Finite Group Databasebeta

← #99718

Subgroups — #99718

Order 512.

474 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#23730C211yesmaximal, characteristic
2#21909C211yesmaximal, characteristic
3#8710C212yesmaximal, complemented
4#7563C211yesmaximal, characteristic
5#7029C212yesmaximal, complemented
6C2×#427—21no—
7C2×#416—21no—
8#2297C2211yescharacteristic
9#2285C2211yescharacteristic
10#2097—21no—
11#2096C411yescharacteristic
12#2095C411yescharacteristic
13#2094C411yescharacteristic
14#2093C411yescharacteristic
15#2092—21no—
16#1924—22no—
17#1923—22no—
18#1797C2212yes—
19#1780—21no—
20#1750—21no—
21C4×C22⋊C8C2212yes—
22C2×#336C2211yescharacteristic
23#1279—22no—
24#1279—22no—
25#1272—21no—
26#1270—21no—
27C22×#41D411yescharacteristic
28C23×C8—21no—
29C2×#117—21no—
30C2×#117D411yescharacteristic
31C22×C42C2311yescharacteristic
32#427—42no—
33#427—42no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#106—21no—
37C2×#106—21no—
38C2×C4⋊C8D412yes—
39C2×C4⋊C8D412yes—
40#416—42no—
41#416—42no—
42C2×#103—21no—
43C2×#103—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—21no—
51C2×#97—21no—
52C2×C4×C8—22no—
53C2×C4×C8—21no—
54#391C2×C411yescharacteristic
55#391D411yescharacteristic
56#390C2×C411yescharacteristic
57#390D411yescharacteristic
58#386—22no—
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61#382—21no—
62#380—22no—
63C4×C4⋊C4—22no—
64C4×C22⋊C4—22no—
65#336—22no—
66#336—22no—
67#336—21no—
68#336—22no—
69#336C2×C411yescharacteristic
70#336C2×C411yescharacteristic
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—42no—
74C2×#41—41no—
75C2×#41—41no—
76C22×Q8C22⋊C411yescharacteristic
77C22×D4—21no—
78C22×D4C22⋊C411yescharacteristic
79C23×C4—21no—
80C23×C4C2×D411yescharacteristic
81C23×C4#4112yes—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91#117—41no—
92#117—41no—
93#117—41no—
94#117—41no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4C22⋊C411yescharacteristic
102C2×C4⋊C4C22⋊C411yescharacteristic
103C2×C22⋊C4—22no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—41no—
107C2×C42—21no—
108C2×C42—42no—
109C2×C42—41no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113#107—42no—
114#107—42no—
115#106—42no—
116#106—42no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119#103—42no—
120#103—42no—
121#102—42no—
122#102—42no—
123C22⋊C8—24no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—81no—
128C22⋊C8—81no—
129C22⋊C8—82no—
130#97—81no—
131C4×C8—42no—
132C4×C8—42no—
133C4×C8—81no—
134#95—22no—
135#95—22no—
136#95—42no—
137C24—21no—
138C24#12311yescharacteristic
139#41—44no—
140#41—44no—
141#41—82no—
142#41—82no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—42no—
152C2×D4—81no—
153C2×D4—81no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C4×D412yes—
172C22×C4#12011yescharacteristic
173C22×C4#12112yes—
174C22×C4#13512yes—
175C22×C4#13712yes—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—44no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—82no—
192C2×C8—82no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C22⋊C4—81no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C42—22no—
209C42—22no—
210C42—41no—
211C42—41no—
212C42—41no—
213C42—41no—
214C42—81no—
215C42—81no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—41no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23#38611yescharacteristic
226C23#48512yes—
227Q8—42no—
228Q8—81no—
229D4—42no—
230D4—81no—
231D4—84no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4#41611yescharacteristic
259C2×C4#41911yescharacteristic
260C2×C4#44312yes—
261C2×C4#44412yes—
262C2×C4#45212yes—
263C2×C4#45311yescharacteristic
264C2×C4#45311yescharacteristic
265C2×C4#45812yes—
266C8—42no—
267C8—42no—
268C8—84no—
269C8—82no—
270C8—82no—
271C22—24no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22—82no—
280C22—81no—
281C22#187311yescharacteristic
282C22#187612yes—
283C22#188112yes—
284C22#188711yescharacteristic
285C22#189011yescharacteristic
286C4—21no—
287C4—21no—
288C4—22no—
289C4—22no—
290C4—42no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—161no—
295C4—161no—
296C4#161812yes—
297C4#161812yes—
298C2—42no—
299C2—161no—
300C2#899412yes—
301C2#900711yescharacteristic
302C2#902411yescharacteristic
303C2#903812yes—
304C2#906911yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8710C2semidirect
5. #7029C2semidirect

Maximal subgroups

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

subgroupindex
#237302
#219092
#87102
#75632
#70292