Finite Group Databasebeta

← #100140

Subgroups — #100140

Order 512.

526 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 94 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
1#21916C211yesmaximal, characteristic
2#14003C211yesmaximal, characteristic
3#8701C211yesmaximal, characteristic
4#7595C212yesmaximal
5#7017C212yesmaximal
6C2×#391C411yescharacteristic
7C2×#391C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×#382C411yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1917—24no—
12#1817C2212yes—
13#1817C2212yes—
14#1797—21no—
15#1749—21no—
16#1443—21no—
17#1443—21no—
18C4⋊#95C411yescharacteristic
19#1434—21no—
20#1434—21no—
21C4×#95C2211yescharacteristic
22#1280—21no—
23#1280—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42C2311yescharacteristic
34C2×C4⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×#97—22no—
38#391—22no—
39#391—22no—
40#391—22no—
41#391—22no—
42#391—42no—
43#382—42no—
44C4×C4⋊C4—22no—
45C4×C4⋊C4—22no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95D411yescharacteristic
59C2×#95D411yescharacteristic
60#336—22no—
61#336—22no—
62#336—22no—
63#336—22no—
64#336—22no—
65#336D414yes—
66C22×Q8—21no—
67C22×Q8—22no—
68C22×Q8—22no—
69C22×Q8—41no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8M1612yes—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—24no—
86C22×C8—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42M1612yes—
112C2×C42C2×D412yes—
113C2×C42#4112yes—
114C4⋊C8—82no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#97—82no—
120#95—24no—
121#95—24no—
122#95—22no—
123#95—22no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—42no—
134#95—41no—
135C24C2×C22⋊C411yescharacteristic
136C2×Q8—24no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—82no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8—82no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
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—41no—
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—41no—
191C22×C4—41no—
192C22×C4—41no—
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—41no—
204C22×C4—41no—
205C22×C4#10012yes—
206C22×C4C4×D412yes—
207C22×C4#12011yescharacteristic
208C22×C4#12112yes—
209C22×C4#12311yescharacteristic
210C22×C4C2×M1611yescharacteristic
211C22×C4C2×M1611yescharacteristic
212C22×C4#13112yes—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C42—42no—
224C42—42no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42—81no—
230C42—81no—
231C42—82no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#38611yescharacteristic
237C23#40711yescharacteristic
238C23#40811yescharacteristic
239C23#41011yescharacteristic
240C23#41011yescharacteristic
241C23C2×#10011yescharacteristic
242C23#41311yescharacteristic
243Q8—84no—
244Q8—84no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
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—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4#43512yes—
314C2×C4#43612yes—
315C2×C4#45512yes—
316C8—84no—
317C8—162no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22#146411yescharacteristic
331C22#146611yescharacteristic
332C22#146911yescharacteristic
333C22#147311yescharacteristic
334C22#187011yescharacteristic
335C22#188411yescharacteristic
336C22#188511yescharacteristic
337C22#189911yescharacteristic
338C22#190011yescharacteristic
339C22#190911yescharacteristic
340C22#191211yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—41no—
347C4—82no—
348C4—82no—
349C4—81no—
350C4—81no—
351C4—82no—
352C4—81no—
353C4—161no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2#896111yescharacteristic
359C2#908411yescharacteristic
360C2#910411yescharacteristic
361C2#944411yescharacteristic
362C2#944811yescharacteristic
363C2#945111yescharacteristic
364C2#946011yescharacteristic
subgroupG/H
center, socleC23#386
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
#219162
#140032
#87012
#75952
#70172