Finite Group Databasebeta

← #125188

Subgroups — #125188

Order 512.

636 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 108 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#15135C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3#7907C211yesmaximal, characteristic
4#7123C214yesmaximal, complemented
5C22×C22⋊C8—21no—
6C2×#399—21no—
7C2×#389—21no—
8C2×#384C411yescharacteristic
9C2×#380—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#323—21no—
13#1445—21no—
14#1444C2212yes—
15#1439C412yes—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C24×C4D411yescharacteristic
21C23×C8—41no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4C2×C412yes—
27C22×C22⋊C4—22no—
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C4⋊C8—41no—
33C2×C22⋊C8—24no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—41no—
37C2×C22⋊C8—42no—
38#399—42no—
39#399—42no—
40#389—44no—
41#384—41no—
42#384—41no—
43#380—44no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—21no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95C2×C412yes—
50C2×#95D411yescharacteristic
51C2×#95C2311yescharacteristic
52#336—41no—
53#323—42no—
54C25C22⋊C411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4SD1612yes—
66C23×C4Q1612yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C22×C8—42no—
71C22×C8—44no—
72C22×C8—42no—
73C22×C8—82no—
74C22×C8—82no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—82no—
90C2×C4⋊C4—82no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—44no—
97C2×C22⋊C4—44no—
98C2×C22⋊C4—82no—
99C2×C22⋊C4D812yes—
100C2×C22⋊C4SD1612yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C4⋊C8—82no—
108C22⋊C8—24no—
109C22⋊C8—44no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#95—24no—
114#95—41no—
115#95—41no—
116#95—44no—
117#95—44no—
118#95—42no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—41no—
124C24—41no—
125C24—41no—
126C24—41no—
127C24#10112yes—
128C24#10212yes—
129C24#10312yes—
130C24C2×C22⋊C411yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
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—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
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—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—84no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4#9911yescharacteristic
180C22×C4#9911yescharacteristic
181C22×C4#10212yes—
182C22×C4#10312yes—
183C22×C4#10412yes—
184C22×C4#13112yes—
185C22×C4C2×D811yescharacteristic
186C22×C4C2×SD1611yescharacteristic
187C22×C4C2×SD1611yescharacteristic
188C22×C4C2×Q1611yescharacteristic
189C2×C8—44no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—88no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C4⋊C4—48no—
197C4⋊C4—82no—
198C4⋊C4—84no—
199C4⋊C4—84no—
200C4⋊C4—84no—
201C4⋊C4—84no—
202C4⋊C4—82no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—44no—
206C22⋊C4—88no—
207C42—42no—
208C42—42no—
209C42—42no—
210C42—81no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23#33214yes—
238C23#40811yescharacteristic
239C23C2×#9911yescharacteristic
240C23C2×#10111yescharacteristic
241C23C2×#10211yescharacteristic
242C23C2×#10311yescharacteristic
243C23C2×#10411yescharacteristic
244C23#42111yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4#32812yes—
297C2×C4#32812yes—
298C2×C4#35212yes—
299C2×C4#35312yes—
300C8—84no—
301C8—162no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22#147211yescharacteristic
326C22#148111yescharacteristic
327C22#149211yescharacteristic
328C22C2×#32811yescharacteristic
329C22#151911yescharacteristic
330C22#153111yescharacteristic
331C22C2×#33211yescharacteristic
332C22C2×#35211yescharacteristic
333C22#212211yescharacteristic
334C22C2×#35311yescharacteristic
335C22#213511yescharacteristic
336C4—22no—
337C4—42no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—84no—
344C4—81no—
345C4—162no—
346C4#135214yes—
347C2—21no—
348C2—21no—
349C2—21no—
350C2—21no—
351C2—44no—
352C2#766511yescharacteristic
353C2#1157011yescharacteristic
354C2#1157611yescharacteristic
355C2C2×#135211yescharacteristic
356C2#1172411yescharacteristic
357C2#1187711yescharacteristic
358C2#1188111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#151352
#85572
#79072
#71232