Finite Group Databasebeta

← #103554

Subgroups — #103554

Order 512.

654 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 100 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#29756C211yesmaximal, characteristic
2#25813C211yesmaximal, characteristic
3C2×#2043C211yesmaximal, characteristic
4#9650C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7668C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#500—21no—
10C2×#483—21no—
11C22×#103—21no—
12C22×#103—21no—
13#2857C2211yescharacteristic
14C2×#389C2211yescharacteristic
15#2084—21no—
16#2066—22no—
17#2057—22no—
18#2043—22no—
19#2043—21no—
20#2043—22no—
21#1990—21no—
22#1872—21no—
23#1852—22no—
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#332—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⋊Q8D412yes—
38C2×C4⋊Q8D411yescharacteristic
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#500—44no—
47#483—42no—
48#483—42no—
49C2×#107—24no—
50C2×#106—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53C2×#103—22no—
54C2×#103—24no—
55C2×#103—24no—
56C2×#103—22no—
57C2×#103—42no—
58C2×#103—44no—
59C2×#103—42no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8D412yes—
63C2×C4×C8—21no—
64#391—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#384D411yescharacteristic
72#380—21no—
73#380—21no—
74C2×#95—21no—
75#340—24no—
76#340—22no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336D412yes—
82#332—44no—
83#324—44no—
84C22×Q8—24no—
85C22×Q8—22no—
86C22×Q8—21no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—42no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—41no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—42no—
105C22×C8—42no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—42no—
114C4⋊Q8—44no—
115C4⋊Q8—44no—
116C22⋊Q8—41no—
117C22⋊Q8—41no—
118C22⋊Q8—41no—
119C22⋊Q8—41no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4D812yes—
145C2×C22⋊C4—41no—
146C2×C42—42no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151#107—48no—
152#106—44no—
153C4⋊C8—42no—
154C4⋊C8—42no—
155#103—48no—
156#103—44no—
157#103—48no—
158#103—82no—
159C22⋊C8—22no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8—42no—
163C22⋊C8SD1614yes—
164C4×C8—42no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169C24#12111yescharacteristic
170C2×Q8—24no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×Q8—82no—
189C2×Q8—81no—
190C2×Q8—81no—
191C2×Q8—82no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—44no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4#12011yescharacteristic
216C22×C4#12011yescharacteristic
217C22×C4#12011yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12711yescharacteristic
221C22×C4C2×D811yescharacteristic
222C22×C4C2×SD1612yes—
223C22×C4#13511yescharacteristic
224C22×C4AGL(1,ℤ/8)11yescharacteristic
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4#13712yes—
227C2×C8—24no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—82no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—41no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23#39211yescharacteristic
275C23#46112yes—
276C23#46611yescharacteristic
277C23#46911yescharacteristic
278C23#47312yes—
279Q8—44no—
280Q8—44no—
281Q8—82no—
282Q8—82no—
283Q8—84no—
284Q8—84no—
285Q8—82no—
286Q8—82no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—162no—
325C2×C4#44711yescharacteristic
326C2×C4#44711yescharacteristic
327C2×C4#44812yes—
328C2×C4#44911yescharacteristic
329C2×C4#44911yescharacteristic
330C2×C4#45412yes—
331C2×C4B(4,𝔽2)12yes—
332C2×C4#47312yes—
333C2×C4#49211yescharacteristic
334C2×C4#49711yescharacteristic
335C8—44no—
336C8—82no—
337C8—82no—
338C8—161no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22#167312yes—
348C22#169312yes—
349C22#199911yescharacteristic
350C22#200311yescharacteristic
351C22#200911yescharacteristic
352C22#201311yescharacteristic
353C22#202011yescharacteristic
354C22#202011yescharacteristic
355C22#202111yescharacteristic
356C4—22no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—82no—
364C4—84no—
365C4—161no—
366C4—161no—
367C4—161no—
368C4#159712yes—
369C4#159812yes—
370C2—22no—
371C2—22no—
372C2#991511yescharacteristic
373C2#994911yescharacteristic
374C2#999111yescharacteristic
375C2#1013511yescharacteristic
376C2#1015911yescharacteristic
377C2#1018511yescharacteristic
378C2#1026411yescharacteristic
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
#297562
#258132
C2×#20432
#96502
#83402
#76682
#70292