Finite Group Databasebeta

← #101216

Subgroups — #101216

Order 512.

550 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 98 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#20429C211yesmaximal, characteristic
2#15121C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7607C212yesmaximal
5#7037C212yesmaximal
6C2×#423—21no—
7C2×C4×M16—21no—
8C2×#389C2211yescharacteristic
9C2×#376C2211yescharacteristic
10#1817C2212yes—
11#1768—24no—
12#1728C2212yes—
13#1445C411yescharacteristic
14#1445C411yescharacteristic
15#1444C411yescharacteristic
16#1442—21no—
17#1442C411yescharacteristic
18#1441—21no—
19#1436—21no—
20#1434—21no—
21#1434C2211yescharacteristic
22#1280—21no—
23#1280—21no—
24#1271—24no—
25C22×M16—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42C2311yescharacteristic
33#423—24no—
34#423—24no—
35C2×C4⋊C8—24no—
36C2×C22⋊C8—22no—
37C4×M16—42no—
38C4×M16—42no—
39C2×#97—22no—
40C2×C4×C8—22no—
41#389—22no—
42#389—22no—
43#376—22no—
44#376—22no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61#336—22no—
62#336—22no—
63#336—22no—
64#336D414yes—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C22×C8—24no—
83C22×C8—42no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42SD1612yes—
106C2×C42Q1612yes—
107C2×C42C2×D412yes—
108C2×C42#4112yes—
109C4⋊C8—216no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112#97—44no—
113C4×C8—44no—
114#95—22no—
115#95—22no—
116#95—22no—
117#95—22no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—81no—
130#95—81no—
131#95—81no—
132C24C2×C22⋊C411yescharacteristic
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
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—41no—
174C22×C4—41no—
175C22×C4—41no—
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—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4#10312yes—
190C22×C4#10312yes—
191C22×C4#10412yes—
192C22×C4#10412yes—
193C22×C4C4×D412yes—
194C22×C4#12011yescharacteristic
195C22×C4#12112yes—
196C22×C4#12311yescharacteristic
197C22×C4C2×SD1611yescharacteristic
198C22×C4C2×Q1611yescharacteristic
199C22×C4AGL(1,ℤ/8)11yescharacteristic
200C22×C4#13711yescharacteristic
201M16—44no—
202M16—44no—
203M16—44no—
204M16—44no—
205C2×C8—28no—
206C2×C8—44no—
207C2×C8—44no—
208C2×C8—82no—
209C2×C8—82no—
210C2×C8—82no—
211C4⋊C4—24no—
212C4⋊C4—42no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C42—24no—
217C42—24no—
218C42—22no—
219C42—22no—
220C42—22no—
221C42—22no—
222C42—42no—
223C42—42no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23#38611yescharacteristic
229C23#41011yescharacteristic
230C23#41011yescharacteristic
231C23C2×#10311yescharacteristic
232C23#41711yescharacteristic
233C23C2×#10411yescharacteristic
234C23C2×#10411yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
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—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4#45011yescharacteristic
306C2×C4#45111yescharacteristic
307C2×C4#48311yescharacteristic
308C2×C4#48411yescharacteristic
309C8—48no—
310C8—162no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22#150311yescharacteristic
324C22#150311yescharacteristic
325C22#150911yescharacteristic
326C22#150911yescharacteristic
327C22#187211yescharacteristic
328C22#188411yescharacteristic
329C22#189111yescharacteristic
330C22#189611yescharacteristic
331C22#189911yescharacteristic
332C22#191011yescharacteristic
333C22#191511yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—82no—
345C4—82no—
346C4—161no—
347C4—161no—
348C2—21no—
349C2—21no—
350C2—21no—
351C2—21no—
352C2#903311yescharacteristic
353C2#914411yescharacteristic
354C2#919311yescharacteristic
355C2#938811yescharacteristic
356C2#944611yescharacteristic
357C2#949211yescharacteristic
358C2#951411yescharacteristic
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

Maximal subgroups

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

subgroupindex
#204292
#151212
#86992
#76072
#70372