Finite Group Databasebeta

← #46821340

Subgroups — #46821340

Order 34432.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575486C211yesmaximal, characteristic, complemented
2C2×#22575484C211yesmaximal, characteristic, complemented
3C2×#22575483C211yesmaximal, characteristic, complemented
4#31771151C211yesmaximal, characteristic
5C2×#22575451C211yesmaximal, characteristic, complemented
6#31770998C212yesmaximal, complemented
7#31770996C212yesmaximal, complemented
8#31770983C212yesmaximal, complemented
9#31770976C212yesmaximal, complemented
10M16×Dic269C211yesmaximal, characteristic, complemented
11C2×#137×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×Dic538C2211yescharacteristic
14#22575486—24no—
15#22575484—24no—
16#22575483—24no—
17C2×Dic1076C2212yes—
18C2×#16864392C2212yes—
19C2×#16864392C2212yes—
20C2×#16864391C2212yes—
21C2×#16864390C2212yes—
22C2×#16864389C2212yes—
23C2×#16864386C2211yescharacteristic
24#22575451—22no—
25#22575451C2212yes—
26#22575451C2212yes—
27#22575451C2211yescharacteristic
28#22575449C2211yescharacteristic
29#22575442C2211yescharacteristic
30#22575440C2211yescharacteristic
31#22575425—22no—
32C4×C22⋊D269—22no—
33C2×#16864381—22no—
34C2×C4×Dic269C2211yescharacteristic
35#22575363C2212yes—
36C8×Dic269C2212yes—
37C2×#41×C269C2211yescharacteristic
38C22×Q8×C269C2211yescharacteristic
39#137×C269—24no—
40C2×Q16×C269C2212yes—
41C2×SD16×C269C2212yes—
42C2×M16×C269C2211yescharacteristic
43#16864402—22no—
44#16864402—22no—
45C2×Dic538—21no—
46C2×Dic538—22no—
47C2×Dic538—22no—
48C2×Dic538C2311yescharacteristic
49C2×Dic538C2312yes—
50C2×D1076C2311yescharacteristic
51C2×C22⋊D269—22no—
52C2×C22⋊D269—21no—
53C22×Dic269—21no—
54C22×Dic269D412yes—
55C2×C4×D269—21no—
56Dic1076—24no—
57#16864392—24no—
58#16864392—24no—
59#16864391—24no—
60#16864390—24no—
61#16864389—24no—
62#16864386D414yescomplemented
63C2×#14524130C2312yes—
64#16864381—24no—
65#16864381—24no—
66#16864381—22no—
67C22⋊Dic269—22no—
68#16864379—22no—
69C4×Dic269D412yes—
70C4×Dic269C2311yescharacteristic
71C4×Dic269C2311yescharacteristic
72#41×C269—22no—
73#41×C269—22no—
74C2×Q8×C269—21no—
75C2×Q8×C269—22no—
76C2×Q8×C269—22no—
77C2×Q8×C269C2311yescharacteristic
78C2×Q8×C269C2312yes—
79C2×D4×C269—21no—
80C2×D4×C269C2311yescharacteristic
81C22×C4×C269—21no—
82C22×C4×C269—21no—
83C22×C4×C269C2311yescharacteristic
84Q16×C269—24no—
85SD16×C269—24no—
86M16×C269D414yescomplemented
87C2×C8×C269C2312yes—
88C22×D269—21no—
89Dic538—22no—
90Dic538—22no—
91Dic538—21no—
92Dic538—21no—
93Dic538—42no—
94D1076—21no—
95D1076—21no—
96C22⋊D269—42no—
97C22⋊D269—44no—
98C2×Dic269—24no—
99C2×Dic269—21no—
100C2×Dic269—22no—
101C2×Dic269—42no—
102C2×Dic269C2×D412yes—
103C2×Dic269C2×D412yes—
104C4×D269—42no—
105#14524130C2×D414yes—
106C23×C269—21no—
107C23×C269C2×D411yescharacteristic
108Q8×C269—22no—
109Q8×C269—21no—
110Q8×C269—22no—
111Q8×C269—21no—
112Q8×C269—42no—
113D4×C269—21no—
114D4×C269—21no—
115D4×C269—42no—
116C2×C4×C269—21no—
117C2×C4×C269—22no—
118C2×C4×C269—42no—
119C2×C4×C269—42no—
120C2×C4×C269C2×D411yescharacteristic
121C2×C4×C269C2×D412yes—
122C2×C4×C269C2×D412yes—
123C2×C4×C269C2411yescharacteristic
124C8×C269C2×D414yes—
125C22×C269—22no—
126C22×C269—41no—
127C22×C269—42no—
128C22×C269#12712yes—
129C22×C269C22×D411yescharacteristic
130C2×D269—41no—
131C2×D269—42no—
132C4×C269—41no—
133C4×C269—42no—
134C4×C269#12712yes—
135C4×C269C22×D411yescharacteristic
136C4×C269C22×D411yescharacteristic
137Dic269—22no—
138Dic269—41no—
139Dic269—42no—
140Dic269#13714yescomplemented
141C2×C269—22no—
142C2×C269—81no—
