Finite Group Databasebeta

← #110232

Subgroups — #110232

Order 512.

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

normal subgroups only

This group has 860 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×#1533C212yesmaximal
3#14153C211yesmaximal, characteristic
4C2×#1285C212yesmaximal, complemented
5#7912C211yesmaximal, characteristic, complemented
6C22×#124C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2212yes—
9C2×#391C412yes—
10C2×#390C412yes—
11C2×C4×C4⋊C4C2211yescharacteristic
12C2×C4×C4⋊C4C2211yescharacteristic
13#1771—24no—
14#1749—21no—
15C4×C22⋊C8—21no—
16#1533—24no—
17#1533C418yescomplemented
18C2×#329—24no—
19#1439—22no—
20#1437—21no—
21#1437—21no—
22#1285—24no—
23#1280—22no—
24C23×Q8C2×C411yescharacteristic
25C23×D4C2×C411yescharacteristic
26C2×#124—22no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124—22no—
30C2×#124—22no—
31C2×#124C2×C414yes—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—22no—
37C22×C22⋊C4—22no—
38C22×C42C2×C412yes—
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432C2×C414yes—
48#432C2×C414yes—
49#432D414yes—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×#97—22no—
55C2×#97—22no—
56C2×#97D414yes—
57C2×C4×C8—22no—
58#391—44no—
59#390—44no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95D412yes—
68#336—22no—
69#336—22no—
70#329—216no—
71C25—21no—
72C22×Q8—21no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8C4212yes—
78C22×D4—21no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—42no—
82C22×D4—42no—
83C22×D4C4212yes—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—24no—
92C22×C8—42no—
93#124—24no—
94#124—24no—
95#124—24no—
96#124—24no—
97#124—42no—
98#124—42no—
99#124—42no—
100#124—42no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—44no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—82no—
120C2×C22⋊C4—82no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—24no—
126C2×C42—42no—
127C2×C42—42no—
128C2×C42—81no—
129C2×C42—81no—
130C2×C42C22×C412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C2×C42#4112yes—
134C2×C42#4112yes—
135C2×C42#4112yes—
136C4⋊C8—24no—
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8—42no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144#97—24no—
145#97—24no—
146#97—42no—
147#97—82no—
148#97C22⋊C418yescomplemented
149C4×C8—82no—
150#95—24no—
151#95—44no—
152#95—44no—
153#95—42no—
154C24—22no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—41no—
160C24—41no—
161C24—42no—
162C24—42no—
163C24C2×C22⋊C411yescharacteristic
164C2×Q8—24no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—42no—
168C2×Q8—84no—
169C2×Q8—82no—
170C2×Q8—82no—
171C2×D4—24no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—84no—
176C2×D4—82no—
177C2×D4—82no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4#10014yes—
205C22×C4C2×C4211yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4C4×D412yes—
208C22×C4C4×D412yes—
209C22×C4C4×D412yes—
210C22×C4C4×D412yes—
211C22×C4#12011yescharacteristic
212C22×C4#12112yes—
213C22×C4#12112yes—
214C22×C4#12311yescharacteristic
215C22×C4#12411yescharacteristic
216C22×C4#12711yescharacteristic
217C2×C8—28no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—84no—
222C4⋊C4—48no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—88no—
230C22⋊C4—88no—
231C42—42no—
232C42—42no—
233C42—81no—
234C42—81no—
235C42—84no—
236C42C2×C22⋊C412yes—
237C42#11712yes—
238C42C4×D412yes—
239C42C4×D412yes—
240C42#12112yes—
241C42#12312yes—
242C42#12412yes—
243C42#12612yes—
244C23—22no—
245C23—21no—
246C23—21no—
247C23—44no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—44no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23—82no—
261C23—82no—
262C23—82no—
263C23—82no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23C4×C22⋊C411yescharacteristic
269C23#38611yescharacteristic
270C23#39011yescharacteristic
271C23C2×#10012yes—
272C23#41312yes—
273Q8—84no—
274Q8—84no—
275D4—84no—
276D4—84no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4C4×C22⋊C412yes—
315C2×C4#38612yes—
316C2×C4#38812yes—
317C2×C4#38812yes—
318C2×C4#38812yes—
319C2×C4#39012yes—
320C8—48no—
321C8—84no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—84no—
331C22—84no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—162no—
336C22—162no—
337C22—162no—
338C22—162no—
339C22#143711yescharacteristic
340C22C4×#10012yes—
341C22#188412yes—
342C22#190012yes—
343C22#191111yescharacteristic
344C22#191211yescharacteristic
345C22#196511yescharacteristic
346C22#196711yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—84no—
355C4—82no—
356C4—82no—
357C2—21no—
358C2—22no—
359C2—21no—
360C2—162no—
361C2#944712yes—
362C2#944812yes—
363C2#1094911yescharacteristic
364C2#1095911yescharacteristic
365C2#1096211yescharacteristic
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. C2×#1285C2semidirect
5. #7912C2semidirect
17. #1533C4semidirect
148. #97C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205342
C2×#15332
#141532
C2×#12852
#79122