Finite Group Databasebeta

← #97020

Subgroups — #97020

Order 512.

742 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2320C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7595C212yesmaximal
5#7593C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7#2320—24no—
8#2320—24no—
9C2×#391C411yescharacteristic
10C2×#391C2211yescharacteristic
11C2×#389C411yescharacteristic
12C2×#389C2211yescharacteristic
13C2×#387—21no—
14C2×#387—21no—
15C2×#383—21no—
16C2×#382—21no—
17C2×C4×C4⋊C4C411yescharacteristic
18C2×#376C2211yescharacteristic
19#1817C2212yes—
20#1817C2212yes—
21#1280—21no—
22#1280—21no—
23#1280—21no—
24#1280—21no—
25#1272—24no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8C814yescomplemented
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C42—21no—
41C22×C42—21no—
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2311yescharacteristic
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47#391—22no—
48#391—22no—
49#391—22no—
50#391D414yes—
51#389—22no—
52#389—22no—
53#389—22no—
54#389D414yes—
55#387—28no—
56#387—28no—
57#383—28no—
58#382—28no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61#376—22no—
62#376—22no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#336—22no—
77#336—22no—
78C22×Q8—21no—
79C22×Q8—21no—
80C22×Q8—21no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8C22⋊C412yes—
84C22×Q8C2×C812yes—
85C22×Q8M1612yes—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C22×C8—42no—
98C22×C8—42no—
99C4×Q8—28no—
100C4×Q8—28no—
101C4×Q8—28no—
102C4×Q8—28no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4C22⋊C412yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C2×C812yes—
121C2×C4⋊C4M1612yes—
122C2×C42—24no—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—24no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42—82no—
135C2×C42C2×C812yes—
136C2×C42M1612yes—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C22⋊C8—44no—
140C22⋊C8—44no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95—22no—
152C24C2×C22⋊C411yescharacteristic
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8C22⋊C814yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C22⋊C812yes—
208C22×C4C22⋊C812yes—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4#12012yes—
212C22×C4#12012yes—
213C22×C4C22×C811yescharacteristic
214C22×C4C2×M1611yescharacteristic
215C22×C4C2×M1611yescharacteristic
216C22×C4C2×M1611yescharacteristic
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—216no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4C22⋊C814yes—
234C42—216no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—84no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23#37911yescharacteristic
249C23C2×C22⋊C811yescharacteristic
250C23#40711yescharacteristic
251C23#41012yes—
252C23#41312yes—
253Q8—44no—
254Q8—44no—
255Q8—44no—
256Q8—44no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C2×C22⋊C811yescharacteristic
310C2×C4C2×C22⋊C811yescharacteristic
311C2×C4#40711yescharacteristic
312C2×C4#40711yescharacteristic
313C8—162no—
314C8—162no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22#145812yes—
324C22#146412yes—
325C22#177911yescharacteristic
326C22#178211yescharacteristic
327C22#178511yescharacteristic
328C22#179511yescharacteristic
329C22#179611yescharacteristic
330C22#179611yescharacteristic
331C22#179611yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—42no—
338C4—44no—
339C4—42no—
340C4—42no—
341C4—44no—
342C4—82no—
343C4—81no—
344C4—82no—
345C4—81no—
346C2—22no—
347C2—22no—
348C2#830011yescharacteristic
349C2#832311yescharacteristic
350C2#832411yescharacteristic
351C2#851911yescharacteristic
352C2#852111yescharacteristic
353C2#852311yescharacteristic
354C2#852711yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
33. C2×C4×Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23202
#86992
#75952
#75932