Finite Group Databasebeta

← #100997

Subgroups — #100997

Order 512.

666 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 124 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#21951C211yesmaximal, characteristic
2#15180C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic
4#8340C212yesmaximal
5#7039C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C2×#431—21no—
8C2×#391—21no—
9C2×#391—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1939—24no—
12#1797—21no—
13#1797C2212yes—
14#1797C2212yes—
15C2×#332—24no—
16C2×#324—21no—
17#1442—21no—
18#1442C412yes—
19#1441—21no—
20#1441C412yes—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8—21no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—44no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42C2311yescharacteristic
35#431—42no—
36#431—42no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8D414yes—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C4×C8—22no—
44#391—44no—
45#391—42no—
46#391—42no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C412yes—
56C2×#95D411yescharacteristic
57#336—22no—
58#336—22no—
59#336—22no—
60#332—48no—
61#324—42no—
62C22×Q8—24no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8—41no—
71C22×Q8—41no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—24no—
82C22×C8—42no—
83C4⋊Q8—28no—
84C4⋊Q8—44no—
85C4⋊Q8—48no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4SD1612yes—
100C2×C4⋊C4Q1612yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42D812yes—
110C2×C42SD1612yes—
111C2×C42C2×D412yes—
112C2×C42#4112yes—
113C4⋊C8—44no—
114C4⋊C8—44no—
115C4⋊C8—44no—
116C4⋊C8—82no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C4×C8—44no—
122#95—22no—
123#95—22no—
124#95—42no—
125#95—44no—
126#95—44no—
127#95—44no—
128#95—44no—
129#95—42no—
130C24C2×C22⋊C411yescharacteristic
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—82no—
138C2×Q8—82no—
139C2×Q8—82no—
140C2×Q8—82no—
141C2×Q8—82no—
142C2×Q8—81no—
143C2×Q8—81no—
144C2×Q8—82no—
145C2×Q8—81no—
146C2×Q8—81no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#10112yes—
187C22×C4#10212yes—
188C22×C4#10212yes—
189C22×C4#10312yes—
190C22×C4#10312yes—
191C22×C4C4×D412yes—
192C22×C4#12011yescharacteristic
193C22×C4#12112yes—
194C22×C4#12311yescharacteristic
195C22×C4C2×D811yescharacteristic
196C22×C4C2×SD1611yescharacteristic
197C22×C4C2×SD1611yescharacteristic
198C22×C4C2×Q1611yescharacteristic
199C22×C4#13512yes—
200C22×C4AGL(1,ℤ/8)11yescharacteristic
201C22×C4#13711yescharacteristic
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—28no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C42—22no—
217C42—22no—
218C42—24no—
219C42—22no—
220C42—22no—
221C42—42no—
222C42—42no—
223C42—81no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23#38611yescharacteristic
229C23C2×#10111yescharacteristic
230C23#41311yescharacteristic
231C23C2×#10211yescharacteristic
232C23C2×#10311yescharacteristic
233C23#41611yescharacteristic
234C23#41711yescharacteristic
235Q8—84no—
236Q8—84no—
237Q8—84no—
238Q8—84no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4#33214yescomplemented
291C2×C4#44711yescharacteristic
292C2×C4#44811yescharacteristic
293C2×C4#45412yes—
294C2×C4B(4,𝔽2)12yes—
295C2×C4#46512yes—
296C2×C4#46712yes—
297C2×C4#48011yescharacteristic
298C2×C4#48111yescharacteristic
299C8—84no—
300C8—162no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22C2×#33211yescharacteristic
314C22#154011yescharacteristic
315C22#155011yescharacteristic
316C22#155211yescharacteristic
317C22#187411yescharacteristic
318C22#188811yescharacteristic
319C22#188911yescharacteristic
320C22#189211yescharacteristic
321C22#190111yescharacteristic
322C22#190911yescharacteristic
323C22#191511yescharacteristic
324C4—21no—
325C4—21no—
326C4—24no—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—81no—
333C4—84no—
334C4—162no—
335C4#159812yes—
336C4#159912yes—
337C2—21no—
338C2—21no—
339C2—21no—
340C2—21no—
341C2#901911yescharacteristic
342C2#913111yescharacteristic
343C2#919511yescharacteristic
344C2#939611yescharacteristic
345C2#945611yescharacteristic
346C2#946711yescharacteristic
347C2#950611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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
290. C2×C4#332semidirect

Maximal subgroups

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

subgroupindex
#219512
#151802
#85582
#83402
#70392