Finite Group Databasebeta

← #100084

Subgroups — #100084

Order 512.

590 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 104 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#20534C211yesmaximal, characteristic
2#14241C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#7575C212yesmaximal
5#7028C212yesmaximal
6C2×#432—21no—
7C2×#395C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797C2212yes—
11#1771—24no—
12#1749—21no—
13C4×C22⋊C8C2212yes—
14C2×#324—21no—
15#1446C411yescharacteristic
16#1444C411yescharacteristic
17#1441C411yescharacteristic
18#1436—21no—
19#1436—21no—
20#1436—21no—
21C4⋊#95—21no—
22C4×#95C2211yescharacteristic
23#1280—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42C2311yescharacteristic
35#432—24no—
36#432—24no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×#97—22no—
42C2×#97—22no—
43C2×C4×C8—22no—
44#395—22no—
45#395—22no—
46#391—22no—
47#391—22no—
48C4×C4⋊C4—22no—
49C4×C4⋊C4—22no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336D414yes—
66#324—42no—
67C22×Q8—21no—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—81no—
71C22×Q8—81no—
72C22×Q8M1612yes—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—24no—
85C22×C8—42no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4—81no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—42no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—81no—
112C2×C42—81no—
113C2×C42—81no—
114C2×C42C2×D412yes—
115C2×C42#4112yes—
116C4⋊C8—28no—
117C4⋊C8—82no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122#97—28no—
123#97—82no—
124C4×C8—82no—
125#95—24no—
126#95—24no—
127#95—22no—
128#95—22no—
129#95—21no—
130#95—22no—
131#95—42no—
132#95—42no—
133#95—41no—
134#95—42no—
135#95—41no—
136#95—81no—
137#95—81no—
138#95—81no—
139#95M1612yes—
140C24C2×C22⋊C411yescharacteristic
141C2×Q8—24no—
142C2×Q8—42no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8—82no—
146C2×Q8—82no—
147C2×Q8—84no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
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—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4#10412yes—
205C22×C4#10412yes—
206C22×C4C4×D412yes—
207C22×C4#12011yescharacteristic
208C22×C4#12112yes—
209C22×C4#12311yescharacteristic
210C22×C4C2×M1611yescharacteristic
211C22×C4C2×M1611yescharacteristic
212C22×C4#13112yes—
213C22×C4AGL(1,ℤ/8)11yescharacteristic
214C22×C4#13712yes—
215C22×C4#13711yescharacteristic
216C2×C8—28no—
217C2×C8—44no—
218C2×C8—44no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C42—22no—
230C42—22no—
231C42—24no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—81no—
236C42—82no—
237C42—82no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23#38611yescharacteristic
243C23#40711yescharacteristic
244C23#40811yescharacteristic
245C23#41711yescharacteristic
246C23#41911yescharacteristic
247C23C2×#10411yescharacteristic
248C23C2×#10411yescharacteristic
249Q8—84no—
250Q8—84no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
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—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
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—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4#43512yes—
319C2×C4#43612yes—
320C2×C4#45612yes—
321C2×C4#46812yes—
322C2×C4#47012yes—
323C8—48no—
324C8—162no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22#148311yescharacteristic
338C22#148511yescharacteristic
339C22#148911yescharacteristic
340C22#149111yescharacteristic
341C22#187011yescharacteristic
342C22#188411yescharacteristic
343C22#188611yescharacteristic
344C22#189411yescharacteristic
345C22#189611yescharacteristic
346C22#191311yescharacteristic
347C22#191411yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—42no—
355C4—42no—
356C4—41no—
357C4—82no—
358C4—81no—
359C4—82no—
360C4—81no—
361C4—161no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2#896211yescharacteristic
367C2#907711yescharacteristic
368C2#910911yescharacteristic
369C2#938511yescharacteristic
370C2#938711yescharacteristic
371C2#940011yescharacteristic
372C2#940811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
#205342
#142412
#85602
#75752
#70282