Finite Group Databasebeta

← #97039

Subgroups — #97039

Order 512.

686 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2644C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic, complemented
3#8699C211yesmaximal, characteristic, complemented
4#7593C212yesmaximal, complemented
5#7592C212yesmaximal
6#2644—24no—
7#2644—24no—
8C2×#397—21no—
9C2×#397—21no—
10C2×#397—21no—
11C2×#396—21no—
12C2×#390C411yescharacteristic
13C2×#390C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1817C2212yes—
20#1817C2212yes—
21#1280—21no—
22#1280—21no—
23#1280—21no—
24#1280—21no—
25#1271—24no—
26#1270—24no—
27C23×D4C2×C411yescharacteristic
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4—21no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45#397—28no—
46#397—28no—
47#397—28no—
48#396—28no—
49#390—22no—
50#390—22no—
51#390—22no—
52#390—22no—
53#390—22no—
54#390D414yes—
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389D414yes—
61#376—22no—
62#376—22no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95D411yescharacteristic
67C2×#95D411yescharacteristic
68C2×#95D411yescharacteristic
69C2×#95D411yescharacteristic
70#336—22no—
71#336—22no—
72#336—22no—
73#336—22no—
74C25—21no—
75C22×D4—21no—
76C22×D4—24no—
77C22×D4—42no—
78C22×D4—42no—
79C22×D4C22⋊C412yes—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C22×C8—42no—
91C22×C8—42no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—82no—
102C2×C4⋊C4—82no—
103C2×C4⋊C4—82no—
104C2×C4⋊C4C22⋊C412yes—
105C2×C4⋊C4C22⋊C412yes—
106C2×C4⋊C4C22⋊C412yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—82no—
114C2×C22⋊C4—82no—
115C2×C22⋊C4—82no—
116C2×C22⋊C4—82no—
117C2×C22⋊C4—82no—
118C2×C22⋊C4—82no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—24no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42C2×D412yes—
127C2×C42C2×D412yes—
128C22⋊C8—44no—
129C22⋊C8—44no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—22no—
134#95—22no—
135#95—22no—
136#95—22no—
137C24—22no—
138C24—42no—
139C24—42no—
140C24—42no—
141C24—42no—
142C24—42no—
143C24—42no—
144C24—42no—
145C24C2×C22⋊C411yescharacteristic
146C2×D4—24no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—88no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
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—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4#12012yes—
192C22×C4#12012yes—
193C22×C4#13114yes—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C4⋊C4—24no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—84no—
207C4⋊C4—84no—
208C4⋊C4—84no—
209C22⋊C4—84no—
210C22⋊C4—84no—
211C22⋊C4—84no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C42—42no—
216C42—42no—
217C42—42no—
218C42—42no—
219C42—42no—
220C42—42no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—84no—
233C23—84no—
234C23—84no—
235C23—84no—
236C23—84no—
237C23—84no—
238C23—84no—
239C23#37911yescharacteristic
240C23#40812yes—
241C23#41012yes—
242C23#41312yes—
243D4—84no—
244D4—84no—
245C2×C4—28no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#40812yes—
287C2×C4#40812yes—
288C8—162no—
289C8—162no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22—168no—
304C22#147314yes—
305C22#178011yescharacteristic
306C22#178211yescharacteristic
307C22#178511yescharacteristic
308C22#179511yescharacteristic
309C22#179511yescharacteristic
310C22#179511yescharacteristic
311C22#179611yescharacteristic
312C4—22no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—82no—
317C4—81no—
318C4—81no—
319C4—82no—
320C4—81no—
321C4—81no—
322C4—82no—
323C2—24no—
324C2—162no—
325C2#830911yescharacteristic
326C2#832511yescharacteristic
327C2#832611yescharacteristic
328C2#851811yescharacteristic
329C2#851911yescharacteristic
330C2#852311yescharacteristic
331C2#852611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8699C2semidirect
3. #8699C2semidirect
4. #7593C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26442
#86992
#75932
#75922