Finite Group Databasebeta

← #135362

Subgroups — #135362

Order 512.

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

normal subgroups only

This group has 680 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#13957C211yesmaximal, characteristic
2#10726C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7250C212yesmaximal
5#7250C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#382C411yescharacteristic, complemented
8C2×#381—21no—
9C2×#376C411yescharacteristic, complemented
10#1817—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#324—21no—
14C2×#323C2212yes—
15#1441C411yescharacteristic, complemented
16#1441C411yescharacteristic, complemented
17#1439C2211yescharacteristic
18#1439C2211yescharacteristic
19#1437—21no—
20#1436—21no—
21#1434—21no—
22C22⋊#95—21no—
23C24×C4D411yescharacteristic
24C2×#117—24no—
25C2×#117—42no—
26C2×#117—42no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C22⋊C4—22no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4D411yescharacteristic
35C22×C22⋊C4C2311yescharacteristic
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C2×C22⋊C8—22no—
40#382—42no—
41#381—24no—
42#381—42no—
43#376—42no—
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×#95—21no—
53C2×#95—42no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56#342—24no—
57#342—42no—
58#336—42no—
59#324—42no—
60#323—24no—
61C25C2×D411yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—22no—
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×C4—42no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—42no—
82#117—416no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4—81no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—81no—
110C2×C22⋊C4—81no—
111C2×C22⋊C4#4114yes—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—41no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
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—48no—
131#95—42no—
132#95—42no—
133#95—81no—
134#95—81no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24C2×C22⋊C411yescharacteristic
144C24#12312yes—
145C24#13512yes—
146C24AGL(1,ℤ/8)11yescharacteristic
147C24#13711yescharacteristic
148C22×C4—24no—
149C22×C4—24no—
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—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
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—44no—
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—41no—
215C22×C4—41no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4#9912yes—
223C22×C4#9912yes—
224C22×C4#10412yes—
225C22×C4#11712yes—
226C22×C4#12312yes—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—84no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C42—28no—
244C42—44no—
245C42—48no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—21no—
258C23—42no—
259C23—41no—
260C23—41no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23#38111yescharacteristic
277C23C2×#9911yescharacteristic
278C23C2×#9911yescharacteristic
279C23#41611yescharacteristic
280C23#41711yescharacteristic
281C23C2×#10411yescharacteristic
282C23#42111yescharacteristic
283C23#48212yes—
284C23#48512yes—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
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—84no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—162no—
353C2×C4—162no—
354C2×C4—162no—
355C2×C4—162no—
356C8—162no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—82no—
382C22—81no—
383C22—81no—
384C22—82no—
385C22#150111yescharacteristic
386C22#150411yescharacteristic
387C22#151311yescharacteristic
388C22#151611yescharacteristic
389C22#181911yescharacteristic
390C22#182511yescharacteristic
391C22#182611yescharacteristic
392C22#212211yescharacteristic
393C22#212311yescharacteristic
394C22#212611yescharacteristic
395C22#212611yescharacteristic
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—82no—
402C4—82no—
403C4—81no—
404C4—164no—
405C4—161no—
406C4—161no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—42no—
413C2#871511yescharacteristic
414C2#1175811yescharacteristic
415C2#1176011yescharacteristic
416C2#1184411yescharacteristic
417C2#1184611yescharacteristic
418C2#1285511yescharacteristic
419C2#1285711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#382C4semidirect
9. C2×#376C4semidirect
15. #1441C4semidirect
16. #1441C4semidirect

Maximal subgroups

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

subgroupindex
#139572
#107262
#83002
#72502