Finite Group Databasebeta

← #135359

Subgroups — #135359

Order 512.

626 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 92 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#13850C211yesmaximal, characteristic
2#10763C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7251C212yesmaximal
5#7251C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C22×#101—21no—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×#376C411yescharacteristic
10#1817—21no—
11C2×#343C2212yes—
12#1728—21no—
13C2×#324—21no—
14C2×#324C2212yes—
15#1446—21no—
16#1445—21no—
17#1441C411yescharacteristic
18#1441C411yescharacteristic
19#1441C2211yescharacteristic
20#1441C2211yescharacteristic
21C4×#95—21no—
22C4×#95—21no—
23C23×Q8D411yescharacteristic
24C22×M16—22no—
25C2×C4×Q8—24no—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C2×C4×Q8—42no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4C2311yescharacteristic
34C22×C42—21no—
35C22×C42—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C2×#101—24no—
40C2×#101—42no—
41C2×#101—42no—
42C2×C22⋊C8—22no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45#376—42no—
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×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58#343—24no—
59#336—41no—
60#336—41no—
61#336—41no—
62#336—41no—
63#324—24no—
64#324—42no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—22no—
69C22×Q8—21no—
70C22×Q8—21no—
71C22×Q8—41no—
72C22×Q8—41no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C2×M16—24no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C22×C8—42no—
91C22×C8—42no—
92C4×Q8—48no—
93C4×Q8—48no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4#4114yes—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—22no—
111C2×C42—24no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—41no—
125C2×C42—41no—
126#101—48no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#95—41no—
130#95—41no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—41no—
141#95—41no—
142C24C2×C22⋊C411yescharacteristic
143C2×Q8—24no—
144C2×Q8—24no—
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
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—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
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—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4#9912yes—
215C22×C4#9911yescharacteristic
216C22×C4#9911yescharacteristic
217C22×C4#10412yes—
218C22×C4#11712yes—
219C22×C4#12312yes—
220C22×C4#12312yes—
221C22×C4#13512yes—
222C22×C4AGL(1,ℤ/8)11yescharacteristic
223C22×C4#13711yescharacteristic
224M16—44no—
225M16—44no—
226M16—82no—
227M16—82no—
228C2×C8—44no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C42—24no—
242C42—24no—
243C42—42no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—48no—
248C42—82no—
249C42—82no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23#38111yescharacteristic
255C23C2×#9911yescharacteristic
256C23C2×#9911yescharacteristic
257C23#41611yescharacteristic
258C23#41711yescharacteristic
259C23C2×#10411yescharacteristic
260C23#42111yescharacteristic
261Q8—48no—
262Q8—48no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
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—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4#35512yes—
335C2×C4#35612yes—
336C2×C4#48212yes—
337C2×C4#48512yes—
338C8—84no—
339C8—162no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22#150111yescharacteristic
353C22#150411yescharacteristic
354C22#151311yescharacteristic
355C22#151611yescharacteristic
356C22#181911yescharacteristic
357C22#182511yescharacteristic
358C22#182611yescharacteristic
359C22#212011yescharacteristic
360C22#212011yescharacteristic
361C22C2×#35511yescharacteristic
362C22C2×#35611yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—82no—
372C4—82no—
373C4—82no—
374C4—81no—
375C4—82no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2#871511yescharacteristic
381C2#1173411yescharacteristic
382C2#1173611yescharacteristic
383C2#1182911yescharacteristic
384C2#1183111yescharacteristic
385C2#1285311yescharacteristic
386C2#1286011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#138502
#107632
#83032
#72512