Finite Group Databasebeta

← #46821113

Subgroups — #46821113

Order 34432.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771224C211yesmaximal, characteristic
2#31771081C211yesmaximal, characteristic, complemented
3#31770945C211yesmaximal, characteristic, complemented
4#31770927C211yesmaximal, characteristic
5#31770901C211yesmaximal, characteristic, complemented
6#31770890C211yesmaximal, characteristic, complemented
7#31770824C211yesmaximal, characteristic, complemented
8#31770759C211yesmaximal, characteristic, complemented
9#31770742C211yesmaximal, characteristic
10#31770709C211yesmaximal, characteristic, complemented
11#31770706C211yesmaximal, characteristic, complemented
12Q16×Dic269C211yesmaximal, characteristic, complemented
13#31770696C211yesmaximal, characteristic, complemented
14#31770665C211yesmaximal, characteristic, complemented
15#464×C269C211yesmaximal, characteristic, complemented
16C2×Q8×D269C2211yescharacteristic
17C2×#16864392C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19#22575453C2211yescharacteristic
20#22575446—21no—
21#22575439—21no—
22#22575437—21no—
23#22575430—21no—
24#22575429C2211yescharacteristic
25#22575424C2211yescharacteristic
26Q8×Dic269C2211yescharacteristic
27Q8×Dic269C2211yescharacteristic
28C4×Dic538—21no—
29C4×Dic538C2211yescharacteristic
30#22575418—21no—
31#22575418C2211yescharacteristic
32D4×Dic269C2211yescharacteristic
33#22575411—21no—
34#22575411C2211yescharacteristic
35#22575410—21no—
36#22575410C2211yescharacteristic
37#22575387C2211yescharacteristic
38#22575387C2211yescharacteristic
39#22575384C2211yescharacteristic
40#22575384C2211yescharacteristic
41C4⋊#14524130C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575365C2211yescharacteristic
49#22575365C2211yescharacteristic
50#22575363C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×Q16×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54#124×C269C2211yescharacteristic
55C4×Q8×C269C2211yescharacteristic
56C4⋊C8×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59C2×Dic538—21no—
60C2×Dic538—21no—
61C2×Dic538C2311yescharacteristic
62Q8×D269—22no—
63Q8×D269—22no—
64C2×C22⋊D269—21no—
65C22×Dic269—21no—
66C2×C4×D269—21no—
67C2×C4×D269C2311yescharacteristic
68#16864392—22no—
69#16864390—22no—
70C2×#14524130C2311yescharacteristic
71C2×#14524130C2311yescharacteristic
72C4⋊Dic269—21no—
73C4⋊Dic269—21no—
74C4⋊Dic269—21no—
75C4⋊Dic269C2311yescharacteristic
76C4⋊Dic269C2311yescharacteristic
77C4⋊Dic269C2311yescharacteristic
78#16864381—21no—
79#16864381—21no—
80#16864381—21no—
81#16864381D411yescharacteristic
82#16864381D411yescharacteristic
83C22⋊Dic269—21no—
84C22⋊Dic269—21no—
85#16864379—21no—
86#16864379—21no—
87#16864379D411yescharacteristic
88#16864379D411yescharacteristic
89C4×Dic269—21no—
90C4×Dic269—21no—
91C4×Dic269—21no—
92C4×Dic269—21no—
93C4×Dic269—21no—
94C4×Dic269C2311yescharacteristic
95C2×Q8×C269C2311yescharacteristic
96C2×Q8×C269C2311yescharacteristic
97C2×D4×C269C2311yescharacteristic
98Q16×C269—22no—
99SD16×C269—22no—
100C2×C8×C269C2311yescharacteristic
101C2×C8×C269C2311yescharacteristic
102C4⋊C4×C269—21no—
103C4⋊C4×C269C2311yescharacteristic
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C42×C269—21no—
107C42×C269C2311yescharacteristic
108C22×D269C2×D411yescharacteristic
109Dic538—24no—
110Dic538—21no—
111Dic538—81no—
112Dic538#4112yes—
113C22⋊D269—81no—
114C2×Dic269—21no—
115C2×Dic269—21no—
116C2×Dic269—21no—
117C2×Dic269—21no—
118C2×Dic269—81no—
119C2×Dic269C2×D411yescharacteristic
120C2×Dic269C2×D411yescharacteristic
121C2×Dic269C2×D411yescharacteristic
122C4×D269—24no—
123C4×D269—21no—
124C4×D269—21no—
125#14524130—21no—
126#14524130—21no—
127C23×C269—21no—
128Q8×C269—21no—
129Q8×C269—21no—
130Q8×C269—21no—
131Q8×C269#4112yes—
132D4×C269—21no—
133D4×C269—21no—
134C2×C4×C269—21no—
135C2×C4×C269—21no—
136C2×C4×C269—21no—
137C2×C4×C269C2×D411yescharacteristic
138C2×C4×C269C2×D411yescharacteristic
139C2×C4×C269C2411yescharacteristic
140C8×C269—21no—
141C8×C269—21no—
142C22×C269—41no—
143C22×C269—41no—
144C22×C269—41no—
145C22×C269C22×D411yescharacteristic
146C2×D269—21no—
147C2×D269—21no—
148C2×D269—21no—
149C4×C269—22no—
150C4×C269—21no—
151C4×C269—21no—
152C4×C269—41no—
153C4×C269—41no—
154C4×C269C2×#4111yescharacteristic
155C4×C269C2×#4111yescharacteristic
