Finite Group Databasebeta

← #149486

Subgroups — #149486

Order 512.

848 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C2×#2809C211yesmaximal, characteristic
2C2×#1394C212yesmaximal
3C2×#1394C211yesmaximal, characteristic
4#10750C212yesmaximal
5#10750C211yesmaximal, characteristic
6C2×#519C2211yescharacteristic
7C22×#101C2212yes—
8C22×#101C2211yescharacteristic
9#2809—22no—
10#2809—24no—
11#2809C414yescomplemented
12C2×#400—21no—
13C2×#398—22no—
14C2×#391C2212yes—
15C2×#391C2211yescharacteristic
16C2×#389—21no—
17C2×#387—22no—
18#1736—22no—
19#1736—21no—
20C2×#339—22no—
21C2×#339—21no—
22#1394—22no—
23#1394—24no—
24C23×Q8C2311yescharacteristic
25C22×M16—22no—
26C22×M16—21no—
27#519—22no—
28#519—21no—
29#519—42no—
30#519Q812yes—
31C2×C4×Q8—42no—
32C2×#117—22no—
33C2×#117—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D411yescharacteristic
40C22×C42D412yes—
41C22×C42D411yescharacteristic
42C2×#101—22no—
43C2×#101—22no—
44C2×#101—22no—
45C2×#101—22no—
46C2×#101—22no—
47C2×#101—22no—
48C2×#101—22no—
49C2×#101—21no—
50C2×#101Q814yes—
51C2×#101Q812yes—
52#400—24no—
53#400—24no—
54#398—28no—
55#398—28no—
56#391—22no—
57#391—22no—
58#391—22no—
59#391C2×C412yes—
60#391C2×C414yes—
61#391D412yes—
62#391D414yes—
63#389—24no—
64#389—24no—
65#387—28no—
66#387—28no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72#340—44no—
73#340—42no—
74#339—42no—
75#339—44no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8C22×C411yescharacteristic
81C22×Q8C2×D412yes—
82C22×Q8C2×D411yescharacteristic
83C22×Q8C2×Q811yescharacteristic
84C22×Q8C2×Q812yes—
85C22×Q8C2×Q811yescharacteristic
86C22×Q8#4111yescharacteristic
87C22×Q8#4112yes—
88C22×Q8#4111yescharacteristic
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C2×M16—24no—
96C2×M16—22no—
97C2×M16—41no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—41no—
105C2×M16—42no—
106C2×M16—41no—
107C22×C8—42no—
108C22×C8—41no—
109C4×Q8—44no—
110C4×Q8—44no—
111C4×Q8—48no—
112#117—22no—
113#117—24no—
114#117—24no—
115#117—22no—
116#117—41no—
117#117—42no—
118#117—42no—
119#117—41no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—48no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—81no—
135C2×C4⋊C4—81no—
136C2×C4⋊C4—81no—
137C2×C4⋊C4—81no—
138C2×C4⋊C4C22⋊C414yes—
139C2×C4⋊C4C22⋊C412yes—
140C2×C22⋊C4—41no—
141C2×C42—22no—
142C2×C42—21no—
143C2×C42—42no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42—42no—
147C2×C42—41no—
148C2×C42—42no—
149C2×C42C22⋊C414yes—
150C2×C42C22⋊C412yes—
151#101—24no—
152#101—24no—
153#101—24no—
154#101—42no—
155#101—42no—
156#101—42no—
157#95—24no—
158#95—28no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165C24#12011yescharacteristic
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—42no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—41no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—41no—
184C2×Q8C4×Q811yescharacteristic
185C2×Q8C4×Q812yes—
186C2×Q8C4×Q811yescharacteristic
187C2×Q8C22⋊Q812yes—
188C2×Q8C22⋊Q812yes—
189C2×Q8C22⋊Q812yes—
190C2×Q8#12412yes—
191C2×Q8#12411yescharacteristic
192C2×Q8C4⋊Q812yes—
193C2×Q8C4⋊Q811yescharacteristic
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4#12012yes—
225C22×C4#12011yescharacteristic
226C22×C4C22⋊Q812yes—
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q812yes—
229C22×C4C22⋊Q811yescharacteristic
230C22×C4#12412yes—
231C22×C4#12411yescharacteristic
232C22×C4C4⋊Q812yes—
233C22×C4C4⋊Q811yescharacteristic
234M16—42no—
235M16—44no—
236M16—44no—
237M16—42no—
238M16—81no—
239M16—82no—
240M16—81no—
241M16—82no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—81no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—81no—
248C4⋊C4—24no—
249C4⋊C4—28no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—416no—
253C4⋊C4—48no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—82no—
257C4⋊C4—81no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—42no—
263C42—44no—
264C42—42no—
265C42—44no—
266C42—42no—
267C42—44no—
268C42—81no—
269C42—82no—
270C23—22no—
271C23—22no—
272C23#37911yescharacteristic
273C23#39311yescharacteristic
274C23#39511yescharacteristic
275C23B(4,𝔽2)11yescharacteristic
276C23B(4,𝔽2)11yescharacteristic
277C23#45811yescharacteristic
278C23#45811yescharacteristic
279Q8—41no—
280Q8—41no—
281Q8—42no—
282Q8—42no—
283Q8—42no—
284Q8—42no—
285Q8—42no—
286Q8—41no—
287Q8—41no—
288Q8—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—82no—
330C2×C4—84no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—84no—
335C2×C4#39112yes—
336C2×C4#39111yescharacteristic
337C2×C4#39112yes—
338C2×C4#39111yescharacteristic
339C2×C4#39312yes—
340C2×C4#39311yescharacteristic
341C2×C4#39512yes—
342C2×C4#39511yescharacteristic
343C8—82no—
344C8—84no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#144611yescharacteristic
354C22#179011yescharacteristic
355C22#179011yescharacteristic
356C22#202911yescharacteristic
357C22#203011yescharacteristic
358C22#206211yescharacteristic
359C22#206211yescharacteristic
360C22#220312yes—
361C22#220412yes—
362C4—21no—
363C4—22no—
364C4—21no—
365C4—22no—
366C4—21no—
367C4—22no—
368C4—21no—
369C4—22no—
370C4—44no—
371C4—84no—
372C4—84no—
373C4—82no—
374C4—84no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2#1139011yescharacteristic
380C2#1277511yescharacteristic
381C2#1277611yescharacteristic
382C2#1337912yes—
383C2#1346512yes—
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC22×Q8C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24#120
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2809C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#13942
#107502