Finite Group Databasebeta

← #46822267

Subgroups — #46822267

Order 34432.

386 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#31771157C211yesmaximal, characteristic, complemented
2#31771150C211yesmaximal, characteristic
3#31770899C211yesmaximal, characteristic, complemented
4#31770898C211yesmaximal, characteristic, complemented
5#31770884C211yesmaximal, characteristic, complemented
6#31770883C211yesmaximal, characteristic, complemented
7#31770853C211yesmaximal, characteristic, complemented
8#31770851C211yesmaximal, characteristic, complemented
9#31770849C211yesmaximal, characteristic, complemented
10#31770809C211yesmaximal, characteristic, complemented
11#31770808C211yesmaximal, characteristic, complemented
12#31770804C211yesmaximal, characteristic, complemented
13#31770803C211yesmaximal, characteristic, complemented
14#31770628C211yesmaximal, characteristic, complemented
15#468×C269C211yesmaximal, characteristic, complemented
16C2×#16864401—21no—
17C2×#16864391C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19C2×#16864389C2211yescharacteristic
20C2×#16864386C2211yescharacteristic
21#22575452C2211yescharacteristic
22#22575452C2211yescharacteristic
23#22575451C2211yescharacteristic
24#22575449C2211yescharacteristic
25#22575443—21no—
26#22575439—21no—
27#22575432C2211yescharacteristic
28#22575430—21no—
29#22575427C2211yescharacteristic
30#22575425—21no—
31Q8×Dic269C2211yescharacteristic
32#22575418—21no—
33#22575418C2211yescharacteristic
34D4×Dic269C2211yescharacteristic
35#22575415C2211yescharacteristic
36#22575413—21no—
37#22575409—21no—
38C2×C4×Dic269C2211yescharacteristic
39#22575387C2211yescharacteristic
40#22575386C2211yescharacteristic
41#22575386C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575382C2211yescharacteristic
44#22575382C2211yescharacteristic
45#22575382C2211yescharacteristic
46#22575378C2211yescharacteristic
47#22575377C2211yescharacteristic
48#22575376C2211yescharacteristic
49#22575375C2211yescharacteristic
50#22575374C2211yescharacteristic
51#22575373C2211yescharacteristic
52C2×SD16×C269C2211yescharacteristic
53C2×M16×C269C2211yescharacteristic
54C22⋊Q8×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#107×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59#16864401—44no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×Dic538C2311yescharacteristic
63C2×D1076C2311yescharacteristic
64C2×C22⋊D269—21no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269—21no—
67C22×Dic269—21no—
68C22×Dic269D411yescharacteristic
69C22×Dic269D411yescharacteristic
70C2×C4×D269—21no—
71#16864391—41no—
72#16864390—41no—
73#16864389—41no—
74#16864386—41no—
75C2×#14524130C2311yescharacteristic
76C2×#14524130C2311yescharacteristic
77C4⋊Dic269—21no—
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381—21no—
86C22⋊Dic269—21no—
87C22⋊Dic269—21no—
88#16864379—21no—
89#16864379—21no—
90C4×Dic269—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—
101M16×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—
110Dic538—21no—
111Dic538—21no—
112Dic538—42no—
113Dic538—42no—
114D1076—21no—
115D1076—21no—
116C22⋊D269—42no—
117C22⋊D269—42no—
118C22⋊D269—42no—
119C2×Dic269—22no—
120C2×Dic269—22no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—42no—
125C2×Dic269C2×D411yescharacteristic
126C2×Dic269C2×D411yescharacteristic
127C2×Dic269C2×D411yescharacteristic
128C2×Dic269C2×D411yescharacteristic
129C4×D269—42no—
130#14524130—21no—
131#14524130—21no—
132C23×C269—21no—
133C23×C269C2×D411yescharacteristic
134Q8×C269—21no—
135Q8×C269—21no—
136D4×C269—21no—
137D4×C269—21no—
138D4×C269—42no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2411yescharacteristic
146C8×C269—21no—
147C8×C269—21no—
148C22×C269—21no—
149C22×C269—21no—
150C22×C269—21no—
151C22×C269—41no—
152C22×C269—41no—
153C22×C269—41no—
154C22×C269C22×D411yescharacteristic
155C2×D269—41no—
156C2×D269—41no—
157C2×D269—41no—
158C4×C269—21no—
159C4×C269—41no—
160C4×C269—41no—
161C4×C269—41no—
162C4×C269H(5,𝔽2)11yescharacteristic
163C4×C2692−1+411yescharacteristic
164Dic269—21no—
165Dic269—21no—
166Dic269—41no—
167Dic269—41no—
168Dic269—41no—
169Dic269AGL(1,ℤ/8)12yes—
170Dic269#13712yes—
171C2×C269—41no—
172C2×C269—81no—
173C2×C269#53711yescharacteristic
