Finite Group Databasebeta

← #119714

Subgroups — #119714

Order 512.

683 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 683 conjugacy classes of subgroups — too many to draw. Only its 118 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#17398C211yesmaximal, characteristic
2#17398C211yesmaximal, characteristic
3#14711C211yesmaximal, characteristic
4#7975C212yesmaximal, complemented
5#7867C212yesmaximal, complemented
6C22×#126C2211yescharacteristic
7#2920C2212yes—
8#2920C2212yes—
9C2×#383C2211yescharacteristic
10C2×C43C2211yescharacteristic
11#1473—24no—
12#1473—24no—
13#1463—21no—
14#1463—21no—
15#1462—21no—
16#1462—21no—
17#1437—21no—
18#1437—21no—
19#1437C412yes—
20#1436—21no—
21#1436—21no—
22#1436C412yes—
23#1288—24no—
24#1276—24no—
25C2×#126—22no—
26C2×#126—22no—
27C2×#126—22no—
28C2×#126—22no—
29C2×#126—22no—
30C2×#126—22no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C22⋊C4—22no—
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4D411yescharacteristic
37C22×C22⋊C4D411yescharacteristic
38C22×C42—21no—
39C22×C42—21no—
40C22×C42C2×C412yes—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#432—22no—
45#432D414yes—
46#431—22no—
47#431—22no—
48#431—22no—
49#431D414yes—
50#423—22no—
51#423—22no—
52#407—22no—
53#407—22no—
54C4×M16—22no—
55C4×M16—22no—
56#383—22no—
57#383—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95C2×C412yes—
64C43—22no—
65C43—22no—
66#324—24no—
67#324—24no—
68#323—24no—
69#323—24no—
70C25C22⋊C411yescharacteristic
71C23×C4—22no—
72C23×C4—22no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C2×M16—42no—
80C2×M16—42no—
81#126—42no—
82#126—42no—
83#126—42no—
84#126—42no—
85#126—42no—
86#126—42no—
87#126—42no—
88#126—42no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—44no—
116C2×C42—24no—
117C2×C42—44no—
118C2×C42—44no—
119C2×C42—44no—
120C2×C42—41no—
121C2×C42—42no—
122C2×C42—41no—
123C2×C42C2×D412yes—
124C2×C42C2×D412yes—
125C2×C42#4112yes—
126C2×C42#4112yes—
127C2×C42#4112yes—
128C2×C42#4112yes—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#97—24no—
138#97—42no—
139C4×C8—24no—
140C4×C8—42no—
141#95—24no—
142#95—44no—
143#95—44no—
144#95—44no—
145#95—41no—
146#95—42no—
147#95—41no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24C2×C22⋊C411yescharacteristic
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4#11711yescharacteristic
185C22×C4#11711yescharacteristic
186C22×C4C4×D412yes—
187C22×C4C4×D412yes—
188C22×C4#12011yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12112yes—
191C22×C4#12112yes—
192C22×C4#12312yes—
193C22×C4#12312yes—
194C22×C4#12311yescharacteristic
195C22×C4#12311yescharacteristic
196M16—84no—
197M16—84no—
198C2×C8—44no—
199C2×C8—44no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—88no—
213C42—28no—
214C42—44no—
215C42—44no—
216C42—48no—
217C42—44no—
218C42#12112yes—
219C42#12112yes—
220C42#12312yes—
221C42#12312yes—
222C42#12412yes—
223C42#12412yes—
224C42#12612yes—
225C42#12612yes—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—41no—
233C23—42no—
234C23—42no—
235C23—82no—
236C23—82no—
237C23—82no—
238C23—82no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23#38111yescharacteristic
244C23#38611yescharacteristic
245C23#38611yescharacteristic
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—28no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4#38812yes—
285C2×C4#38812yes—
286C2×C4#39412yes—
287C2×C4#39412yes—
288C2×C4#39712yes—
289C2×C4#39712yes—
290C8—84no—
291C8—84no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—84no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—162no—
307C22—162no—
308C22#144011yescharacteristic
309C22#182712yes—
310C22#188412yes—
311C22#189112yes—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—48no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—162no—
323C2—22no—
324C2—22no—
325C2—22no—
326C2—161no—
327C2#1114811yescharacteristic
328C2#1116011yescharacteristic
329C2#1117511yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7975C2semidirect
5. #7867C2semidirect

Maximal subgroups

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

subgroupindex
#173982
#147112
#79752
#78672