C2211yescharacteristic
22#22575449C2211yescharacteristic
23#22575448—21no—
24#22575444C2211yescharacteristic
25#22575442C2211yescharacteristic
26#22575439—21no—
27#22575437—21no—
28#22575427C2211yescharacteristic
29#22575427C2211yescharacteristic
30#22575425—21no—
31Q8×Dic269C2211yescharacteristic
32D4×Dic269C2211yescharacteristic
33#22575415C2211yescharacteristic
34#22575415C2211yescharacteristic
35#22575413—21no—
36#22575413—21no—
37#22575411—21no—
38C2×C4×Dic269C2211yescharacteristic
39#22575387C2211yescharacteristic
40#22575386C2211yescharacteristic
41#22575384C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575382C2211yescharacteristic
44#22575382C2211yescharacteristic
45#22575382C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575375C2211yescharacteristic
48#22575375C2211yescharacteristic
49#22575374C2211yescharacteristic
50#22575373C2211yescharacteristic
51#22575373C2211yescharacteristic
52C2×SD16×C269C2211yescharacteristic
53C22×C8×C269C2211yescharacteristic
54C22⋊Q8×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#106×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59#16864401—44no—
60C2×Dic538—21no—
61C2×D1076C2311yescharacteristic
62C2×D1076C2311yescharacteristic
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269—21no—
67C22×Dic269—21no—
68C22×Dic269D411yescharacteristic
69C22×Dic269D411yescharacteristic
70C2×C4×D269—21no—
71C2×C4×D269—21no—
72#16864389—41no—
73D2152—41no—
74#16864387—41no—
75C2×#14524130—41no—
76C2×#14524130C2311yescharacteristic
77C2×#14524130C2311yescharacteristic
78C4⋊Dic269—21no—
79C4⋊Dic269—21no—
80C4⋊Dic269C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83#16864381—21no—
84C22⋊Dic269—21no—
85#16864379—21no—
86#16864379—21no—
87#16864379—21no—
88#16864379—21no—
89#16864379—21no—
90#16864379—21no—
91C4×Dic269—21no—
92C4×Dic269D411yescharacteristic
93C4×Dic269D411yescharacteristic
94C4×Dic269C2311yescharacteristic
95C4×Dic269C2311yescharacteristic
96C2×Q8×C269C2311yescharacteristic
97C2×D4×C269—21no—
98C2×D4×C269C2311yescharacteristic
99C22×C4×C269C2311yescharacteristic
100SD16×C269—41no—
101C2×C8×C269—41no—
102C2×C8×C269C2311yescharacteristic
103C2×C8×C269C2311yescharacteristic
104C4⋊C4×C269—21no—
105C4⋊C4×C269C2311yescharacteristic
106C4⋊C4×C269C2311yescharacteristic
107C22⋊C4×C269—21no—
108C22⋊C4×C269—21no—
109C22×D269—21no—
110C22×D269—21no—
111Dic538—42no—
112D1076—21no—
113D1076—21no—
114D1076—21no—
115D1076—21no—
116C22⋊D269—42no—
117C22⋊D269—42no—
118C22⋊D269—42no—
119C22⋊D269—42no—
120C2×Dic269—22no—
121C2×Dic269—22no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—42no—
125C2×Dic269C2×D411yescharacteristic
126C2×Dic269C2×D411yescharacteristic
127C2×Dic269C2×D411yescharacteristic
128C2×Dic269C2×D411yescharacteristic
129C4×D269—42no—
130C4×D269—42no—
131#14524130—21no—
132#14524130—21no—
133C23×C269—21no—
134C23×C269C2×D411yescharacteristic
135Q8×C269—21no—
136Q8×C269—21no—
137D4×C269—21no—
138D4×C269—21no—
139D4×C269—42no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2411yescharacteristic
147C8×C269—21no—
148C8×C269—21no—
149C22×C269—21no—
150C22×C269—21no—
151C22×C269—41no—
152C22×C269—41no—
153C22×C269—41no—
154C22×C269AGL(1,ℤ/8)12yes—
155C22×C269C22×D411yescharacteristic
156C2×D269—41no—
157C2×D269—41no—
158C2×D269—41no—
159C2×D269—41no—
160C2×D269—41no—
161C2×D269—41no—
162C4×C269—21no—
163C4×C269—41no—
164C4×C269—41no—
165C4×C269—41no—
166C4×C269H(5,𝔽2)11yescharacteristic
167C4×C2692−1+411yescharacteristic
168Dic269—21no—
169Dic269—21no—
170Dic269—21no—
171Dic269—41no—
172Dic269—41no—
173Dic269#13512yes—
174C2×C269—22no—
175C2×C269—81no—
176C2×C269#53711yescharacteristic
177C2×C269C2×#13511yescharacteristic
178C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
179D269—81no—
180D269—81no—
181C269#324711yescharacteristic, complemented, Sylow
182#3247—2691nomaximal, Sylow
183#532—2691no—
184#529—2691no—
185#478—2691no—
186#478—2691no—
187#477—2691no—
188#476—2691no—
189#466—2691no—
190#466—2691no—
191#465—2691no—