174C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C269C2×#13711yescharacteristic
176D269—81no—
177C269#325411yescharacteristic, complemented, Sylow
178#3254—2691nomaximal, Sylow
179#533—2691no—
180#532—2691no—
181#477—2691no—
182#477—2691no—
183#474—2691no—
184#474—2691no—
185#468—2691no—
186#468—2691no—
187#468—2691no—
188#468D26911yescharacteristic, complemented
189#461—2691no—
190#461—2691no—
191#460—2691no—
192#460—2691no—
193#423—2691no—
194C2×#41—5381no—
195C2×SD16—2691no—
196C2×SD16—2691no—
197C2×SD16—2691no—
198C2×SD16C2×D26911yescharacteristic
199C2×M16—2691no—
200C2×M16C2×D26911yescharacteristic
201C4⋊Q8—2691no—
202C4⋊Q8—2691no—
203C4⋊Q8—2691no—
204#127—2691no—
205#125—5381no—
206#124—5381no—
207C22⋊Q8—2691no—
208C22⋊Q8—5381no—
209C22⋊Q8C2×D26911yescharacteristic
210#121—2691no—
211#121—5381no—
212#121C2×D26911yescharacteristic
213C4×Q8—2691no—
214C4×Q8—2691no—
215C4×Q8—5381no—
216C4×D4—2691no—
217C4×D4—2691no—
218C4×D4—5381no—
219#117—5381no—
220C2×C42—2691no—
221#107—2691no—
222#107—2691no—
223#107—2691no—
224#107C2×D26911yescharacteristic
225C4⋊C8—2691no—
226C4⋊C8—2691no—
227C4⋊C8—2691no—
228C4⋊C8—2691no—
229#103—2691no—
230#103—2691no—
231#103—2691no—
232#103C2×D26911yescharacteristic
233#102—2691no—
234#102—2691no—
235#102—2691no—
236#102C2×D26911yescharacteristic
237#41—10764no—
238C2×Q8—2691no—
239C2×Q8—5381no—
240C2×Q8—5381no—
241C2×Q8C22×D26911yescharacteristic
242C2×D4—21no—
243C2×D4—2691no—
244C2×D4—5381no—
245C2×D4—5381no—
246C2×D4—5381no—
247C2×D4C22×D26911yescharacteristic
248C22×C4—2691no—
249C22×C4—2691no—
250C22×C4—5381no—
251C22×C4—5381no—
252C22×C4C22×D26911yescharacteristic
253SD16—41no—
254SD16—10761no—
255SD16—10761no—
256SD16—10761no—
257M16—41no—
258M16—10761no—
259C2×C8—2691no—
260C2×C8—2691no—
261C2×C8C22×D26911yescharacteristic
262C2×C8C22×D26911yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—2691no—
265C4⋊C4—2691no—
266C4⋊C4—5381no—
267C4⋊C4—5381no—
268C4⋊C4—5381no—
269C4⋊C4—5381no—
270C4⋊C4—5381no—
271C4⋊C4—5381no—
272C4⋊C4—5381no—
273C4⋊C4C22×D26911yescharacteristic
274C4⋊C4C22×D26911yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—5381no—
278C22⋊C4—5381no—
279C22⋊C4—5381no—
280C22⋊C4—5381no—
281C42—2691no—
282C42—2691no—
283C42—2691no—
284C42—2691no—
285C42—5381no—
286C42—5381no—
287C23—21no—
288C23—5381no—
289C23D4×D26911yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5381no—
293Q8—5381no—
294Q8—10762no—
295Q8—10762no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—5381no—
300D4—5381no—
301D4—10762no—
302D4—10762no—
303D4—10762no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2691no—
310C2×C4—2691no—
311C2×C4—2691no—
312C2×C4—2691no—
313C2×C4—5382no—
314C2×C4—5382no—
315C2×C4—5381no—
316C2×C4—5381no—
317C2×C4—5381no—
318C2×C4—10762no—
319C2×C4—10762no—
320C2×C4D4×D26911yescharacteristic
321C2×C4C23×D26911yescharacteristic
322C8—21no—
323C8—21no—
324C8—5381no—
325C8—5381no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—10761no—
333C22—10761no—
334C22—10761no—
335C22C2×D4×D26911yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—2692no—
341C4—2692no—
342C4—5381no—
343C4—5381no—
344C4—10761no—
345C4—10761no—
346C4—10761no—
347C4#2257549811yescharacteristic
348C4#2257550011yescharacteristic
349C2—41no—
350C2—81no—
351C2—21521no—
352C2#3177118211yescharacteristic
353C2AGL(1,ℤ/8)×D26911yescharacteristic
354C2#137×D26911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#468×C269C2
socleC22×C269C22×D4
prime radical#31771150C2
hypercenter#468D269
hypocenterC269#3254
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #31771157C2semidirect
3. #31770899C2semidirect
4. #31770898C2semidirect
5. #31770884C2semidirect
6. #31770883C2semidirect
7. #31770853C2semidirect
8. #31770851C2semidirect
9. #31770849C2semidirect
10. #31770809C2semidirect
11. #31770808C2semidirect
12. #31770804C2semidirect
13. #31770803C2semidirect
14. #31770628C2semidirect
15. #468×C269C2semidirect
177. C269#3254semidirect
188. #468D269semidirect

Maximal subgroups

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

subgroupindex
#317711572
#317711502
#317708992
#317708982
#317708842
#317708832
#317708532
#317708512
#317708492
#317708092
#317708082
#317708042
#317708032
#317706282
#468×C2692
#3254269