Finite Group Databasebeta

← #140683

Subgroups — #140683

Order 512.

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

normal subgroups only

This group has 724 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#13904C211yesmaximal, characteristic
2C2×#1317C214yesmaximal
3#8559C211yesmaximal, characteristic
4#8152C211yesmaximal, characteristic
5C23×C4⋊C4C411yescharacteristic
6C2×#381C412yes—
7C2×#381C2212yes—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11#1749—21no—
12C2×#323C2212yes—
13C2×#323C2212yes—
14#1445—21no—
15#1445—21no—
16C4⋊#95—21no—
17C4⋊#95—21no—
18#1317—28no—
19#1280—21no—
20#1280—21no—
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C412yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—24no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4—41no—
29C22×C4⋊C4—42no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4C2311yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×#97—41no—
39#381—24no—
40#381—24no—
41#381—44no—
42#380—42no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95D411yescharacteristic
51C2×#95D411yescharacteristic
52#336—41no—
53#336—41no—
54#323—22no—
55#323—24no—
56#323D414yes—
57C25C22×C411yescharacteristic
58C23×C4—22no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—24no—
65C23×C4—24no—
66C23×C4—22no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4M1614yes—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—42no—
81C22×C8—42no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—48no—
86C2×C4⋊C4—48no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4C2×D412yes—
103C2×C22⋊C4#4112yes—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—42no—
109C2×C42—42no—
110C4⋊C8—82no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115#97—82no—
116#95—41no—
117#95—41no—
118#95—44no—
119#95—42no—
120#95—44no—
121#95—44no—
122#95—42no—
123#95—42no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24—41no—
132C24C2×C22⋊C411yescharacteristic
133C24C4×D412yes—
134C24C2×M1612yes—
135C24#13112yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—24no—
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—24no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—28no—
155C22×C4—48no—
156C22×C4—48no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
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—44no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4#9911yescharacteristic
194C22×C4#9911yescharacteristic
195C22×C4#10012yes—
196C22×C4#10012yes—
197C22×C4#12112yes—
198C22×C4#12411yescharacteristic
199C22×C4#12711yescharacteristic
200C2×C8—44no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8—82no—
205C2×C8—82no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—48no—
209C4⋊C4—84no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—44no—
213C22⋊C4—84no—
214C42—42no—
215C42—42no—
216C42—81no—
217C42—81no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#39011yescharacteristic
245C23#40711yescharacteristic
246C23#40811yescharacteristic
247C23C2×#9911yescharacteristic
248C23#41011yescharacteristic
249C23C2×#10011yescharacteristic
250C23C2×#10011yescharacteristic
251C23#43514yes—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—48no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—88no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—88no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#35212yes—
304C2×C4#35212yes—
305C8—84no—
306C8—162no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22#146211yescharacteristic
331C22#146611yescharacteristic
332C22#146911yescharacteristic
333C22#147111yescharacteristic
334C22#195511yescharacteristic
335C22#196411yescharacteristic
336C22#196511yescharacteristic
337C22C2×#35211yescharacteristic
338C22C2×#35211yescharacteristic
339C22#212311yescharacteristic
340C22#212311yescharacteristic
341C4—22no—
342C4—22no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—42no—
348C4—81no—
349C4—84no—
350C4—162no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2—44no—
356C2#965311yescharacteristic
357C2#1150011yescharacteristic
358C2#1150411yescharacteristic
359C2#1155811yescharacteristic
360C2#1156211yescharacteristic
361C2#1321711yescharacteristic
362C2#1322011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutator, prime radicalC25C22×C4
Frattini, cosocleC22×C22⋊C4C23
LCSG ▷ C25 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#139042
C2×#13172
#85592
#81522