Finite Group Databasebeta

← #125621

Subgroups — #125621

Order 512.

496 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 76 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#15045C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7122C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C2×#389C411yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8#1797—21no—
9#1779—21no—
10C2×#323—21no—
11#1442—21no—
12#1442C2211yescharacteristic
13#1442C2211yescharacteristic
14#1439—21no—
15#1439C411yescharacteristic
16#1439C411yescharacteristic
17#1434—21no—
18#1434C411yescharacteristic
19C22⋊#95—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4D411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C4⋊C8—41no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—41no—
37C2×C22⋊C8—41no—
38#389—41no—
39#389—41no—
40C4×C22⋊C4—41no—
41C4×C22⋊C4—41no—
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×#95D411yescharacteristic
54C2×#95C2311yescharacteristic
55#336—41no—
56#323—42no—
57C25C22⋊C411yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—41no—
66C23×C4—42no—
67C23×C4—81no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—81no—
81C2×C4⋊C4—81no—
82C2×C4⋊C4—81no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—81no—
94C2×C22⋊C4—81no—
95C2×C22⋊C4—81no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—42no—
101C2×C42—81no—
102C4⋊C8—82no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—82no—
108C22⋊C8—82no—
109#95—22no—
110#95—22no—
111#95—41no—
112#95—41no—
113#95—42no—
114#95—42no—
115#95—41no—
116#95—41no—
117#95—42no—
118#95—41no—
119#95—41no—
120#95—81no—
121#95—81no—
122#95—81no—
123#95—81no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—41no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24#10112yes—
135C24C2×C22⋊C411yescharacteristic
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
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—42no—
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—81no—
195C22×C4—81no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4#9911yescharacteristic
199C22×C4#9911yescharacteristic
200C22×C4#13112yes—
201C22×C4#13512yes—
202C22×C4AGL(1,ℤ/8)11yescharacteristic
203C22×C4#13711yescharacteristic
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—84no—
218C22⋊C4—84no—
219C42—42no—
220C42—81no—
221C42—81no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23#40811yescharacteristic
251C23C2×#9911yescharacteristic
252C23C2×#10111yescharacteristic
253C23#41611yescharacteristic
254C23#41711yescharacteristic
255C23#42111yescharacteristic
256C23#42111yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—84no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—84no—
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—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4#35412yes—
322C8—162no—
323C8—162no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22#147211yescharacteristic
348C22#148211yescharacteristic
349C22#149511yescharacteristic
350C22#151311yescharacteristic
351C22#151611yescharacteristic
352C22#153411yescharacteristic
353C22#155211yescharacteristic
354C22#212011yescharacteristic
355C22#212111yescharacteristic
356C22C2×#35411yescharacteristic
357C22#213411yescharacteristic
358C4—22no—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—42no—
363C4—41no—
364C4—81no—
365C4—81no—
366C4—82no—
367C4—161no—
368C4—161no—
369C4—161no—
370C2—21no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—82no—
375C2#773111yescharacteristic
376C2#1156911yescharacteristic
377C2#1157811yescharacteristic
378C2#1182311yescharacteristic
379C2#1183211yescharacteristic
380C2#1184511yescharacteristic
381C2#1184711yescharacteristic
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. #7122C2semidirect
5. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#150452
#87002
#85572
#71222