Finite Group Databasebeta

← #103626

Subgroups — #103626

Order 512.

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

normal subgroups only

This group has 566 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#29715C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic
3C2×#2095C211yesmaximal, characteristic
4#9878C211yesmaximal, characteristic
5#8725C211yesmaximal, characteristic
6#7668C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C2×#451—21no—
9C22×#107—21no—
10C22×#103—21no—
11#2857C2211yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#384C2211yescharacteristic
14#2095—22no—
15#2095—22no—
16#2084—22no—
17#2072—21no—
18#2056—22no—
19#2016—21no—
20#1990—21no—
21#1872—21no—
22#1853—22no—
23#1844—21no—
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#336C2211yescharacteristic
28#1296—22no—
29#1280—22no—
30#1271—22no—
31C23×Q8D411yescharacteristic
32C22×Q16—22no—
33C23×C8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C22⋊Q8—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C42C2311yescharacteristic
41#451—42no—
42#451—42no—
43C2×#107—24no—
44C2×#107—44no—
45C2×#106—24no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D412yes—
48C2×#103—22no—
49C2×#103—24no—
50C2×#103—22no—
51C2×#103—44no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8D412yes—
55C2×C4×C8—21no—
56#391—21no—
57#391—21no—
58#391—21no—
59#391—21no—
60#391—21no—
61#389—21no—
62#389—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384D411yescharacteristic
67#384D412yes—
68#380—21no—
69#380—21no—
70C2×#95—21no—
71C2×#95—21no—
72#340—22no—
73#340—24no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336D412yes—
79C22×Q8—21no—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8SD1612yes—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C2×Q16—44no—
95C2×Q16—82no—
96C2×Q16—82no—
97C22×C8—24no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—42no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C4⋊Q8—42no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
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—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4—82no—
129C2×C22⋊C4—41no—
130C2×C42—42no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135#107—48no—
136#106—84no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139#103—48no—
140#103—44no—
141#103—82no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—42no—
146C22⋊C8Q1614yescomplemented
147C4×C8—42no—
148#95—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154C24#12111yescharacteristic
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—44no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—44no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—82no—
170C2×Q8—82no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12711yescharacteristic
199C22×C4C2×SD1611yescharacteristic
200C22×C4C2×Q1612yes—
201C22×C4#13511yescharacteristic
202C22×C4AGL(1,ℤ/8)11yescharacteristic
203C22×C4#13711yescharacteristic
204C22×C4#13712yes—
205Q16—88no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—81no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—82no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C42—22no—
240C42—22no—
241C42—41no—
242C42—41no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23#39211yescharacteristic
248C23#46212yes—
249C23#46511yescharacteristic
250C23#46811yescharacteristic
251C23#47112yes—
252Q8—44no—
253Q8—44no—
254Q8—82no—
255Q8—82no—
256Q8—82no—
257Q8—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—164no—
294C2×C4—162no—
295C2×C4#44811yescharacteristic
296C2×C4#44911yescharacteristic
297C2×C4#45011yescharacteristic
298C2×C4#45112yes—
299C2×C4#45211yescharacteristic
300C2×C4#45312yes—
301C2×C4#45812yes—
302C2×C4#47312yes—
303C2×C4#49411yescharacteristic
304C2×C4#49711yescharacteristic
305C8—44no—
306C8—44no—
307C8—82no—
308C8—161no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22#167212yes—
318C22#169212yes—
319C22#200011yescharacteristic
320C22#200211yescharacteristic
321C22#200811yescharacteristic
322C22#201611yescharacteristic
323C22#201911yescharacteristic
324C22#202011yescharacteristic
325C22#202211yescharacteristic
326C4—22no—
327C4—21no—
328C4—21no—
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—82no—
334C4—161no—
335C4—162no—
336C4—161no—
337C4—161no—
338C4#161312yes—
339C4#161812yes—
340C2—22no—
341C2—22no—
342C2#992011yescharacteristic
343C2#995911yescharacteristic
344C2#999511yescharacteristic
345C2#1012911yescharacteristic
346C2#1015511yescharacteristic
347C2#1018111yescharacteristic
348C2#1027311yescharacteristic
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

Complements

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

NH ≅ G/Ntype
146. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#297152
#279482
C2×#20952
#98782
#87252
#76682
#70292