Finite Group Databasebeta

← #46822754

Subgroups — #46822754

Order 34432.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771313C211yesmaximal, characteristic
2Q8×C22⋊D269C211yesmaximal, characteristic
3#31770954C211yesmaximal, characteristic, complemented
4#31770941C211yesmaximal, characteristic, complemented
5#31770939C211yesmaximal, characteristic, complemented
6#31770898C211yesmaximal, characteristic, complemented
7#31770844C211yesmaximal, characteristic, complemented
8#31770843C211yesmaximal, characteristic, complemented
9#31770813C211yesmaximal, characteristic, complemented
10#31770759C211yesmaximal, characteristic, complemented
11Q16×Dic269C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic
13C2×#22575377C211yesmaximal, characteristic, complemented
14#103×D269C211yesmaximal, characteristic, complemented
15#451×C269C211yesmaximal, characteristic, complemented
16C2×Q8×D269C2211yescharacteristic
17C2×#16864392C2211yescharacteristic
18C22×#14524130C2211yescharacteristic
19C2×C8×D269C2211yescharacteristic
20#22575452C2211yescharacteristic
21#22575451—21no—
22#22575451C2211yescharacteristic
23#22575443—21no—
24#22575433—21no—
25#22575432—21no—
26#22575432C2211yescharacteristic
27#22575430—21no—
28#22575429—21no—
29#22575429C2211yescharacteristic
30#22575429C2211yescharacteristic
31Q8×Dic269C2211yescharacteristic
32Q8×Dic269C2211yescharacteristic
33C4×C22⋊D269—21no—
34C4×C22⋊D269C2211yescharacteristic
35#22575413—21no—
36C2×C4⋊Dic269C2211yescharacteristic
37C4⋊C4×D269C2211yescharacteristic
38#22575387C2211yescharacteristic
39#22575387C2211yescharacteristic
40#22575386C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575377C2211yescharacteristic
48#22575377C2211yescharacteristic
49#22575376C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C22×Q8×C269C2211yescharacteristic
53C2×Q16×C269C2211yescharacteristic
54C2×Q16×C269C2211yescharacteristic
55C22⋊Q8×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59C2×Dic538—21no—
60C2×Dic538—21no—
61C2×Dic538C2311yescharacteristic
62Q8×D269—22no—
63Q8×D269—41no—
64C2×C22⋊D269D411yescharacteristic
65C22×Dic269—21no—
66C2×C4×D269—21no—
67C2×C4×D269—21no—
68C2×C4×D269C2311yescharacteristic
69#16864392—41no—
70C2×#14524130—41no—
71C2×#14524130C2311yescharacteristic
72C2×#14524130C2311yescharacteristic
73C8×D269—41no—
74C4⋊Dic269—21no—
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77C4⋊Dic269C2311yescharacteristic
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381—21no—
86#16864381—21no—
87#16864381D411yescharacteristic
88C22⋊Dic269—21no—
89C22⋊Dic269D411yescharacteristic
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×Q8×C269C2311yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269C2311yescharacteristic
104Q16×C269—22no—
105Q16×C269—41no—
106C2×C8×C269C2311yescharacteristic
107C2×C8×C269C2311yescharacteristic
108C4⋊C4×C269—21no—
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22×D269C2×D411yescharacteristic
112Dic538—21no—
113Dic538—21no—
114Dic538—42no—
115Dic538—81no—
116C22⋊D269Q1614yescomplemented
117C2×Dic269—21no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—81no—
122C2×Dic269C2×D411yescharacteristic
123C2×Dic269C2×D411yescharacteristic
124C2×Dic269C2×D411yescharacteristic
125C4×D269—21no—
126C4×D269—21no—
127C4×D269—42no—
128C4×D269—81no—
129#14524130—21no—
130#14524130—21no—
131C23×C269C2×D411yescharacteristic
132Q8×C269—21no—
133Q8×C269—21no—
134Q8×C269—21no—
135Q8×C269—21no—
136Q8×C269—21no—
137Q8×C269—42no—
138Q8×C269#4112yes—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—42no—
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2411yescharacteristic
148C8×C269—21no—
149C8×C269—21no—
150C22×C269—21no—
151C22×C269—21no—
152C22×C269C2×Q1612yes—
153C22×C269C22×D411yescharacteristic
154C2×D269—21no—
155C2×D269—21no—
156C2×D269C2×Q1612yes—
157C4×C269—21no—
158C4×C269—22no—
159C4×C269—41no—
160C4×C269—41no—
161C4×C269—41no—
162C4×C269C2×#4111yescharacteristic
163C4×C269H(5,𝔽2)11yescharacteristic
164Dic269—21no—
165Dic269—21no—
166Dic269—41no—
167Dic269—41no—
168Dic269—41no—
169Dic269—41no—
170Dic269C2×Q1612yes—
171C2×C269—22no—
172C2×C269#54611yescharacteristic
