Finite Group Databasebeta

← #97613

Subgroups — #97613

Order 512.

676 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 124 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×#2511C211yesmaximal, characteristic
2#8707C212yesmaximal
3#8558C211yesmaximal, characteristic
4C4×#324C211yesmaximal, characteristic
5#7575C212yesmaximal
6#2511—24no—
7#2511—24no—
8C2×#400—21no—
9C2×#400—21no—
10C2×#391C411yescharacteristic
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#389C411yescharacteristic
14C2×#387—21no—
15C2×#387—21no—
16C2×#384C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1272—24no—
27C23×Q8C2×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×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42C2311yescharacteristic
40C2×C4⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C4×C8—22no—
44#400—28no—
45#400—28no—
46#391—22no—
47#391—22no—
48#391—22no—
49#391—22no—
50#391—22no—
51#391D414yes—
52#389—22no—
53#389—22no—
54#387—28no—
55#387—28no—
56#384—22no—
57#384—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4D414yes—
62C2×#95—21no—
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—42no—
76C22×Q8—21no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×Q8—42no—
80C22×Q8—42no—
81C22×Q8C22⋊C412yes—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—42no—
94C22×C8—42no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4C22⋊C412yes—
115C2×C4⋊C4C22⋊C412yes—
116C2×C4⋊C4C22⋊C412yes—
117C2×C4⋊C4SD1612yes—
118C2×C4⋊C4Q1612yes—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—24no—
126C2×C42—42no—
127C2×C42—42no—
128C2×C42—82no—
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C4⋊C8—44no—
132C22⋊C8—24no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C4×C8—44no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—22no—
146#95—22no—
147C24C2×C22⋊C411yescharacteristic
148C2×Q8—24no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—84no—
155C2×Q8—84no—
156C22×C4—22no—
157C22×C4—22no—
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—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4#10312yes—
204C22×C4#10312yes—
205C22×C4#10412yes—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4#12012yes—
209C22×C4#12012yes—
210C22×C4#13112yes—
211C22×C4#13112yes—
212C22×C4C2×SD1611yescharacteristic
213C22×C4C2×Q1611yescharacteristic
214C22×C4#13712yes—
215C2×C8—44no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—48no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C42—24no—
234C42—42no—
235C42—42no—
236C42—42no—
237C42—42no—
238C42—42no—
239C42—84no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23#37911yescharacteristic
245C23#40811yescharacteristic
246C23#40811yescharacteristic
247C23C2×#10311yescharacteristic
248C23#41911yescharacteristic
249C23C2×#10411yescharacteristic
250C23#42111yescharacteristic
251Q8—84no—
252Q8—84no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4#40812yes—
310C2×C4#40812yes—
311C2×C4#41012yes—
312C2×C4#41312yes—
313C2×C4#44812yes—
314C2×C4#45112yes—
315C8—84no—
316C8—162no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22#148111yescharacteristic
330C22#148511yescharacteristic
331C22#149311yescharacteristic
332C22#149711yescharacteristic
333C22#178011yescharacteristic
334C22#178211yescharacteristic
335C22#178511yescharacteristic
336C22#178711yescharacteristic
337C22#178911yescharacteristic
338C22#179511yescharacteristic
339C22#179611yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—82no—
348C4—81no—
349C4—82no—
350C4—82no—
351C4—82no—
352C4#147314yes—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2#830911yescharacteristic
358C2#831411yescharacteristic
359C2#832511yescharacteristic
360C2#836311yescharacteristic
361C2#837011yescharacteristic
362C2#838011yescharacteristic
363C2#839211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ 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×#25112
#87072
#85582
C4×#3242
#75752