Finite Group Databasebeta

← #142210

Subgroups — #142210

Order 512.

572 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 90 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#15017C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7250C212yesmaximal
5#7250C212yesmaximal
6C2×#382C411yescharacteristic, complemented
7C2×C4×C22⋊C4C2211yescharacteristic
8#1797—21no—
9C2×#342—21no—
10C2×#342C2212yes—
11C2×#323C2212yes—
12#1444C411yescharacteristic, complemented
13#1443—21no—
14#1442C411yescharacteristic, complemented
15#1439—21no—
16#1439C2211yescharacteristic
17#1439C2211yescharacteristic
18#1436—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×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36#382—41no—
37#382—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—22no—
56#342—42no—
57#342Q814yes—
58#336—41no—
59#323—24no—
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—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—81no—
83C2×C4⋊C4—81no—
84C2×C4⋊C4—81no—
85C2×C4⋊C4—81no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
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—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—81no—
103C2×C22⋊C4—81no—
104C2×C22⋊C4—81no—
105C2×C22⋊C4C2×Q812yes—
106C2×C22⋊C4#4112yes—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—42no—
112C2×C42—81no—
113C4⋊C8—82no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—41no—
126#95—41no—
127#95—81no—
128#95—81no—
129#95—81no—
130#95—82no—
131#95—82no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—41no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24C2×C22⋊C411yescharacteristic
141C24C22⋊Q812yes—
142C24#13512yes—
143C24AGL(1,ℤ/8)11yescharacteristic
144C24#13711yescharacteristic
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—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
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—41no—
178C22×C4—41no—
179C22×C4—41no—
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—44no—
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—81no—
206C22×C4—81no—
207C22×C4—84no—
208C22×C4—84no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4#9911yescharacteristic
216C22×C4#9911yescharacteristic
217C22×C4C4×Q812yes—
218C22×C4#12411yescharacteristic
219C22×C4C4⋊Q811yescharacteristic
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—84no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—84no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C42—42no—
238C42—42no—
239C42—42no—
240C42—81no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23#39111yescharacteristic
272C23C2×#9911yescharacteristic
273C23#41011yescharacteristic
274C23#41611yescharacteristic
275C23#41711yescharacteristic
276C23#42111yescharacteristic
277C23#42111yescharacteristic
278C23#47812yes—
279C23#47912yes—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
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—84no—
313C2×C4—81no—
314C2×C4—84no—
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—84no—
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—84no—
339C2×C4—84no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—162no—
348C2×C4—162no—
349C2×C4—162no—
350C2×C4—162no—
351C2×C4#35212yes—
352C8—162no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—22no—
367C22—42no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—81no—
379C22—81no—
380C22—82no—
381C22#151211yescharacteristic
382C22#151311yescharacteristic
383C22#151611yescharacteristic
384C22#152111yescharacteristic
385C22#198911yescharacteristic
386C22#199511yescharacteristic
387C22#199611yescharacteristic
388C22C2×#35211yescharacteristic
389C22#212211yescharacteristic
390C22#212611yescharacteristic
391C22#212711yescharacteristic
392C4—22no—
393C4—41no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—161no—
402C4—164no—
403C4—161no—
404C4—161no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—42no—
410C2—42no—
411C2#991311yescharacteristic
412C2#1182211yescharacteristic
413C2#1182511yescharacteristic
414C2#1184111yescharacteristic
415C2#1184411yescharacteristic
416C2#1329511yescharacteristic
417C2#1330011yescharacteristic
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×#382C4semidirect
12. #1444C4semidirect
14. #1442C4semidirect
19. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#150172
#108012
#85572
#72502