Finite Group Databasebeta

← #103619

Subgroups — #103619

Order 512.

614 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 96 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#29966C211yesmaximal, characteristic
2#25902C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic
4#9650C211yesmaximal, characteristic
5#8342C211yesmaximal, characteristic
6#7665C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#487—21no—
10C22×#106—21no—
11#2857C2211yescharacteristic
12C2×#389C2211yescharacteristic
13#2099—21no—
14#2086—22no—
15#2058—22no—
16#2058—22no—
17#2057—21no—
18#2024—22no—
19#1990—21no—
20#1872—22no—
21#1852—21no—
22#1823—21no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#333—22no—
29C2×#324—22no—
30#1271—22no—
31C23×Q8—21no—
32C23×C8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D411yescharacteristic
38C2×C4⋊Q8D412yes—
39C2×C22⋊Q8—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42C2311yescharacteristic
46#487—44no—
47C2×#107—22no—
48C2×#106—24no—
49C2×#106—44no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D412yes—
52C2×#103—24no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8D412yes—
56C2×C4×C8—21no—
57#391—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#384D411yescharacteristic
65#380—21no—
66#380—21no—
67C2×#95—21no—
68#340—22no—
69#340—24no—
70#340—22no—
71#340—22no—
72#340—24no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336D412yes—
78#333—44no—
79#324—44no—
80C22×Q8—21no—
81C22×Q8—22no—
82C22×Q8—44no—
83C22×Q8—41no—
84C22×Q8—82no—
85C22×Q8—82no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—41no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C22×C8—24no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—42no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—44no—
106C4⋊Q8—44no—
107C22⋊Q8—42no—
108C22⋊Q8—42no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4Q1612yes—
135C2×C22⋊C4—41no—
136C2×C42—42no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141#107—82no—
142#106—48no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145#103—48no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—42no—
151C4×C8—42no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156C24#12111yescharacteristic
157C2×Q8—24no—
158C2×Q8—44no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—82no—
166C2×Q8—816no—
167C2×Q8—82no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—44no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4#12011yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12711yescharacteristic
196C22×C4C2×Q1611yescharacteristic
197C22×C4#13511yescharacteristic
198C22×C4#13512yes—
199C22×C4AGL(1,ℤ/8)12yes—
200C22×C4#13711yescharacteristic
201C22×C4#13711yescharacteristic
202C2×C8—24no—
203C2×C8—42no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—81no—
233C4⋊C4—82no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C22⋊C4—82no—
239C42—22no—
240C42—22no—
241C42—41no—
242C42—41no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23#39211yescharacteristic
248C23#46312yes—
249C23#46711yescharacteristic
250C23#47011yescharacteristic
251C23#47112yes—
252Q8—88no—
253Q8—84no—
254Q8—84no—
255Q8—84no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—88no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—162no—
293C2×C4#44912yes—
294C2×C4#45111yescharacteristic
295C2×C4#45111yescharacteristic
296C2×C4#45211yescharacteristic
297C2×C4#45211yescharacteristic
298C2×C4#45612yes—
299C2×C4B(4,𝔽2)12yes—
300C2×C4#47112yes—
301C2×C4#49511yescharacteristic
302C2×C4#49611yescharacteristic
303C8—44no—
304C8—82no—
305C8—82no—
306C8—161no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22#168312yes—
316C22#170312yes—
317C22#200111yescharacteristic
318C22#200511yescharacteristic
319C22#200911yescharacteristic
320C22#201211yescharacteristic
321C22#201911yescharacteristic
322C22#201911yescharacteristic
323C22#202111yescharacteristic
324C4—22no—
325C4—21no—
326C4—21no—
327C4—22no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—84no—
332C4—82no—
333C4—161no—
334C4—161no—
335C4—161no—
336C4#160312yes—
337C4#160512yes—
338C2—22no—
339C2—22no—
340C2#992511yescharacteristic
341C2#994811yescharacteristic
342C2#999011yescharacteristic
343C2#1014311yescharacteristic
344C2#1016611yescharacteristic
345C2#1019111yescharacteristic
346C2#1027511yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#299662
#259022
C2×#20582
#96502
#83422
#76652
#70292