Finite Group Databasebeta

← #99229

Subgroups — #99229

Order 512.

584 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 88 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#13961C211yesmaximal, characteristic
2#10763C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7037C212yesmaximal
5#7037C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C22×#101—21no—
8C2×#383C411yescharacteristic
9C2×#376C411yescharacteristic
10#1817C2212yes—
11C2×#343—21no—
12#1728C2212yes—
13C2×#324—21no—
14C2×#324—21no—
15#1446—21no—
16#1445—21no—
17#1436—21no—
18#1436—21no—
19#1434C411yescharacteristic
20#1434C411yescharacteristic
21#1434C2211yescharacteristic
22#1434C2211yescharacteristic
23C23×Q8D411yescharacteristic
24C22×M16—22no—
25C2×C4×Q8—24no—
26C2×C4×Q8—42no—
27C2×C4×Q8—42no—
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42C2311yescharacteristic
35C2×#101—24no—
36C2×#101—42no—
37C2×#101—42no—
38C2×C22⋊C8—22no—
39#383—42no—
40#376—42no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56#343—42no—
57#336—24no—
58#336—24no—
59#324—42no—
60#324—42no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—21no—
66C22×Q8—21no—
67C22×Q8—41no—
68C22×Q8—41no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C2×M16—24no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—42no—
86C22×C8—42no—
87C22×C8—42no—
88C4×Q8—416no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42#4114yes—
110#101—48no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—41no—
128#95—41no—
129C24C2×C22⋊C411yescharacteristic
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
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—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
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#9912yes—
202C22×C4#9912yes—
203C22×C4#10412yes—
204C22×C4#11712yes—
205C22×C4#12312yes—
206C22×C4#12312yes—
207C22×C4#13512yes—
208C22×C4AGL(1,ℤ/8)11yescharacteristic
209C22×C4#13711yescharacteristic
210M16—44no—
211M16—44no—
212M16—82no—
213M16—82no—
214C2×C8—44no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C42—28no—
224C42—44no—
225C42—44no—
226C42—48no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#38111yescharacteristic
232C23C2×#9911yescharacteristic
233C23C2×#9911yescharacteristic
234C23#41611yescharacteristic
235C23#41711yescharacteristic
236C23C2×#10411yescharacteristic
237C23#42111yescharacteristic
238Q8—48no—
239Q8—48no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
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—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
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—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
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#48212yes—
311C2×C4#48512yes—
312C8—84no—
313C8—162no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22#150111yescharacteristic
327C22#150411yescharacteristic
328C22#151311yescharacteristic
329C22#151611yescharacteristic
330C22#181911yescharacteristic
331C22#182511yescharacteristic
332C22#182611yescharacteristic
333C22#182711yescharacteristic
334C22#182711yescharacteristic
335C22#183011yescharacteristic
336C22#183011yescharacteristic
337C4—22no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—82no—
345C4—82no—
346C4—81no—
347C4—161no—
348C4—161no—
349C2—21no—
350C2—21no—
351C2—21no—
352C2—21no—
353C2#871511yescharacteristic
354C2#873911yescharacteristic
355C2#875411yescharacteristic
356C2#876811yescharacteristic
357C2#876911yescharacteristic
358C2#877411yescharacteristic
359C2#877511yescharacteristic
subgroupG/H
center, socleC23#381
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
#139612
#107632
#83032
#70372