Finite Group Databasebeta

← #111908

Subgroups — #111908

Order 512.

1282 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1282 conjugacy classes of subgroups — too many to draw. Only its 158 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×#2548C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic, complemented
3C2×#1555C212yesmaximal
4C2×#1467C211yesmaximal, characteristic, complemented
5C2×#1285C212yesmaximal, complemented
6C22×#127C2211yescharacteristic
7C22×#124C412yes—
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2212yes—
10C2×#432C2212yes—
11#2548—24no—
12#2548—24no—
13C2×#394—22no—
14C2×#390—22no—
15C2×#390C412yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1985—24no—
18#1817—22no—
19#1555—24no—
20#1555—24no—
21C2×#331—24no—
22#1467—24no—
23C2×#324—22no—
24#1285—24no—
25C23×Q8C2×C411yescharacteristic
26C23×D4—21no—
27C23×D4—21no—
28C23×D4C2×C412yes—
29C2×#127—22no—
30C2×#127—22no—
31C2×#127—22no—
32C2×#127—22no—
33C2×#127—22no—
34C2×#127D414yes—
35C2×#124—24no—
36C2×#124—24no—
37C2×#124—48no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—24no—
49C22×C42C2×C412yes—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#432—22no—
55#432—22no—
56#432—22no—
57#432—22no—
58#432—22no—
59#432D414yes—
60C2×C4⋊C8—22no—
61C2×C4⋊C8D414yes—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×#97—22no—
65C2×#97—22no—
66#394—216no—
67#390—216no—
68#390—24no—
69#390—24no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C2×#95—21no—
73#336—28no—
74#331—216no—
75#324—28no—
76C25—22no—
77C22×Q8—24no—
78C22×Q8—21no—
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—42no—
83C22×D4—22no—
84C22×D4—24no—
85C22×D4—24no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—44no—
89C22×D4—44no—
90C22×D4—48no—
91C22×D4—41no—
92C22×D4—41no—
93C22×D4—42no—
94C22×D4—42no—
95C22×D4—41no—
96C22×D4—41no—
97C22×D4—84no—
98C22×D4C22⋊C414yes—
99C23×C4—22no—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C22×C8—24no—
106C22×C8—42no—
107#127—24no—
108#127—24no—
109#127—24no—
110#127—24no—
111#127—42no—
112#127—42no—
113#127—42no—
114#127—42no—
115#124—416no—
116#124—416no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×Q8—42no—
122C4×Q8—42no—
123C4×Q8—42no—
124C4×Q8—42no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—44no—
135C2×C22⋊C4—216no—
136C2×C22⋊C4—216no—
137C2×C22⋊C4—48no—
138C2×C22⋊C4—48no—
139C2×C22⋊C4—88no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—24no—
143C2×C42—24no—
144C2×C42—22no—
145C2×C42—44no—
146C2×C42C22⋊C414yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C2×C42#4112yes—
152C2×C42#4112yes—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—24no—
156C4⋊C8—42no—
157C22⋊C8—24no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161#97—24no—
162#97—42no—
163#97—42no—
164#97—42no—
165#95—24no—
166C24—24no—
167C24—48no—
168C24—48no—
169C24—42no—
170C24—42no—
171C24—44no—
172C24—44no—
173C24C2×C22⋊C411yescharacteristic
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—48no—
177C2×Q8—48no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×D4—28no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—416no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—88no—
194C2×D4—88no—
195C2×D4—84no—
196C2×D4—88no—
197C2×D4—88no—
198C2×D4—88no—
199C2×D4—84no—
200C2×D4—84no—
201C2×D4—84no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4#10014yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C4×D412yes—
224C22×C4C4×D412yes—
225C22×C4#12012yes—
226C22×C4#12011yescharacteristic
227C22×C4#12012yes—
228C22×C4#12011yescharacteristic
229C22×C4#12112yes—
230C22×C4#12112yes—
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—48no—
236C2×C8—84no—
237C4⋊C4—48no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C22⋊C4—432no—
243C22⋊C4—816no—
244C42—44no—
245C42—44no—
246C42—44no—
247C42—44no—
248C42#12012yes—
249C42#12112yes—
250C42#12112yes—
251C42#12112yes—
252C42#12112yes—
253C42#12312yes—
254C42#12412yes—
255C42#12712yes—
256C23—22no—
257C23—22no—
258C23—48no—
259C23—48no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—84no—
264C23—84no—
265C23—816no—
266C23—816no—
267C23—84no—
268C23—84no—
269C23—84no—
270C23—84no—
271C23#37911yescharacteristic
272C23#38611yescharacteristic
273C23#38611yescharacteristic
274C23C2×#10012yes—
275C23#41312yes—
276Q8—48no—
277Q8—48no—
278D4—88no—
279D4—88no—
280D4—88no—
281D4—88no—
282D4—88no—
283D4—88no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—88no—
311C2×C4—82no—
312C2×C4#38612yes—
313C2×C4#39012yes—
314C2×C4#39212yes—
315C2×C4#39212yes—
316C2×C4#39412yes—
317C2×C4#39412yes—
318C2×C4#45414yes—
319C8—84no—
320C8—84no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—24no—
326C22—24no—
327C22—88no—
328C22—88no—
329C22—84no—
330C22—84no—
331C22—84no—
332C22—168no—
333C22—168no—
334C22#143811yescharacteristic
335C22#178311yescharacteristic
336C22#178511yescharacteristic
337C22#188812yes—
338C22#190012yes—
339C22#190012yes—
340C22#191012yes—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—44no—
347C4—84no—
348C4—82no—
349C4#200314yes—
350C2—21no—
351C2—21no—
352C2—22no—
353C2—164no—
354C2#945814yes—
355C2#1096711yescharacteristic
356C2#1097611yescharacteristic
357C2#1098711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21822C2semidirect
4. C2×#1467C2semidirect
5. C2×#1285C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
#218222
C2×#15552
C2×#14672
C2×#12852