Finite Group Databasebeta

← #46821174

Subgroups — #46821174

Order 34432.

382 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#31771209C211yesmaximal, characteristic
2#31771081C211yesmaximal, characteristic, complemented
3#31770928C211yesmaximal, characteristic, complemented
4#31770927C211yesmaximal, characteristic, complemented
5#31770913C211yesmaximal, characteristic, complemented
6#31770912C211yesmaximal, characteristic, complemented
7#31770802C211yesmaximal, characteristic, complemented
8#31770742C211yesmaximal, characteristic, complemented
9#31770742C211yesmaximal, characteristic, complemented
10#31770706C211yesmaximal, characteristic, complemented
11#31770702C211yesmaximal, characteristic, complemented
12#31770696C211yesmaximal, characteristic, complemented
13SD16×Dic269C211yesmaximal, characteristic, complemented
14#31770665C211yesmaximal, characteristic, complemented
15#460×C269C211yesmaximal, characteristic, complemented
16C2×Q8×D269C2211yescharacteristic
17C2×#16864390C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19#22575445C2211yescharacteristic
20#22575434—21no—
21#22575434—21no—
22#22575425—21no—
23#22575425—21no—
24#22575424C2211yescharacteristic
25#22575424C2211yescharacteristic
26Q8×Dic269C2211yescharacteristic
27C4×Dic538—21no—
28C4×Dic538C2211yescharacteristic
29#22575418—21no—
30#22575418C2211yescharacteristic
31D4×Dic269C2211yescharacteristic
32D4×Dic269C2211yescharacteristic
33#22575411—21no—
34#22575411C2211yescharacteristic
35#22575410—21no—
36#22575410C2211yescharacteristic
37#22575387C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575384C2211yescharacteristic
41C4⋊#14524130C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575374C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575365C2211yescharacteristic
49#22575365C2211yescharacteristic
50#22575363C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×SD16×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54#127×C269C2211yescharacteristic
55C4×Q8×C269C2211yescharacteristic
56C4⋊C8×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59C2×Dic538—21no—
60C2×Dic538C2311yescharacteristic
61Q8×D269—22no—
62Q8×D269—22no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65C22×Dic269—21no—
66C22×Dic269—21no—
67C2×C4×D269—21no—
68C2×C4×D269C2311yescharacteristic
69#16864390—22no—
70#16864390—22no—
71C2×#14524130C2311yescharacteristic
72C2×#14524130C2311yescharacteristic
73C4⋊Dic269—21no—
74C4⋊Dic269C2311yescharacteristic
75C4⋊Dic269C2311yescharacteristic
76C4⋊Dic269C2311yescharacteristic
77#16864381—21no—
78#16864381—21no—
79#16864381—21no—
80#16864381—21no—
81#16864381D411yescharacteristic
82#16864381D411yescharacteristic
83C22⋊Dic269—21no—
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269—21no—
87#16864379—21no—
88#16864379D411yescharacteristic
89#16864379D411yescharacteristic
90C4×Dic269—21no—
91C4×Dic269—21no—
92C4×Dic269—21no—
93C4×Dic269—21no—
94C4×Dic269C2311yescharacteristic
95C2×Q8×C269C2311yescharacteristic
96C2×D4×C269—21no—
97C2×D4×C269—21no—
98C2×D4×C269C2311yescharacteristic
99C2×D4×C269C2311yescharacteristic
100SD16×C269—22no—
101SD16×C269—22no—
102C2×C8×C269C2311yescharacteristic
103C2×C8×C269C2311yescharacteristic
104C4⋊C4×C269—21no—
105C4⋊C4×C269C2311yescharacteristic
106C42×C269—21no—
107C42×C269C2311yescharacteristic
108C22×D269C2×D411yescharacteristic
109Dic538—24no—
110Dic538—21no—
111Dic538#4112yes—
112C22⋊D269—81no—
113C22⋊D269—81no—
114C2×Dic269—21no—
115C2×Dic269—21no—
116C2×Dic269—21no—
117C2×Dic269—21no—
118C2×Dic269—81no—
119C2×Dic269—81no—
120C2×Dic269C2×D411yescharacteristic
121C2×Dic269C2×D411yescharacteristic
122C2×Dic269C2×D411yescharacteristic
123C4×D269—24no—
124C4×D269—21no—
125C4×D269—21no—
126#14524130—21no—
127#14524130—21no—
128C23×C269—21no—
129C23×C269—21no—
130Q8×C269—21no—
131Q8×C269#4112yes—
132D4×C269—21no—
133D4×C269—21no—
134D4×C269—21no—
135D4×C269—21no—
136D4×C269—81no—
137D4×C269—81no—
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×C269—41no—
149C22×C269—41no—
150C22×C269—41no—
151C22×C269C22×D411yescharacteristic
