Finite Group Databasebeta

← #101212

Subgroups — #101212

Order 512.

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

normal subgroups only

This group has 566 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#15119C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7609C212yesmaximal
5#7037C212yesmaximal
6C2×#423—21no—
7C2×C4×M16—21no—
8C2×#391C2211yescharacteristic
9C2×#376C2211yescharacteristic
10#1817C2212yes—
11#1768—24no—
12#1728C2212yes—
13#1446C411yescharacteristic
14#1446C411yescharacteristic
15#1445—21no—
16#1444C411yescharacteristic
17#1443—21no—
18#1442C411yescharacteristic
19#1436—21no—
20#1434—21no—
21#1434C2211yescharacteristic
22#1280—21no—
23#1280—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C42D411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42C2311yescharacteristic
32#423—24no—
33#423—24no—
34C2×C4⋊C8—24no—
35C2×C22⋊C8—22no—
36C4×M16—42no—
37C4×M16—42no—
38C2×#97—22no—
39C2×C4×C8—22no—
40#391—22no—
41#391—22no—
42#376—22no—
43#376—22no—
44C2×#95—21no—
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—
65C22×Q8—22no—
66C22×Q8—42no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—81no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C22×C8—24no—
88C22×C8—42no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42SD1612yes—
105C2×C42Q1612yes—
106C2×C42C2×D412yes—
107C2×C42#4112yes—
108C4⋊C8—216no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—44no—
112C4×C8—44no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—22no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—41no—
125#95—42no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—81no—
130#95—81no—
131#95—81no—
132C24C2×C22⋊C411yescharacteristic
133C2×Q8—24no—
134C2×Q8—42no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—84no—
138C2×Q8—84no—
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—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
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—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#10312yes—
196C22×C4#10312yes—
197C22×C4#10412yes—
198C22×C4#10412yes—
199C22×C4C4×D412yes—
200C22×C4#12011yescharacteristic
201C22×C4#12112yes—
202C22×C4#12311yescharacteristic
203C22×C4C2×SD1611yescharacteristic
204C22×C4C2×Q1611yescharacteristic
205C22×C4AGL(1,ℤ/8)11yescharacteristic
206C22×C4#13711yescharacteristic
207M16—44no—
208M16—44no—
209M16—44no—
210M16—44no—
211C2×C8—28no—
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C4⋊C4—82no—
218C4⋊C4—84no—
219C42—24no—
220C42—24no—
221C42—22no—
222C42—22no—
223C42—22no—
224C42—22no—
225C42—42no—
226C42—42no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#38611yescharacteristic
232C23#41011yescharacteristic
233C23#41011yescharacteristic
234C23C2×#10311yescharacteristic
235C23#41711yescharacteristic
236C23C2×#10411yescharacteristic
237C23C2×#10411yescharacteristic
238Q8—84no—
239Q8—84no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
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—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#45011yescharacteristic
311C2×C4#45111yescharacteristic
312C2×C4#48311yescharacteristic
313C2×C4#48411yescharacteristic
314C8—48no—
315C8—162no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#150311yescharacteristic
329C22#150311yescharacteristic
330C22#150911yescharacteristic
331C22#150911yescharacteristic
332C22#187211yescharacteristic
333C22#188411yescharacteristic
334C22#189111yescharacteristic
335C22#189611yescharacteristic
336C22#189911yescharacteristic
337C22#190911yescharacteristic
338C22#191411yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—82no—
350C4—82no—
351C4—161no—
352C4—161no—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2#903311yescharacteristic
358C2#914411yescharacteristic
359C2#919111yescharacteristic
360C2#938711yescharacteristic
361C2#944411yescharacteristic
362C2#949111yescharacteristic
363C2#951211yescharacteristic
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
#151192
#86992
#76092
#70372