156Dic269—22no—
157Dic269—21no—
158Dic269—21no—
159Dic269—21no—
160Dic269—41no—
161Dic269—41no—
162Dic269—41no—
163C2×C269—81no—
164C2×C269#52511yescharacteristic
165C2×C269#57711yescharacteristic
166C2×C269#57811yescharacteristic
167D269—41no—
168C269#312111yescharacteristic, complemented, Sylow
169#3121—2691nomaximal, Sylow
170#542—2691no—
171#519—2691no—
172#485—2691no—
173#482—2691no—
174#477—2691no—
175#475—2691no—
176#464—2691no—
177#464D26911yescharacteristic, complemented
178#451—2691no—
179#448—2691no—
180#441—2691no—
181#440—2691no—
182C4×Q16—2691no—
183C4×SD16—2691no—
184#433—2691no—
185C22×Q8—2691no—
186C2×Q16—2691no—
187C2×Q16C2×D26911yescharacteristic
188C2×SD16—2691no—
189C2×SD16C2×D26911yescharacteristic
190C4⋊Q8—2691no—
191#126—5381no—
192#124—5381no—
193#124C2×D26911yescharacteristic
194#123—5381no—
195C22⋊Q8—2691no—
196C22⋊Q8—5381no—
197#121—2691no—
198C4×Q8—2691no—
199C4×Q8—2691no—
200C4×Q8—2691no—
201C4×Q8—2691no—
202C4×Q8—5381no—
203C4×Q8—5381no—
204C4×Q8C2×D26911yescharacteristic
205C4×D4—2691no—
206#117—2691no—
207#117—2691no—
208#117—5381no—
209#117—5381no—
210#107—2691no—
211#107—2691no—
212#106—2691no—
213#106—2691no—
214C4⋊C8—2691no—
215C4⋊C8C2×D26911yescharacteristic
216#103—2691no—
217#103—2691no—
218#103—2691no—
219#103—2691no—
220#103—2691no—
221#103C2×D26911yescharacteristic
222#102—2691no—
223#102C2×D26911yescharacteristic
224C22⋊C8—2691no—
225C22⋊C8—2691no—
226#97—2691no—
227C4×C8—2691no—
228C2×Q8—2691no—
229C2×Q8—5381no—
230C2×Q8—5382no—
231C2×Q8—5382no—
232C2×Q8—5381no—
233C2×Q8C22×D26911yescharacteristic
234C2×Q8C22×D26911yescharacteristic
235C2×D4—5381no—
236C2×D4C22×D26911yescharacteristic
237C22×C4—2691no—
238C22×C4—5381no—
239C22×C4—5381no—
240Q16—22no—
241Q16—5382no—
242SD16—22no—
243SD16—5382no—
244C2×C8—2691no—
245C2×C8—2691no—
246C2×C8C22×D26911yescharacteristic
247C2×C8C22×D26911yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—2691no—
250C4⋊C4—2691no—
251C4⋊C4—2691no—
252C4⋊C4—2691no—
253C4⋊C4—2691no—
254C4⋊C4—5381no—
255C4⋊C4—5381no—
256C4⋊C4—5381no—
257C4⋊C4—5381no—
258C4⋊C4—5381no—
259C4⋊C4—5381no—
260C4⋊C4C22×D26911yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—2691no—
264C22⋊C4—2691no—
265C22⋊C4—5381no—
266C22⋊C4—5381no—
267C22⋊C4—5381no—
268C22⋊C4—5381no—
269C42—21no—
270C42—2691no—
271C42—5381no—
272C42—5381no—
273C42—5381no—
274C42—5381no—
275C42—5381no—
276C42C22×D26911yescharacteristic
277C23—21no—
278C23—2691no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—2692no—
283Q8—5384no—
284Q8—5381no—
285Q8—21521no—
286Q8#1686440112yes—
287D4—21no—
288D4—21no—
289D4—21521no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2691no—
294C2×C4—2691no—
295C2×C4—2691no—
296C2×C4—5384no—
297C2×C4—5381no—
298C2×C4—5381no—
299C2×C4—5381no—
300C2×C4—5381no—
301C2×C4—5381no—
302C2×C4—5381no—
303C2×C4—21521no—
304C2×C4D4×D26911yescharacteristic
305C2×C4D4×D26911yescharacteristic
306C2×C4C23×D26911yescharacteristic
307C8—21no—
308C8—21no—
309C8—5381no—
310C8—5381no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—5381no—
315C22—5381no—
316C22—5381no—
317C22C2×D4×D26911yescharacteristic
318C4—21no—
319C4—21no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—5381no—
324C4—5381no—
325C4—5381no—
326C4—5382no—
327C4—10761no—
328C4—10761no—
329C4—10761no—
330C4#41×D26911yescharacteristic
331C4C2×#1686440111yescharacteristic
332C2—81no—
333C2—10761no—
334C2#3177111511yescharacteristic
335C2#3177141411yescharacteristic
336C2#3177141911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#464×C269C2
socleC22×C269C22×D4
prime radical#22575424C22
hypercenter#464D269
hypocenterC269#3121
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #31771081C2semidirect
3. #31770945C2semidirect
5. #31770901C2semidirect
6. #31770890C2semidirect
7. #31770824C2semidirect
8. #31770759C2semidirect
10. #31770709C2semidirect
11. #31770706C2semidirect
12. Q16×Dic269C2semidirect
13. #31770696C2semidirect
14. #31770665C2semidirect
15. #464×C269C2semidirect
168. C269#3121semidirect
177. #464D269semidirect

Maximal subgroups

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

subgroupindex
#317712242
#317710812
#317709452
#317709272
#317709012
#317708902
#317708242
#317707592
#317707422
#317707092
#317707062
Q16×Dic2692
#317706962
#317706652
#464×C2692
#3121269