Finite Group Databasebeta

← #101232

Subgroups — #101232

Order 512.

494 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 94 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#20753C211yesmaximal, characteristic
2#15121C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7607C212yesmaximal
5#7037C212yesmaximal
6C2×#423—21no—
7C2×#389C2211yescharacteristic
8C2×#376C2211yescharacteristic
9#1832—24no—
10#1817C2212yes—
11#1728—21no—
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—42no—
34#423—42no—
35C2×C4⋊C8—24no—
36C2×C22⋊C8—22no—
37#389—22no—
38#389—22no—
39#376—22no—
40#376—22no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95D411yescharacteristic
57#336—24no—
58#336—22no—
59#336—22no—
60#336—22no—
61#336D414yes—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C22×C8—24no—
80C22×C8—42no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42SD1612yes—
103C2×C42Q1612yes—
104C2×C42C2×D412yes—
105C2×C42#4112yes—
106C4⋊C8—48no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#95—22no—
110#95—22no—
111#95—22no—
112#95—22no—
113#95—41no—
114#95—41no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—81no—
125#95—81no—
126#95—81no—
127C24C2×C22⋊C411yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
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#10312yes—
187C22×C4#10312yes—
188C22×C4C4×D412yes—
189C22×C4#12011yescharacteristic
190C22×C4#12112yes—
191C22×C4#12311yescharacteristic
192C22×C4C2×SD1611yescharacteristic
193C22×C4C2×Q1611yescharacteristic
194C22×C4AGL(1,ℤ/8)11yescharacteristic
195C22×C4#13711yescharacteristic
196M16—82no—
197M16—82no—
198M16—82no—
199M16—82no—
200C2×C8—44no—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—82no—
204C2×C8—82no—
205C2×C8—82no—
206C4⋊C4—24no—
207C4⋊C4—42no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—22no—
215C42—42no—
216C42—42no—
217C42—42no—
218C42—42no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23#38611yescharacteristic
224C23#41011yescharacteristic
225C23#41011yescharacteristic
226C23C2×#10311yescharacteristic
227C23#41711yescharacteristic
228C23#42111yescharacteristic
229C23#42111yescharacteristic
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
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—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#45011yescharacteristic
301C2×C4#45111yescharacteristic
302C2×C4#48311yescharacteristic
303C2×C4#48411yescharacteristic
304C8—84no—
305C8—162no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22#151211yescharacteristic
319C22#151211yescharacteristic
320C22#151811yescharacteristic
321C22#151811yescharacteristic
322C22#187211yescharacteristic
323C22#188411yescharacteristic
324C22#189111yescharacteristic
325C22#189711yescharacteristic
326C22#189911yescharacteristic
327C22#191011yescharacteristic
328C22#191511yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—82no—
340C4—81no—
341C4—161no—
342C4—161no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2#903311yescharacteristic
348C2#915211yescharacteristic
349C2#919311yescharacteristic
350C2#939011yescharacteristic
351C2#944611yescharacteristic
352C2#949411yescharacteristic
353C2#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
#207532
#151212
#86992
#76072
#70372