Finite Group Databasebeta

← #416702

Subgroups — #416702

Order 512.

1248 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1248 conjugacy classes of subgroups — too many to draw. Only its 234 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#35181C211yesmaximal, characteristic, complemented
2#21204C212yesmaximal
3#14153C211yesmaximal, characteristic
4#11064C214yesmaximal, complemented
5#11063C214yesmaximal
6#10851C212yesmaximal, complemented
7C4×#343C211yesmaximal, characteristic, complemented
8C22×#124C2211yescharacteristic
9C2×#419C2214yes—
10C2×#418C2214yes—
11C2×C4×M16C2212yes—
12#2851—22no—
13#2646—22no—
14#2571—22no—
15#2570—24no—
16#2552—22no—
17#2317C2214yes—
18C2×#391C2212yes—
19C2×#390C2212yes—
20#2285C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1959—22no—
24#1959—22no—
25#1958—24no—
26C2×#343C2214yes—
27C2×#339C2214yes—
28C2×#339C2214yes—
29#1439—22no—
30#1437—22no—
31C23×Q8C2311yescharacteristic
32C23×D4C2311yescharacteristic
33C22×M16C2314yes—
34C2×#125—22no—
35C2×#124—22no—
36C2×#124—22no—
37C2×#124—22no—
38C2×#124—24no—
39C2×#124Q814yes—
40C2×#117C2314yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2312yes—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—24no—
46C22×C42D412yes—
47C22×C42C2312yes—
48C22×C42C2311yescharacteristic
49#419—28no—
50#419—44no—
51#418—28no—
52#418—44no—
53C2×#103—24no—
54C2×#103—24no—
55C2×#102—24no—
56C2×#102—24no—
57C4×M16—24no—
58C4×M16—24no—
59C2×#97—22no—
60C2×C4×C8—22no—
61#400—28no—
62#398—28no—
63#391—44no—
64#390—44no—
65#389—24no—
66#389—24no—
67#389—24no—
68#387—28no—
69#385—28no—
70#384—24no—
71#384D412yes—
72#384D412yes—
73#382—22no—
74#382—24no—
75#382D414yes—
76#380—22no—
77#380—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—24no—
83C2×#95—21no—
84C2×#95—24no—
85C2×#95D412yes—
86#343—28no—
87#339—44no—
88#339—44no—
89C25—21no—
90C22×Q8—21no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—24no—
94C22×Q8—24no—
95C22×Q8C2×Q812yes—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—24no—
99C22×D4—21no—
100C22×D4—24no—
101C22×D4C2×Q812yes—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4C2×D414yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16#4118yes—
114C22×C8—24no—
115#125—84no—
116#124—28no—
117#124—28no—
118#124—44no—
119#124—44no—
120#117—24no—
121#117—24no—
122#117—24no—
123#117—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—48no—
147C2×C22⋊C4—48no—
148C2×C22⋊C4—84no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—22no—
155C2×C42—82no—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×Q812yes—
159C2×C42#4114yes—
160C2×C42#4112yes—
161#103—48no—
162#103—48no—
163#102—48no—
164#102—48no—
165#97—44no—
166C4×C8—44no—
167#95—28no—
168#95—28no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—48no—
173#95—42no—
174C24—22no—
175C24—44no—
176C24—42no—
177C24—42no—
178C24—41no—
179C24—41no—
180C24—42no—
181C24—42no—
182C24C22×D411yescharacteristic
183C2×Q8—24no—
184C2×Q8—28no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×D4—24no—
192C2×D4—28no—
193C2×D4—42no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4#12011yescharacteristic
222C22×C4#12012yes—
223C22×C4#12011yescharacteristic
224C22×C4C22⋊Q812yes—
225C22×C4C22⋊Q812yes—
226C22×C4#12412yes—
227C22×C4#12412yes—
228C22×C4C22×D412yes—
229C22×C4C22×Q811yescharacteristic
230C22×C4C2×#4112yes—
231C22×C4C2×#4111yescharacteristic
232C22×C4C2×#4114yes—
233C22×C4H(5,𝔽2)14yes—
234M16—28no—
235M16—28no—
236M16—44no—
237M16—44no—
238C2×C8—28no—
239C2×C8—44no—
240C2×C8—44no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—48no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—88no—
253C4⋊C4—88no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—48no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—816no—
260C42—44no—
261C42—44no—
262C42—44no—
263C42—48no—
264C42—82no—
265C42C22⋊Q814yes—
266C42C22⋊Q814yes—
267C42#12414yes—
268C42#12414yes—
269C23—22no—
270C23—21no—
271C23—21no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23—42no—
276C23—84no—
277C23—84no—
278C23—84no—
279C23—84no—
280C23—82no—
281C23—82no—
282C23—81no—
283C23—81no—
284C23—82no—
285C23—82no—
286C23—81no—
287C23—81no—
288C23C2×#12011yescharacteristic
289C23C2×C22⋊Q811yescharacteristic
290C23C2×#12411yescharacteristic
291C23#53212yes—
292C23#53512yes—
293Q8—48no—
294Q8—84no—
295D4—48no—
296D4—84no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4#39314yes—
331C2×C4#39312yes—
332C2×C4#39312yes—
333C2×C4C2×C22⋊Q812yes—
334C2×C4C2×#12412yes—
335C2×C4#52914yes—
336C2×C4#54014yes—
337C2×C4#54614yes—
338C2×C4#55014yes—
339C2×C4#55314yes—
340C2×C4#55414yes—
341C8—48no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—88no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—162no—
355C22—162no—
356C22—162no—
357C22—162no—
358C22C2×#39311yescharacteristic
359C22#243112yes—
360C22#256612yes—
361C22#264111yescharacteristic
362C22#286111yescharacteristic
363C22#301812yes—
364C22#301911yescharacteristic
365C22#302011yescharacteristic
366C4—24no—
367C4—22no—
368C4—22no—
369C4—84no—
370C4—84no—
371C4—84no—
372C4—84no—
373C4—82no—
374C4#245414yes—
375C4#249014yes—
376C2—21no—
377C2—22no—
378C2—21no—
379C2—162no—
380C2#1522911yescharacteristic
381C2#2346111yescharacteristic
382C2#2346211yescharacteristic
383C2#2562512yes—
384C2#2562612yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
4. #11064C2semidirect
6. #10851C2semidirect
7. C4×#343C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#212042
#141532
#110642
#110632
#108512
C4×#3432