Finite Group Databasebeta

← #97660

Subgroups — #97660

Order 512.

856 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 160 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
1C4×#559C211yesmaximal, characteristic
2#9638C211yesmaximal, characteristic
3#8564C212yesmaximal, complemented
4C4×Q8⋊C8C212yesmaximal, complemented
5#7569C211yesmaximal, characteristic
6C2×#559C411yescharacteristic
7#2374—22no—
8#2367—22no—
9#2366C411yescharacteristic
10#2357—22no—
11#2355C412yes—
12C4×#127C2211yescharacteristic
13C4×#124—22no—
14C4×#123—22no—
15C42×Q8C2212yes—
16C42×D4—22no—
17#1842C2211yescharacteristic
18C4×C4⋊C8C2212yes—
19C42×C8C2211yescharacteristic
20C2×Q8⋊C8—22no—
21C2×Q8⋊C8—22no—
22C2×D4⋊C8—22no—
23#1279—22no—
24#1279—22no—
25#1270—22no—
26#559—22no—
27#559—24no—
28#559C814yescomplemented
29C2×#127C2×C411yescharacteristic
30C2×#124—22no—
31C2×#123—22no—
32C2×C4×Q8C2×C412yes—
33C2×C4×Q8D412yes—
34C2×C4×Q8D412yes—
35C2×C4×D4—22no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4D412yes—
39C22×C42—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—22no—
43C2×C4×C8—21no—
44C2×C4×C8—22no—
45C2×C4×C8—22no—
46#391—22no—
47#390—22no—
48#390—22no—
49#390D412yes—
50#388—24no—
51#388—24no—
52#388—24no—
53#387—24no—
54#386—22no—
55#385—22no—
56#384C2×C411yescharacteristic
57#382—22no—
58#381—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4C2×C412yes—
65C4×C4⋊C4D412yes—
66C4×C4⋊C4D412yes—
67C4×C22⋊C4—24no—
68C4×C22⋊C4—24no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4—22no—
71C43—22no—
72C43C2311yescharacteristic
73#336—22no—
74Q8⋊C8—28no—
75Q8⋊C8—44no—
76D4⋊C8—44no—
77C22×Q8C22⋊C412yes—
78C22×D4—22no—
79C22×D4C22⋊C412yes—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C22×C8—41no—
84C22×C8—42no—
85C22×C8—41no—
86#127—22no—
87#127C2×C812yes—
88#127M1612yes—
89#124—28no—
90#124—28no—
91#123—28no—
92#123—28no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—24no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—42no—
101C4×Q8C2×C814yes—
102C4×Q8M1614yes—
103C4×D4—28no—
104C4×D4—28no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—44no—
108C4×D4—44no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4C22⋊C412yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C22⋊C412yes—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—24no—
133C2×C42—22no—
134C2×C42—82no—
135C2×C42—82no—
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C4⋊C8—24no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145C4⋊C8—42no—
146C4×C8—22no—
147C4×C8—24no—
148C4×C8—42no—
149C4×C8—41no—
150C4×C8—42no—
151C4×C8—42no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—22no—
156#95—22no—
157#95—22no—
158C24—22no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8C22⋊C814yes—
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4C22⋊C814yes—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—48no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C2×C22⋊C412yes—
191C22×C4#12011yescharacteristic
192C22×C4#12012yes—
193C22×C4#12011yescharacteristic
194C2×C8—42no—
195C2×C8—44no—
196C2×C8—44no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—81no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—44no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4C22⋊C814yes—
214C4⋊C4#10414yes—
215C22⋊C4—216no—
216C22⋊C4—28no—
217C22⋊C4—48no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—24no—
223C42—24no—
224C42—22no—
225C42—22no—
226C42—21no—
227C42—21no—
228C42—41no—
229C42—44no—
230C42—42no—
231C42—42no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—42no—
236C42—82no—
237C42—84no—
238C42C22×C811yescharacteristic
239C42C2×M1611yescharacteristic
240C42C2×M1612yes—
241C42#13512yes—
242C42#13512yes—
243C42AGL(1,ℤ/8)12yes—
244C42#13712yes—
245C23—24no—
246C23—42no—
247C23—42no—
248C23—44no—
249C23—44no—
250C23#37911yescharacteristic
251Q8—24no—
252Q8—24no—
253Q8—42no—
254Q8—42no—
255D4—44no—
256D4—44no—
257D4—44no—
258D4—44no—
259D4—44no—
260D4—82no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—88no—
282C2×C4—88no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4C2×C22⋊C811yescharacteristic
287C2×C4C2×C22⋊C812yes—
288C2×C4#40711yescharacteristic
289C2×C4#40712yes—
290C2×C4#41612yes—
291C2×C4#41712yes—
292C2×C4C2×#10412yes—
293C2×C4#42112yes—
294C2×C4#44912yes—
295C2×C4#44912yes—
296C2×C4#45212yes—
297C2×C4#45212yes—
298C2×C4#45311yescharacteristic
299C2×C4#45511yescharacteristic
300C8—44no—
301C8—84no—
302C8—161no—
303C22—48no—
304C22—44no—
305C22—84no—
306C22—82no—
307C22—82no—
308C22#177911yescharacteristic
309C22#178812yes—
310C22#178912yes—
311C22#179211yescharacteristic
312C22#179411yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—44no—
319C4—41no—
320C4—44no—
321C4—82no—
322C4—82no—
323C4#147612yes—
324C4#147712yes—
325C4#148812yes—
326C4#148912yes—
327C4#162011yescharacteristic
328C4#163011yescharacteristic
329C4#164012yes—
330C2—84no—
331C2#831212yes—
332C2#831811yescharacteristic
333C2#843712yes—
334C2#844011yescharacteristic
335C2#844511yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC23#379
prime radicalC2×#127C2×C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8564C2semidirect
4. C4×Q8⋊C8C2semidirect
28. #559C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5592
#96382
#85642
C4×Q8⋊C82
#75692