Finite Group Databasebeta

← #98555

Subgroups — #98555

Order 512.

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

normal subgroups only

This group has 656 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
1C2×#2511C211yesmaximal, characteristic
2#8724C212yesmaximal
3#8560C211yesmaximal, characteristic
4#8560C211yesmaximal, characteristic
5#7643C212yesmaximal
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×#384C2211yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23#1296—24no—
24#1280—21no—
25#1280—21no—
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×C4⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊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#384D414yes—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
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—42no—
75#324—42no—
76C22×Q8—21no—
77C22×Q8—42no—
78C22×Q8—42no—
79C22×Q8—41no—
80C22×Q8—41no—
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—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—82no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4C22⋊C412yes—
117C2×C4⋊C4C22⋊C412yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4SD1612yes—
120C2×C4⋊C4Q1612yes—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—24no—
126C2×C42—42no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—42no—
130C2×C42—82no—
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C4⋊C8—44no—
134C4⋊C8—44no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—22no—
148#95—22no—
149C24C2×C22⋊C411yescharacteristic
150C2×Q8—24no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—84no—
157C2×Q8—84no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
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—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
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—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4#10312yes—
207C22×C4#10312yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4#12012yes—
211C22×C4#12012yes—
212C22×C4C2×SD1611yescharacteristic
213C22×C4C2×Q1611yescharacteristic
214C22×C4#13512yes—
215C22×C4AGL(1,ℤ/8)11yescharacteristic
216C22×C4#13712yes—
217C22×C4#13711yescharacteristic
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C42—24no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—42no—
244C42—84no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23#37911yescharacteristic
250C23#41311yescharacteristic
251C23#41311yescharacteristic
252C23C2×#10311yescharacteristic
253C23#41611yescharacteristic
254C23#41711yescharacteristic
255C23#41911yescharacteristic
256Q8—84no—
257Q8—84no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4#41012yes—
317C2×C4#41312yes—
318C2×C4#44812yes—
319C2×C4#44912yes—
320C2×C4#45112yes—
321C2×C4#45212yes—
322C8—162no—
323C8—162no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22#154211yescharacteristic
337C22#155011yescharacteristic
338C22#155411yescharacteristic
339C22#155911yescharacteristic
340C22#178211yescharacteristic
341C22#178511yescharacteristic
342C22#178511yescharacteristic
343C22#178711yescharacteristic
344C22#178811yescharacteristic
345C22#179511yescharacteristic
346C22#179611yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—81no—
356C4—82no—
357C4—82no—
358C4—81no—
359C4—82no—
360C4—82no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2#834311yescharacteristic
366C2#838011yescharacteristic
367C2#838211yescharacteristic
368C2#838611yescharacteristic
369C2#839211yescharacteristic
370C2#852311yescharacteristic
371C2#855611yescharacteristic
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×#25112
#87242
#85602
#76432