Finite Group Databasebeta

← #135368

Subgroups — #135368

Order 512.

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

normal subgroups only

This group has 646 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#13938C211yesmaximal, characteristic
2#10728C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7251C212yesmaximal, complemented
5#7251C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7C2×#407—21no—
8C2×#383C411yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10#1817—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13C2×#324C2212yes—
14C2×#323—21no—
15#1441C2211yescharacteristic
16#1441C2211yescharacteristic
17#1439C411yescharacteristic
18#1439C411yescharacteristic
19#1437—21no—
20#1436—21no—
21#1434—21no—
22C22⋊#95—21no—
23C24×C4D411yescharacteristic
24C22×M16—22no—
25C2×#117—24no—
26C2×#117—42no—
27C2×#117—42no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4D411yescharacteristic
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39#407—24no—
40#407—42no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—42no—
43#383—42no—
44C4×C4⋊C4—42no—
45C2×#95—21no—
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×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56#343—24no—
57#343—42no—
58#336—42no—
59#324—24no—
60#323—42no—
61C25C22⋊C411yescharacteristic
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—41no—
73C23×C4—41no—
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—
88#117—416no—
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#4114yes—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—81no—
110C2×C22⋊C4—81no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—41no—
117C2×C42—42no—
118C2×C42—81no—
119C2×C42—81no—
120C22⋊C8—48no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#95—41no—
124#95—41no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—41no—
132#95—41no—
133#95—81no—
134#95—81no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—41no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24#10412yes—
146C24C2×C22⋊C411yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4—24no—
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—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
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—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4#9912yes—
213C22×C4#9912yes—
214C22×C4#11712yes—
215C22×C4#12312yes—
216C22×C4#12312yes—
217C22×C4#13512yes—
218C22×C4AGL(1,ℤ/8)11yescharacteristic
219C22×C4#13711yescharacteristic
220M16—44no—
221M16—44no—
222M16—82no—
223M16—82no—
224C2×C8—44no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C42—28no—
241C42—44no—
242C42—84no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23#38111yescharacteristic
271C23C2×#9911yescharacteristic
272C23C2×#9911yescharacteristic
273C23#41611yescharacteristic
274C23#41711yescharacteristic
275C23C2×#10411yescharacteristic
276C23#42111yescharacteristic
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
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—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
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—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4#48212yes—
343C2×C4#48512yes—
344C8—84no—
345C8—162no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#150111yescharacteristic
370C22#150411yescharacteristic
371C22#151311yescharacteristic
372C22#151611yescharacteristic
373C22#181911yescharacteristic
374C22#182511yescharacteristic
375C22#182611yescharacteristic
376C22#212011yescharacteristic
377C22#212011yescharacteristic
378C22#213411yescharacteristic
379C22#213511yescharacteristic
380C4—22no—
381C4—22no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—81no—
387C4—82no—
388C4—82no—
389C4—161no—
390C4—161no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—82no—
396C2#871511yescharacteristic
397C2#1176111yescharacteristic
398C2#1176311yescharacteristic
399C2#1184711yescharacteristic
400C2#1184911yescharacteristic
401C2#1285311yescharacteristic
402C2#1286311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7251C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
#139382
#107282
#83002
#72512