Finite Group Databasebeta

← #98518

Subgroups — #98518

Order 512.

648 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2570C211yesmaximal, characteristic
2#8719C212yesmaximal
3#8153C211yesmaximal, characteristic
4#8153C211yesmaximal, characteristic
5#7627C212yesmaximal
6#2570—24no—
7#2570—24no—
8C2×#400—21no—
9C2×#398—21no—
10C2×#389C411yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#387—21no—
13C2×#387—21no—
14C2×#384C411yescharacteristic
15C2×#382C411yescharacteristic
16C2×#382C2211yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1749C2212yes—
20#1749C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23#1295—24no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C22×C4⋊C4—21no—
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×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C42—21no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—22no—
44C2×#97—22no—
45#400—28no—
46#398—28no—
47#389—22no—
48#389—22no—
49#389—22no—
50#389D414yes—
51#387—28no—
52#387—28no—
53#384—22no—
54#384—22no—
55#382—22no—
56#382—22no—
57#382—22no—
58#382D414yes—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72#336—22no—
73#336—22no—
74#324—24no—
75#324—24no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C22×C8—42no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C22⋊C412yes—
115C2×C4⋊C4C22⋊C412yes—
116C2×C4⋊C4C22⋊C412yes—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—24no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—82no—
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#97—44no—
134#97—44no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—22no—
143#95—22no—
144C24C2×C22⋊C411yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#10112yes—
196C22×C4#10112yes—
197C22×C4#10412yes—
198C22×C4#10412yes—
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4#12012yes—
202C22×C4#12012yes—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C4⋊C4—24no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—48no—
217C4⋊C4—48no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C42—24no—
224C42—24no—
225C42—24no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42—84no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23#37911yescharacteristic
235C23C2×#10111yescharacteristic
236C23C2×#10111yescharacteristic
237C23C2×#10411yescharacteristic
238C23C2×#10411yescharacteristic
239C23#42111yescharacteristic
240C23#42111yescharacteristic
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
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—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#41012yes—
300C2×C4#41312yes—
301C8—84no—
302C8—84no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22#152611yescharacteristic
316C22#153111yescharacteristic
317C22#153111yescharacteristic
318C22#153711yescharacteristic
319C22#178211yescharacteristic
320C22#178411yescharacteristic
321C22#178511yescharacteristic
322C22#178911yescharacteristic
323C22#178911yescharacteristic
324C22#179511yescharacteristic
325C22#179611yescharacteristic
326C4—22no—
327C4—22no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4—82no—
335C4—82no—
336C4—82no—
337C4—82no—
338C2—21no—
339C2—21no—
340C2—21no—
341C2—21no—
342C2#833811yescharacteristic
343C2#836211yescharacteristic
344C2#836311yescharacteristic
345C2#836811yescharacteristic
346C2#837011yescharacteristic
347C2#852511yescharacteristic
348C2#855411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25702
#87192
#81532
#76272