Finite Group Databasebeta

← #213930

Subgroups — #213930

Order 512.

1038 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 186 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#36707C211yesmaximal, characteristic
2#21757C212yesmaximal, complemented
3#9513C214yesmaximal, complemented
4#9511C214yesmaximal, complemented
5#8755C212yesmaximal, complemented
6#8555C211yesmaximal, characteristic, complemented
7#8555C211yesmaximal, characteristic, complemented
8C2×#537C2211yescharacteristic
9C22×C22⋊Q8C2211yescharacteristic
10#2879C2214yes—
11#2847C412yes—
12#2692—24no—
13#2676—24no—
14#2618C412yes—
15#2539C412yes—
16#2436C412yes—
17#2406C2212yes—
18#2404C2212yes—
19#2313C2211yescharacteristic
20#1901—24no—
21#1900—24no—
22#1821C2212yes—
23#1821C2212yes—
24#1785C2214yes—
25#1463C2212yes—
26#1463C2212yes—
27#1462C2212yes—
28#1462C2212yes—
29#1306C2218yes—
30C22×#41C2312yes—
31C23×Q8C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C22×M16—24no—
34#537—42no—
35#537—42no—
36C2×#124—22no—
37C2×#123C2×C414yes—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—42no—
40C2×C22⋊Q8—42no—
41C2×C22⋊Q8D414yes—
42C2×#121—22no—
43C2×#121C2×C412yes—
44C2×#121C2×C412yes—
45C2×#121D414yes—
46C2×#120C2×C414yes—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2311yescharacteristic
52C2×#101—24no—
53C2×#100—24no—
54#408—44no—
55#408—44no—
56#407—24no—
57#407C2314yes—
58#407C2314yes—
59#399—22no—
60#398—22no—
61#396—24no—
62#395—22no—
63#394—22no—
64#394—22no—
65#393—22no—
66#393—24no—
67#392—22no—
68#392—22no—
69#391—22no—
70#390—24no—
71#387—24no—
72#386—24no—
73#386—24no—
74#386C2×C414yes—
75#386C2×C414yes—
76#385—24no—
77#381C2312yes—
78#380—22no—
79#380C2×C412yes—
80C4×C22⋊C4C2×C412yes—
81#343—24no—
82#343—24no—
83#324—42no—
84#324—42no—
85#323—42no—
86#323—42no—
87C25C22×C411yescharacteristic
88C2×#41—24no—
89C2×#41—24no—
90C2×#41—84no—
91C22×Q8—22no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—82no—
95C22×Q8C2×D412yes—
96C22×D4—24no—
97C22×D4C22⋊C412yes—
98C22×D4C22⋊C412yes—
99C22×D4C22×C414yes—
100C22×D4C2×D412yes—
101C23×C4—22no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—82no—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D414yes—
111C23×C4C2411yescharacteristic
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—44no—
115C2×M16—84no—
116C2×M16—84no—
117C22×C8—44no—
118#124—84no—
119#123—84no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—48no—
123C22⋊Q8—48no—
124#121—44no—
125#121—44no—
126#121—42no—
127#121—42no—
128#121—42no—
129#121—42no—
130#121—84no—
131#120—84no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C2×D414yes—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4C22⋊C414yes—
156C2×C22⋊C4C22⋊C414yes—
157C2×C22⋊C4C22×C414yes—
158C2×C42—22no—
159#101—84no—
160#100—84no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165C22⋊C8—48no—
166#95—22no—
167#95—24no—
168#95—22no—
169#95—24no—
170#95—22no—
171#95—22no—
172C24—21no—
173C24—21no—
174C24—21no—
175C24—42no—
176C24—42no—
177C24—81no—
178C24C2×C22⋊C414yes—
179C24C23×C411yescharacteristic
180#41—84no—
181#41—84no—
182#41—84no—
183#41—84no—
184C2×Q8—24no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—48no—
202C2×D4—84no—
203C2×D4—84no—
204C2×D4—82no—
205C2×D4—82no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C2×C22⋊C414yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C22×D412yes—
241C22×C4H(5,𝔽2)12yes—
242C22×C4H(5,𝔽2)12yes—
243M16—44no—
244M16—44no—
245M16—44no—
246M16—44no—
247M16—88no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—88no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—84no—
258C4⋊C4—82no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—48no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C22⋊C4—84no—
268C42—82no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—44no—
283C23—44no—
284C23—41no—
285C23—42no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—88no—
291C23C22×C22⋊C411yescharacteristic
292C23#51811yescharacteristic
293C23#51811yescharacteristic
294Q8—84no—
295Q8—88no—
296D4—84no—
297D4—84no—
298D4—84no—
299D4—162no—
300D4—162no—
301D4—168no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—88no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#53414yes—
340C8—84no—
341C8—84no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22—42no—
354C22—44no—
355C22—44no—
356C22—84no—
357C22—84no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—81no—
364C22—81no—
365C22#230811yescharacteristic
366C22#288312yes—
367C4—24no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—44no—
372C4—84no—
373C4—81no—
374C4—81no—
375C2—22no—
376C2—41no—
377C2—41no—
378C2—84no—
379C2—82no—
380C2#2175011yescharacteristic
381C2#2177811yescharacteristic
382C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radical#36707C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21757C2semidirect
3. #9513C2semidirect
4. #9511C2semidirect
5. #8755C2semidirect
6. #8555C2semidirect
7. #8555C2semidirect

Maximal subgroups

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

subgroupindex
#367072
#217572
#95132
#95112
#87552
#85552