Finite Group Databasebeta

← #111134

Subgroups — #111134

Order 512.

850 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 166 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#20534C211yesmaximal, characteristic, complemented
2C2×#1527C212yesmaximal
3#14288C211yesmaximal, characteristic
4C4×#343C211yesmaximal, characteristic, complemented
5C2×#1288C212yesmaximal, complemented
6C22×#124C2211yescharacteristic
7C2×#432C2212yes—
8C2×C4×M16C2212yes—
9C2×#383C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1771—24no—
12#1749—21no—
13C4×C22⋊C8—21no—
14C2×#343—22no—
15#1527—24no—
16#1527C418yescomplemented
17C2×#329—24no—
18#1446C412yes—
19#1439—22no—
20#1438C412yes—
21#1437—21no—
22#1437—21no—
23#1288—24no—
24C23×Q8C2×C411yescharacteristic
25C23×D4C2×C411yescharacteristic
26C22×M16—22no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124—22no—
30C2×#124—22no—
31C2×#124—22no—
32C2×#124C2×C414yes—
33C22×C4⋊C4D412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—22no—
37C22×C22⋊C4—22no—
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42C2311yescharacteristic
41#432—22no—
42#432—22no—
43#432C2×C414yes—
44#432D414yes—
45C2×C4⋊C8—22no—
46C2×C22⋊C8—22no—
47C4×M16—22no—
48C4×M16—22no—
49C4×M16—22no—
50C4×M16C2×C414yes—
51C2×#97—22no—
52C2×#97—22no—
53C2×#97D414yes—
54C2×C4×C8—22no—
55C2×C4×C8—22no—
56#383—22no—
57#383—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95C2×C412yes—
65C2×#95C2×C412yes—
66#343—28no—
67#336—22no—
68#336—22no—
69#329—216no—
70C25—21no—
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—82no—
75C22×Q8C4212yes—
76C22×D4—21no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—82no—
80C22×D4C4212yes—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C2×M16—28no—
89C2×M16—24no—
90C2×M16—44no—
91C2×M16—42no—
92C22×C8—24no—
93C22×C8—42no—
94#124—24no—
95#124—24no—
96#124—24no—
97#124—24no—
98#124—42no—
99#124—42no—
100#124—42no—
101#124—42no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—42no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—82no—
116C2×C22⋊C4—82no—
117C2×C22⋊C4—82no—
118C2×C22⋊C4—82no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—81no—
124C2×C42—81no—
125C2×C42C22×C412yes—
126C2×C42C2×D412yes—
127C2×C42C2×D412yes—
128C2×C42#4112yes—
129C2×C42#4112yes—
130C2×C42#4112yes—
131C4⋊C8—24no—
132C4⋊C8—24no—
133C22⋊C8—24no—
134C22⋊C8—24no—
135#97—24no—
136#97—24no—
137#97—42no—
138#97—82no—
139#97C22⋊C418yescomplemented
140C4×C8—24no—
141C4×C8—42no—
142C4×C8—82no—
143#95—24no—
144#95—24no—
145#95—42no—
146#95—41no—
147#95—41no—
148#95—82no—
149#95—82no—
150C24—22no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24—41no—
156C24—41no—
157C24—42no—
158C24—42no—
159C24C2×C22⋊C411yescharacteristic
160C2×Q8—24no—
161C2×Q8—42no—
162C2×Q8—84no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—84no—
166C2×D4—24no—
167C2×D4—42no—
168C2×D4—84no—
169C2×D4—82no—
170C2×D4—84no—
171C2×D4—82no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4#10414yes—
199C22×C4C2×C4211yescharacteristic
200C22×C4#11711yescharacteristic
201C22×C4C4×D412yes—
202C22×C4C4×D412yes—
203C22×C4C4×D412yes—
204C22×C4C4×D412yes—
205C22×C4#12011yescharacteristic
206C22×C4#12112yes—
207C22×C4#12112yes—
208C22×C4#12311yescharacteristic
209C22×C4#12411yescharacteristic
210C22×C4#12711yescharacteristic
211M16—28no—
212M16—48no—
213M16—44no—
214C2×C8—28no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—48no—
218C2×C8—44no—
219C4⋊C4—48no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—88no—
225C22⋊C4—88no—
226C42—42no—
227C42—42no—
228C42—81no—
229C42—81no—
230C42C2×C22⋊C412yes—
231C42#11712yes—
232C42C4×D412yes—
233C42C4×D412yes—
234C42#12112yes—
235C42#12312yes—
236C42#12412yes—
237C42#12612yes—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—44no—
242C23—44no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—82no—
247C23—82no—
248C23—81no—
249C23—81no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—81no—
259C23—81no—
260C23—82no—
261C23—82no—
262C23C4×C22⋊C411yescharacteristic
263C23#38611yescharacteristic
264C23#39011yescharacteristic
265C23C2×#10412yes—
266C23#42112yes—
267Q8—84no—
268Q8—84no—
269D4—84no—
270D4—84no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4C4×C22⋊C412yes—
309C2×C4#38612yes—
310C2×C4#38812yes—
311C2×C4#38812yes—
312C2×C4#38812yes—
313C2×C4#39012yes—
314C8—48no—
315C8—48no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—84no—
325C22—84no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—162no—
330C22—162no—
331C22—162no—
332C22—162no—
333C22#143711yescharacteristic
334C22C4×#10411yescharacteristic
335C22#175911yescharacteristic
336C22#187112yes—
337C22#191111yescharacteristic
338C22#191211yescharacteristic
339C22#191412yes—
340C22#198012yes—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—44no—
348C4—82no—
349C4—82no—
350C4—162no—
351C2—22no—
352C2—21no—
353C2—21no—
354C2—162no—
355C2#918712yes—
356C2#918812yes—
357C2#1093311yescharacteristic
358C2#1093911yescharacteristic
359C2#1094911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×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
1. #20534C2semidirect
4. C4×#343C2semidirect
5. C2×#1288C2semidirect
16. #1527C4semidirect
139. #97C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205342
C2×#15272
#142882
C4×#3432
C2×#12882