Finite Group Databasebeta

← #142062

Subgroups — #142062

Order 512.

612 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 94 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#15195C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7250C214yesmaximal
5C2×#384C411yescharacteristic, complemented
6C2×#380—21no—
7C2×C4×C22⋊C4C2211yescharacteristic
8#1749—21no—
9C2×#342—21no—
10C2×#342C2212yes—
11C2×#323C2212yes—
12#1442—21no—
13#1442C412yescomplemented
14#1439—21no—
15#1439C2212yes—
16C4⋊#95—21no—
17C4⋊#95C411yescharacteristic, complemented
18C22⋊#95—21no—
19#1280—21no—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C412yes—
25C22×C22⋊C4—22no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4D411yescharacteristic
28C22×C22⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C2×C22⋊C8—22no—
32C2×#97—41no—
33#384—41no—
34#384—41no—
35#380—44no—
36C4×C22⋊C4—41no—
37C4×C22⋊C4—41no—
38C2×#95—21no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—41no—
43C2×#95—41no—
44C2×#95C2×C412yes—
45C2×#95C2×C411yescharacteristic
46C2×#95D411yescharacteristic
47#342—24no—
48#342—24no—
49#336—41no—
50#323—22no—
51#323Q814yes—
52C25C2×D411yescharacteristic
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—41no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C22×C8—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—82no—
79C2×C4⋊C4—82no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—82no—
93C2×C22⋊C4—82no—
94C2×C22⋊C4C2×Q812yes—
95C2×C22⋊C4#4112yes—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—42no—
101C2×C42—42no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104#97—82no—
105#95—41no—
106#95—41no—
107#95—42no—
108#95—44no—
109#95—42no—
110#95—82no—
111#95—82no—
112#95—82no—
113#95—82no—
114C24—21no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—41no—
119C24—41no—
120C24—41no—
121C24—41no—
122C24C2×C22⋊C411yescharacteristic
123C24#12411yescharacteristic
124C24C4⋊Q811yescharacteristic
125C24AGL(1,ℤ/8)12yes—
126C24#13712yes—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—48no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—84no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4#9911yescharacteristic
183C22×C4#9911yescharacteristic
184C22×C4#10412yes—
185C22×C4C4×Q812yes—
186C22×C4C22⋊Q812yes—
187C2×C8—44no—
188C2×C8—82no—
189C2×C8—82no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—84no—
194C4⋊C4—84no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C22⋊C4—88no—
202C22⋊C4—82no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—81no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23#39111yescharacteristic
238C23C2×#9911yescharacteristic
239C23#41011yescharacteristic
240C23#41811yescharacteristic
241C23#41911yescharacteristic
242C23C2×#10411yescharacteristic
243C23#42111yescharacteristic
244C23#48912yes—
245C23#50112yes—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—84no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—162no—
297C2×C4—162no—
298C2×C4—162no—
299C2×C4—162no—
300C2×C4#35212yes—
301C2×C4#35412yes—
302C8—84no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—82no—
328C22—81no—
329C22—81no—
330C22—82no—
331C22#150511yescharacteristic
332C22#150811yescharacteristic
333C22#151011yescharacteristic
334C22#151511yescharacteristic
335C22#198611yescharacteristic
336C22#199311yescharacteristic
337C22#199611yescharacteristic
338C22C2×#35211yescharacteristic
339C22#212211yescharacteristic
340C22C2×#35411yescharacteristic
341C22#212811yescharacteristic
342C4—22no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—81no—
348C4—81no—
349C4—162no—
350C4—164no—
351C4—162no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#988911yescharacteristic
359C2#1178611yescharacteristic
360C2#1178711yescharacteristic
361C2#1179711yescharacteristic
362C2#1180011yescharacteristic
363C2#1329211yescharacteristic
364C2#1329911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#384C4semidirect
13. #1442C4semidirect
17. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151952
#108012
#81522
#72502