143C2×C269C2×#12711yescharacteristic
144C2×C269C2×#13712yes—
145D269—81no—
146C269#315211yescharacteristic, complemented, Sylow
147#3152—2691nomaximal, Sylow
148C2×#137—2691no—
149C2×#137—2691no—
150C2×#137—2691no—
151C2×#137D26911yescharacteristic, complemented
152#532—2691no—
153C2×C4⋊Q8—2691no—
154#497—2692no—
155#497—2692no—
156#494—2692no—
157#492—2692no—
158C4×M16—2691no—
159C2×#41—2691no—
160C2×#41C2×D26911yescharacteristic
161C22×Q8—2691no—
162C22×Q8C2×D26911yescharacteristic
163#137—24no—
164#137—5384no—
165#137—5384no—
166#137—5384no—
167C2×Q16—2692no—
168C2×Q16—2692no—
169C2×Q16—2692no—
170C2×Q16C2×D26912yes—
171C2×SD16—2692no—
172C2×SD16—2692no—
173C2×SD16—2692no—
174C2×SD16C2×D26912yes—
175C2×M16—2691no—
176C2×M16C2×D26911yescharacteristic
177C4⋊Q8—2692no—
178C4⋊Q8—2692no—
179C4⋊Q8—2691no—
180C4⋊Q8—5382no—
181#127—2691no—
182#124—2691no—
183#124—2691no—
184#121—5382no—
185C4×D4—5382no—
186C2×C4⋊C4—5382no—
187C2×C42—2691no—
188#97—2692no—
189C4×C8—2692no—
190#41—22no—
191#41—22no—
192#41—5382no—
193#41—5382no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—2691no—
198C2×Q8—2692no—
199C2×Q8—5381no—
200C2×Q8—5382no—
201C2×Q8—5382no—
202C2×Q8C22×D26911yescharacteristic
203C2×Q8C22×D26912yes—
204C2×D4—21no—
205C2×D4—2691no—
206C2×D4—5382no—
207C2×D4—5381no—
208C2×D4C22×D26911yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—2692no—
212C22×C4—5381no—
213C22×C4—5381no—
214C22×C4C22×D26911yescharacteristic
215Q16—24no—
216Q16—5384no—
217Q16—5384no—
218Q16—5384no—
219SD16—24no—
220SD16—5384no—
221SD16—5384no—
222SD16—5384no—
223M16—2694no—
224M16C22⋊D26914yescomplemented
225C2×C8—2692no—
226C2×C8C22×D26912yes—
227C4⋊C4—5384no—
228C4⋊C4—5384no—
229C4⋊C4—5382no—
230C22⋊C4—5382no—
231C22⋊C4—5382no—
232C42—2692no—
233C42—2691no—
234C42—2691no—
235C23—21no—
236C23—5381no—
237C23D4×D26911yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—5382no—
244Q8—5381no—
245Q8—5382no—
246Q8—5381no—
247Q8—10762no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—5381no—
252D4—5381no—
253D4—10764no—
254D4—10762no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—2692no—
260C2×C4—2692no—
261C2×C4—5384no—
262C2×C4—5381no—
263C2×C4—5382no—
264C2×C4—10762no—
265C2×C4—10762no—
266C2×C4D4×D26911yescharacteristic
267C2×C4D4×D26912yes—
268C2×C4C2×C22⋊D26912yes—
269C2×C4C23×D26911yescharacteristic
270C8—2694no—
271C8C2×C22⋊D26914yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—10762no—
276C22—10761no—
277C22#2257544912yes—
278C22C2×D4×D26911yescharacteristic
279C4—41no—
280C4—42no—
281C4—2694no—
282C4—5382no—
283C4—10761no—
284C4—10762no—
285C4#2257544912yes—
286C4C2×D4×D26911yescharacteristic
287C4C22×C22⋊D26911yescharacteristic
288C2—22no—
289C2—81no—
290C2—21521no—
291C2C2×#2257544911yescharacteristic
292C2#137×D26912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC22×C269C22×D4
prime radical#31771151C2
hypercenterC2×#137D269
hypocenterC269#3152
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22575486C2semidirect
2. C2×#22575484C2semidirect
3. C2×#22575483C2semidirect
5. C2×#22575451C2semidirect
6. #31770998C2semidirect
7. #31770996C2semidirect
8. #31770983C2semidirect
9. #31770976C2semidirect
10. M16×Dic269C2semidirect
11. C2×#137×C269C2semidirect
62. #16864386D4semidirect
86. M16×C269D4semidirect
140. Dic269#137semidirect
146. C269#3152semidirect
151. C2×#137D269semidirect
224. M16C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754862
C2×#225754842
C2×#225754832
#317711512
C2×#225754512
#317709982
#317709962
#317709832
#317709762
M16×Dic2692
C2×#137×C2692
#3152269