Finite Group Databasebeta

← #102798

Subgroups — #102798

Order 512.

551 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 112 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#30291C211yesmaximal, characteristic
2#27950C211yesmaximal, characteristic
3#27910C211yesmaximal, characteristic
4#11357C211yesmaximal, characteristic
5#9884C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#7016C211yesmaximal, characteristic
8C2×#484—22no—
9#2857C2211yescharacteristic
10#2857C2211yescharacteristic
11C2×#384C2211yescharacteristic
12#2095—22no—
13#2083—22no—
14#2083—21no—
15#2083—21no—
16#2057—21no—
17#2057—21no—
18#1990—21no—
19#1956—21no—
20#1852—22no—
21#1845—21no—
22#1845—21no—
23#1808—22no—
24C4×C22⋊C8C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—22no—
27C2×#336C2211yescharacteristic
28C2×#328—22no—
29C4⋊#95C2211yescharacteristic
30#1294—22no—
31C23×C8D412yes—
32C2×C4⋊Q8D411yescharacteristic
33C2×C4⋊Q8D411yescharacteristic
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C42C2311yescharacteristic
41#484—44no—
42#484—44no—
43C2×#107—24no—
44C2×#107—22no—
45C2×#107—22no—
46C2×#107—24no—
47C2×#103—24no—
48C2×C22⋊C8D412yes—
49C2×C22⋊C8D412yes—
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52#391—21no—
53#391—21no—
54#389—21no—
55#389—21no—
56#389—21no—
57#389—21no—
58#384—21no—
59#384—21no—
60#384—21no—
61#384D411yescharacteristic
62#384D411yescharacteristic
63#384D412yes—
64#380—21no—
65#380—21no—
66#380—21no—
67#380—21no—
68C2×#95—21no—
69C2×#95—21no—
70#340—22no—
71#340—22no—
72#340—24no—
73#340—44no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#328—44no—
79C22×Q8—21no—
80C22×Q8—21no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8Q1614yes—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C22⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—82no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—42no—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133#107—48no—
134#107—84no—
135#107—82no—
136#107—82no—
137#103—48no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8D814yes—
141C22⋊C8Q1614yes—
142C4×C8—42no—
143C4×C8—42no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—42no—
149#95—42no—
150C24#12711yescharacteristic
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—81no—
156C2×Q8—81no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4#12011yescharacteristic
181C22×C4#12011yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4C2×D812yes—
187C22×C4C2×Q1612yes—
188C22×C4C2×Q1612yes—
189C22×C4AGL(1,ℤ/8)12yes—
190C22×C4#13712yes—
191C22×C4#13712yes—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—44no—
202C2×C8—82no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C42—22no—
231C42—22no—
232C42—41no—
233C42—41no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23#39211yescharacteristic
239C23#49311yescharacteristic
240C23#49411yescharacteristic
241C23#49411yescharacteristic
242C23#49611yescharacteristic
243C23#49711yescharacteristic
244C23#49711yescharacteristic
245Q8—84no—
246Q8—162no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—161no—
284C2×C4—161no—
285C2×C4—161no—
286C2×C4—162no—
287C2×C4—162no—
288C2×C4—161no—
289C2×C4#44712yes—
290C2×C4#45112yes—
291C2×C4#45112yes—
292C2×C4#45311yescharacteristic
293C2×C4#45411yescharacteristic
294C2×C4#45812yes—
295C2×C4#45912yes—
296C2×C4#46212yes—
297C2×C4#46212yes—
298C8—44no—
299C8—44no—
300C8—44no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22#171311yescharacteristic
314C22#172211yescharacteristic
315C22#172311yescharacteristic
316C22#172311yescharacteristic
317C22#200611yescharacteristic
318C22#200811yescharacteristic
319C22#200911yescharacteristic
320C22#201111yescharacteristic
321C22#201611yescharacteristic
322C22#201611yescharacteristic
323C22#202211yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—162no—
330C4—161no—
331C4—161no—
332C4—162no—
333C4—161no—
334C4—161no—
335C4#161412yes—
336C4#161712yes—
337C2—21no—
338C2—21no—
339C2—21no—
340C2—21no—
341C2#995211yescharacteristic
342C2#995911yescharacteristic
343C2#997211yescharacteristic
344C2#1019211yescharacteristic
345C2#1020311yescharacteristic
346C2#1022411yescharacteristic
347C2#1030911yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#127
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
#302912
#279502
#279102
#113572
#98842
#96792
#70162