173C2×C269C22×Q1611yescharacteristic
174C2×C269#57711yescharacteristic
175D269—22no—
176C269#329911yescharacteristic, complemented, Sylow
177#3299—2691nomaximal, Sylow
178#554—2691no—
179D4×Q8—2691no—
180#487—2691no—
181#484—2691no—
182#484—2691no—
183#477—2691no—
184#467—2691no—
185#467—2691no—
186#462—2691no—
187#451—2691no—
188#451D26911yescharacteristic, complemented
189C4×Q16—2691no—
190C8×D4—2691no—
191C2×#103—2691no—
192C2×#103—2691no—
193C22×Q8—2691no—
194C22×Q8C2×D26911yescharacteristic
195C2×Q16—2691no—
196C2×Q16C2×D26911yescharacteristic
197C2×Q16C2×D26911yescharacteristic
198C22×C8—2691no—
199C22×C8—2691no—
200C4⋊Q8—2691no—
201C4⋊Q8—2691no—
202C4⋊Q8—5381no—
203#125—5381no—
204C22⋊Q8—2691no—
205C22⋊Q8—2691no—
206C22⋊Q8—2691no—
207C22⋊Q8—5381no—
208C22⋊Q8—5381no—
209C22⋊Q8—5381no—
210C22⋊Q8—5381no—
211C22⋊Q8C2×D26911yescharacteristic
212C4×Q8—2691no—
213C4×Q8—2691no—
214C4×D4—2691no—
215C4×D4—5381no—
216C4×D4—5381no—
217C2×C4⋊C4—2691no—
218C2×C4⋊C4—2691no—
219#107—2691no—
220#107—2691no—
221#107—2691no—
222C4⋊C8—2691no—
223#103—2691no—
224#103—2691no—
225#103—2691no—
226#103—2691no—
227#103—2691no—
228#103—2691no—
229#103—2691no—
230#103—2691no—
231#103C2×D26911yescharacteristic
232#103C2×D26911yescharacteristic
233C22⋊C8—2691no—
234C22⋊C8C2×D26911yescharacteristic
235C4×C8—2691no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—2691no—
240C2×Q8—5381no—
241C2×Q8—5381no—
242C2×Q8—5382no—
243C2×Q8—10761no—
244C2×Q8C22×D26911yescharacteristic
245C2×Q8C22×D26911yescharacteristic
246C2×Q8C22×D26911yescharacteristic
247C2×D4—2691no—
248C22×C4—21no—
249C22×C4—2691no—
250C22×C4—5381no—
251C22×C4—5381no—
252C22×C4—5381no—
253C22×C4C22×D26911yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—10761no—
257C2×C8—2691no—
258C2×C8—2691no—
259C2×C8—10761no—
260C2×C8—10761no—
261C2×C8C22×D26911yescharacteristic
262C2×C8C22×D26911yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—2691no—
265C4⋊C4—2691no—
266C4⋊C4—2691no—
267C4⋊C4—2691no—
268C4⋊C4—5381no—
269C4⋊C4—5381no—
270C4⋊C4—5381no—
271C4⋊C4—5381no—
272C4⋊C4—5381no—
273C4⋊C4—5381no—
274C4⋊C4—5381no—
275C4⋊C4—5381no—
276C4⋊C4—5381no—
277C4⋊C4—5381no—
278C4⋊C4C22×D26911yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—2691no—
281C22⋊C4—2691no—
282C22⋊C4—5381no—
283C22⋊C4—5381no—
284C22⋊C4—5381no—
285C42—2691no—
286C42—5381no—
287C42—5381no—
288C23—2691no—
289C23D4×D26911yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—5381no—
297Q8—5381no—
298Q8—10762no—
299Q8—21521no—
300Q8#1686440112yes—
301D4—2694no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2691no—
310C2×C4—2691no—
311C2×C4—2691no—
312C2×C4—5381no—
313C2×C4—5381no—
314C2×C4—5381no—
315C2×C4—5381no—
316C2×C4—5381no—
317C2×C4—5381no—
318C2×C4—10762no—
319C2×C4—21521no—
320C2×C4—21521no—
321C2×C4D4×D26911yescharacteristic
322C2×C4C23×D26911yescharacteristic
323C8—21no—
324C8—21no—
325C8—5381no—
326C8—5381no—
327C22—21no—
328C22—21no—
329C22—2692no—
330C22—5381no—
331C22—5381no—
332C22Q16×D26912yes—
333C22C2×D4×D26911yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—2692no—
340C4—5381no—
341C4—5381no—
342C4—10761no—
343C4—10761no—
344C4—10761no—
345C4—10761no—
346C4C2×#1686440111yescharacteristic
347C4#2257549811yescharacteristic
348C2—22no—
349C2—5382no—
350C2#3177124211yescharacteristic
351C2C2×Q16×D26911yescharacteristic
352C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#451×C269C2
socleC22×C269C22×D4
prime radicalC2×C22⋊D269D4
hypercenter#451D269
hypocenterC269#3299
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #31770954C2semidirect
4. #31770941C2semidirect
5. #31770939C2semidirect
6. #31770898C2semidirect
7. #31770844C2semidirect
8. #31770843C2semidirect
9. #31770813C2semidirect
10. #31770759C2semidirect
11. Q16×Dic269C2semidirect
13. C2×#22575377C2semidirect
14. #103×D269C2semidirect
15. #451×C269C2semidirect
116. C22⋊D269Q16semidirect
176. C269#3299semidirect
188. #451D269semidirect

Maximal subgroups

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

subgroupindex
#317713132
Q8×C22⋊D2692
#317709542
#317709412
#317709392
#317708982
#317708442
#317708432
#317708132
#317707592
Q16×Dic2692
#317706682
C2×#225753772
#103×D2692
#451×C2692
#3299269