Finite Group Databasebeta

← #98556

Subgroups — #98556

Order 512.

760 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2552C211yesmaximal, characteristic
2#8724C212yesmaximal
3#8560C211yesmaximal, characteristic
4#8560C211yesmaximal, characteristic
5#7641C212yesmaximal
6C22×#125—21no—
7#2552—24no—
8#2552—24no—
9C2×#398—21no—
10C2×#398—21no—
11C2×#389—21no—
12C2×#389C411yescharacteristic
13C2×#389C2211yescharacteristic
14C2×#384C411yescharacteristic
15C2×#384C411yescharacteristic
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#1271—24no—
27C2×#125—28no—
28C2×#125—44no—
29C2×#125—42no—
30C2×#125—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53#398—28no—
54#398—28no—
55#389—28no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389D414yes—
60#384—22no—
61#384—22no—
62#384—22no—
63#384—22no—
64#384—22no—
65#384D414yes—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95D411yescharacteristic
74C2×#95D411yescharacteristic
75#336—22no—
76#336—22no—
77#324—42no—
78#324—42no—
79C23×C4—21no—
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—
92#125—416no—
93#125—416no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4—82no—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4D812yes—
134C2×C4⋊C4SD1612yes—
135C2×C4⋊C4SD1612yes—
136C2×C4⋊C4Q1612yes—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—24no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—42no—
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C4⋊C8—44no—
151C4⋊C8—44no—
152C22⋊C8—42no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—22no—
160#95—22no—
161C24C2×C22⋊C411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—41no—
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×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
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#10112yes—
207C22×C4#10112yes—
208C22×C4#10212yes—
209C22×C4#10212yes—
210C22×C4#10312yes—
211C22×C4#10312yes—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4#12012yes—
215C22×C4#12012yes—
216C22×C4C2×D811yescharacteristic
217C22×C4C2×SD1611yescharacteristic
218C22×C4C2×SD1611yescharacteristic
219C22×C4C2×Q1611yescharacteristic
220C22×C4AGL(1,ℤ/8)12yes—
221C22×C4#13712yes—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—48no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C4⋊C4—84no—
250C42—24no—
251C42—24no—
252C42—24no—
253C42—42no—
254C42—42no—
255C42—48no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#37911yescharacteristic
262C23C2×#10111yescharacteristic
263C23C2×#10111yescharacteristic
264C23C2×#10211yescharacteristic
265C23C2×#10311yescharacteristic
266C23#41811yescharacteristic
267C23#41911yescharacteristic
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#33214yes—
326C2×C4#41012yes—
327C2×C4#41312yes—
328C2×C4#44712yes—
329C2×C4#44812yes—
330C2×C4#45012yes—
331C2×C4#45112yes—
332C8—162no—
333C8—162no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22C2×#33211yescharacteristic
347C22#154611yescharacteristic
348C22#154911yescharacteristic
349C22#155611yescharacteristic
350C22#178211yescharacteristic
351C22#178411yescharacteristic
352C22#178511yescharacteristic
353C22#178611yescharacteristic
354C22#178711yescharacteristic
355C22#179511yescharacteristic
356C22#179611yescharacteristic
357C4—22no—
358C4—22no—
359C4—24no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4—82no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2#834011yescharacteristic
376C2#837611yescharacteristic
377C2#838011yescharacteristic
378C2#838411yescharacteristic
379C2#839211yescharacteristic
380C2#852511yescharacteristic
381C2#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×#25522
#87242
#85602
#76412