152C2×D269—21no—
153C2×D269—21no—
154C2×D269—21no—
155C4×C269—22no—
156C4×C269—21no—
157C4×C269—21no—
158C4×C269—41no—
159C4×C269C2×#4111yescharacteristic
160C4×C269C2×#4111yescharacteristic
161Dic269—22no—
162Dic269—21no—
163Dic269—21no—
164Dic269—21no—
165Dic269—41no—
166Dic269—41no—
167Dic269—41no—
168C2×C269—81no—
169C2×C269—81no—
170C2×C269#52511yescharacteristic
171C2×C269#57711yescharacteristic
172C2×C269#57711yescharacteristic
173D269—41no—
174C269#312611yescharacteristic, complemented, Sylow
175#3126—2691nomaximal, Sylow
176#540—2691no—
177#519—2691no—
178#482—2691no—
179#482—2691no—
180#479—2691no—
181#479—2691no—
182#460—2691no—
183#460D26911yescharacteristic, complemented
184#448—2691no—
185#448—2691no—
186#440—2691no—
187#440—2691no—
188C4×SD16—2691no—
189C4×SD16—2691no—
190#433—2691no—
191C22×Q8—2691no—
192C2×SD16—2691no—
193C2×SD16—2691no—
194C2×SD16C2×D26911yescharacteristic
195C2×SD16C2×D26911yescharacteristic
196#127C2×D26911yescharacteristic
197#125—2691no—
198#123—5381no—
199#123—5381no—
200#121—2691no—
201#121—2691no—
202#121—5381no—
203#121—5381no—
204C4×Q8—2691no—
205C4×Q8—2691no—
206C4×Q8—2691no—
207C4×Q8—5381no—
208C4×Q8—5381no—
209C4×Q8C2×D26911yescharacteristic
210C4×D4—2691no—
211C4×D4—2691no—
212#117—2691no—
213#117—2691no—
214#117—5381no—
215#117—5381no—
216#107—2691no—
217#107—2691no—
218#106—2691no—
219#106—2691no—
220C4⋊C8—2691no—
221C4⋊C8C2×D26911yescharacteristic
222#103—2691no—
223#103—2691no—
224#103—2691no—
225#103—2691no—
226#102—2691no—
227#102—2691no—
228#102C2×D26911yescharacteristic
229#102C2×D26911yescharacteristic
230C22⋊C8—2691no—
231C22⋊C8—2691no—
232#97—2691no—
233C4×C8—2691no—
234C2×Q8—2691no—
235C2×Q8—5381no—
236C2×Q8—5382no—
237C2×Q8—5382no—
238C2×Q8C22×D26911yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—5381no—
242C2×D4—5381no—
243C2×D4C22×D26911yescharacteristic
244C2×D4C22×D26911yescharacteristic
245C22×C4—2691no—
246C22×C4—5381no—
247C22×C4—5381no—
248C22×C4—5381no—
249SD16—22no—
250SD16—22no—
251SD16—5382no—
252SD16—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—2691no—
270C22⋊C4—2691no—
271C22⋊C4—5381no—
272C22⋊C4—5381no—
273C22⋊C4—5381no—
274C22⋊C4—5381no—
275C22⋊C4—5381no—
276C42—21no—
277C42—2691no—
278C42—5381no—
279C42—5381no—
280C42—5381no—
281C42—5381no—
282C42C22×D26911yescharacteristic
283C23—21no—
284C23—21no—
285C23—2691no—
286Q8—21no—
287Q8—2692no—
288Q8—5384no—
289Q8—5381no—
290Q8#1686440112yes—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4—81no—
297D4—21521no—
298D4—21521no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—2691no—
302C2×C4—2691no—
303C2×C4—2691no—
304C2×C4—5384no—
305C2×C4—5381no—
306C2×C4—5381no—
307C2×C4—5381no—
308C2×C4—5381no—
309C2×C4—5381no—
310C2×C4—5381no—
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—41no—
324C22—41no—
325C22—41no—
326C22—5381no—
327C22—5381no—
328C22—5381no—
329C22C2×D4×D26911yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—5381no—
335C4—5381no—
336C4—5381no—
337C4—5382no—
338C4—10761no—
339C4—10761no—
340C4—10761no—
341C4#41×D26911yescharacteristic
342C4C2×#1686440111yescharacteristic
343C2—81no—
344C2—81no—
345C2—10761no—
346C2#3177111511yescharacteristic
347C2#3177141111yescharacteristic
348C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#460×C269C2
socleC22×C269C22×D4
prime radical#31771209C2
hypercenter#460D269
hypocenterC269#3126
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #31771081C2semidirect
3. #31770928C2semidirect
4. #31770927C2semidirect
5. #31770913C2semidirect
6. #31770912C2semidirect
7. #31770802C2semidirect
8. #31770742C2semidirect
9. #31770742C2semidirect
10. #31770706C2semidirect
11. #31770702C2semidirect
12. #31770696C2semidirect
13. SD16×Dic269C2semidirect
14. #31770665C2semidirect
15. #460×C269C2semidirect
174. C269#3126semidirect
183. #460D269semidirect

Maximal subgroups

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

subgroupindex
#317712092
#317710812
#317709282
#317709272
#317709132
#317709122
#317708022
#317707422
#317707062
#317707022
#317706962
SD16×Dic2692
#317706652
#460×C2692
#3126269