Finite Group Databasebeta

← #142033

Subgroups — #142033

Order 512.

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

normal subgroups only

This group has 584 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#15045C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7250C212yesmaximal
5#7250C212yesmaximal
6C2×#389C411yescharacteristic, complemented
7C2×C4×C22⋊C4C2211yescharacteristic
8C4×C22⋊C8—21no—
9C2×#342—21no—
10C2×#342C2212yes—
11C2×#323C2212yes—
12#1442—21no—
13#1442C411yescharacteristic, complemented
14#1442C411yescharacteristic, complemented
15#1439—21no—
16#1439C2211yescharacteristic
17#1439C2211yescharacteristic
18#1434—21no—
19#1434C411yescharacteristic, complemented
20C22⋊#95—21no—
21C22⋊#95—21no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—22no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4C2311yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C4×C8—41no—
36#389—41no—
37#389—41no—
38C4×C22⋊C4—41no—
39C4×C22⋊C4—41no—
40C2×#95—21no—
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—41no—
50C2×#95—41no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95D411yescharacteristic
55#342—24no—
56#342—24no—
57#336—41no—
58#323—22no—
59#323Q814yes—
60C25C2×D411yescharacteristic
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—41no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—81no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—42no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—81no—
84C2×C4⋊C4—81no—
85C2×C4⋊C4—81no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—81no—
101C2×C22⋊C4—81no—
102C2×C22⋊C4—81no—
103C2×C22⋊C4C2×Q812yes—
104C2×C22⋊C4#4112yes—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—42no—
110C2×C42—81no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C4×C8—82no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—41no—
122#95—41no—
123#95—42no—
124#95—81no—
125#95—81no—
126#95—81no—
127#95—81no—
128#95—82no—
129#95—82no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24C2×C22⋊C411yescharacteristic
139C24#12411yescharacteristic
140C24C4⋊Q811yescharacteristic
141C24#13512yes—
142C24#13512yes—
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—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
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—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—44no—
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—81no—
203C22×C4—81no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4#9911yescharacteristic
213C22×C4#9911yescharacteristic
214C22×C4#10412yes—
215C22×C4C4×Q812yes—
216C22×C4C22⋊Q812yes—
217C2×C8—44no—
218C2×C8—82no—
219C2×C8—82no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C4⋊C4—84no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—84no—
230C22⋊C4—82no—
231C22⋊C4—84no—
232C22⋊C4—82no—
233C42—42no—
234C42—81no—
235C42—81no—
236C42—81no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23#39111yescharacteristic
268C23C2×#9911yescharacteristic
269C23#41011yescharacteristic
270C23#41611yescharacteristic
271C23#41611yescharacteristic
272C23C2×#10411yescharacteristic
273C23#42111yescharacteristic
274C23#48812yes—
275C23#49912yes—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
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—84no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—84no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—84no—
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—84no—
335C2×C4—84no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—162no—
342C2×C4—162no—
343C2×C4—162no—
344C2×C4—162no—
345C2×C4#35412yes—
346C8—84no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—82no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22#150011yescharacteristic
376C22#150111yescharacteristic
377C22#151611yescharacteristic
378C22#152111yescharacteristic
379C22#198611yescharacteristic
380C22#199111yescharacteristic
381C22#199611yescharacteristic
382C22#212011yescharacteristic
383C22#212111yescharacteristic
384C22C2×#35411yescharacteristic
385C22#212811yescharacteristic
386C4—22no—
387C4—41no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—82no—
392C4—81no—
393C4—81no—
394C4—161no—
395C4—161no—
396C4—164no—
397C4—161no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—42no—
403C2—42no—
404C2#988611yescharacteristic
405C2#1172811yescharacteristic
406C2#1172911yescharacteristic
407C2#1184011yescharacteristic
408C2#1184511yescharacteristic
409C2#1329411yescharacteristic
410C2#1329911yescharacteristic
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 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#389C4semidirect
13. #1442C4semidirect
14. #1442C4semidirect
19. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#150452
#108012
C4×#3232
#72502