192#465D26911yescharacteristic, complemented
193#463—2691no—
194#463—2691no—
195#460—2691no—
196#459—2691no—
197C2×C4⋊C8—2691no—
198C2×#41—5381no—
199C2×SD16—2691no—
200C2×SD16C2×D26911yescharacteristic
201C2×D8—2691no—
202C2×D8—2691no—
203C22×C8—2691no—
204C22×C8C2×D26911yescharacteristic
205C4⋊Q8—2691no—
206#127—2691no—
207#126—5381no—
208#125—2691no—
209#124—2691no—
210#124—5381no—
211#123—5381no—
212C22⋊Q8C2×D26911yescharacteristic
213#121—2691no—
214#121—2691no—
215#121—5381no—
216#121C2×D26911yescharacteristic
217C4×Q8—2691no—
218C4×D4—2691no—
219C4×D4—2691no—
220C4×D4—2691no—
221C4×D4—5381no—
222C4×D4—5381no—
223#117—5381no—
224C2×C42—2691no—
225#107—2691no—
226#107—2691no—
227#106—2691no—
228#106C2×D26911yescharacteristic
229C4⋊C8—2691no—
230C4⋊C8—2691no—
231C4⋊C8—2691no—
232C4⋊C8—2691no—
233#103—2691no—
234#103C2×D26911yescharacteristic
235#102—2691no—
236#102—2691no—
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102C2×D26911yescharacteristic
241#41—10764no—
242C2×Q8—5381no—
243C2×Q8C22×D26911yescharacteristic
244C2×D4—21no—
245C2×D4—2691no—
246C2×D4—2691no—
247C2×D4—5381no—
248C2×D4—5381no—
249C2×D4—5381no—
250C2×D4—5381no—
251C2×D4C22×D26911yescharacteristic
252C22×C4—2691no—
253C22×C4—2691no—
254C22×C4—5381no—
255C22×C4—5381no—
256C22×C4—5381no—
257C22×C4C22×D26911yescharacteristic
258SD16—41no—
259SD16—10761no—
260D8—10761no—
261D8—10761no—
262C2×C8—41no—
263C2×C8—2691no—
264C2×C8—2691no—
265C2×C8—10761no—
266C2×C8C22×D26911yescharacteristic
267C2×C8C22×D26911yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—2691no—
270C4⋊C4—2691no—
271C4⋊C4—5381no—
272C4⋊C4—5381no—
273C4⋊C4—5381no—
274C4⋊C4—5381no—
275C4⋊C4C22×D26911yescharacteristic
276C4⋊C4C22×D26911yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5381no—
280C22⋊C4—5381no—
281C22⋊C4—5381no—
282C22⋊C4—5381no—
283C22⋊C4—5381no—
284C22⋊C4—5381no—
285C22⋊C4—5381no—
286C42—2691no—
287C42—2691no—
288C42—2691no—
289C42—2691no—
290C42—5381no—
291C23—21no—
292C23—5381no—
293C23—5381no—
294C23D4×D26911yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—10762no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—5381no—
302D4—5381no—
303D4—5381no—
304D4—5381no—
305D4—10762no—
306D4—10762no—
307D4—10762no—
308D4—10762no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—2691no—
315C2×C4—2691no—
316C2×C4—2691no—
317C2×C4—2691no—
318C2×C4—5382no—
319C2×C4—5382no—
320C2×C4—5381no—
321C2×C4—5381no—
322C2×C4—10762no—
323C2×C4—10762no—
324C2×C4—10762no—
325C2×C4D4×D26911yescharacteristic
326C2×C4C23×D26911yescharacteristic
327C8—21no—
328C8—21no—
329C8—5381no—
330C8—5381no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—10761no—
337C22—10761no—
338C22—10761no—
339C22—10761no—
340C22—10761no—
341C22—10761no—
342C22#2257548012yes—
343C22C2×D4×D26911yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—2692no—
349C4—5381no—
350C4—5381no—
351C4—5381no—
352C4—10761no—
353C4—10761no—
354C4#2257549811yescharacteristic
355C4#2257550011yescharacteristic
356C2—22no—
357C2—81no—
358C2—21521no—
359C2—21521no—
360C2#3177118211yescharacteristic
361C2#135×D26911yescharacteristic
362C2C2×#2257548011yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#465×C269C2
socleC22×C269C22×D4
hypercenter#465D269
hypocenterC269#3247
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31771150C2semidirect
2. #31771141C2semidirect
3. #31770904C2semidirect
4. #31770903C2semidirect
5. #31770898C2semidirect
6. #31770894C2semidirect
7. #31770841C2semidirect
8. #31770839C2semidirect
9. #31770836C2semidirect
10. #31770823C2semidirect
11. #31770822C2semidirect
12. #31770803C2semidirect
13. #31770799C2semidirect
14. C2×#22575382C2semidirect
15. #465×C269C2semidirect
181. C269#3247semidirect
192. #465D269semidirect

Maximal subgroups

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

subgroupindex
#317711502
#317711412
#317709042
#317709032
#317708982
#317708942
#317708412
#317708392
#317708362
#317708232
#317708222
#317708032
#317707992
C2×#225753822
#465×C2692
#3247269