Finite Group Databasebeta

← #134474

Subgroups — #134474

Order 512.

530 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 96 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#14226C211yesmaximal, characteristic
2C2×#1346C212yesmaximal
3#10802C211yesmaximal, characteristic
4#8558C211yesmaximal, characteristic
5#7251C212yesmaximal
6C2×#398C2211yescharacteristic
7C2×#389—21no—
8C2×#389C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13#1728—21no—
14C2×#324C2212yes—
15#1445C411yescharacteristic
16#1444C411yescharacteristic
17#1441C2211yescharacteristic
18#1436—21no—
19#1436—21no—
20#1436—21no—
21C4×#95C411yescharacteristic
22#1346—24no—
23#1280—21no—
24C22×M16—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C42—21no—
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38#398—21no—
39#398—21no—
40#398D412yes—
41#398Q812yes—
42#389—44no—
43#389—41no—
44#389—41no—
45C4×C4⋊C4—41no—
46C4×C4⋊C4—41no—
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×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#343—24no—
60#343—22no—
61#343—22no—
62#336—41no—
63#336—41no—
64#336—41no—
65#324—22no—
66#324—22no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16—42no—
84C22×C8—42no—
85C22×C8—42no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4C4⋊C412yes—
107C2×C4⋊C4C2×D411yescharacteristic
108C2×C4⋊C4C2×Q811yescharacteristic
109C2×C4⋊C4#4111yescharacteristic
110C2×C4⋊C4#4111yescharacteristic
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—41no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42—81no—
122C2×C42—81no—
123C2×C42—81no—
124C4⋊C8—82no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127#95—24no—
128#95—21no—
129#95—22no—
130#95—22no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—41no—
138#95—41no—
139#95—42no—
140#95—81no—
141#95C4⋊C412yes—
142C24C2×C22⋊C411yescharacteristic
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
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—22no—
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—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#9911yescharacteristic
202C22×C4#9911yescharacteristic
203C22×C4#10412yes—
204C22×C4C2×C4⋊C411yescharacteristic
205C22×C4#11711yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4#13512yes—
211C22×C4AGL(1,ℤ/8)12yes—
212M16—44no—
213M16—44no—
214M16—82no—
215M16—82no—
216C2×C8—44no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—82no—
234C42—42no—
235C42—42no—
236C42—42no—
237C42—81no—
238C42—81no—
239C42—81no—
240C42—84no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23#38011yescharacteristic
246C23C2×#9911yescharacteristic
247C23#41011yescharacteristic
248C23#41611yescharacteristic
249C23#41811yescharacteristic
250C23C2×#10411yescharacteristic
251C23#42111yescharacteristic
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
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—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
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#35612yes—
325C2×C4#46312yes—
326C2×C4#47812yes—
327C8—84no—
328C8—162no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22#150011yescharacteristic
342C22#150711yescharacteristic
343C22#151411yescharacteristic
344C22#151611yescharacteristic
345C22#179911yescharacteristic
346C22#180411yescharacteristic
347C22#180611yescharacteristic
348C22#212111yescharacteristic
349C22#212311yescharacteristic
350C22C2×#35611yescharacteristic
351C22#213211yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—44no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—82no—
363C4—81no—
364C4—81no—
365C4—82no—
366C4—161no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2#857311yescharacteristic
372C2#1178211yescharacteristic
373C2#1178311yescharacteristic
374C2#1185311yescharacteristic
375C2#1185411yescharacteristic
376C2#1279611yescharacteristic
377C2#1280211yescharacteristic
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 ◁ C2×C4×C4⋊C4 ◁ 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
#142262
C2×#13462
#108022
#85582
#72512