Finite Group Databasebeta

← #99237

Subgroups — #99237

Order 512.

600 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 86 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#13966C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7036C212yesmaximal, complemented
5#7036C212yesmaximal, complemented
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8C22×#95C411yescharacteristic
9#1749C2212yes—
10#1749C2212yes—
11C2×#323—21no—
12C2×#323—21no—
13#1442—21no—
14#1442—21no—
15#1434C411yescharacteristic
16#1434C411yescharacteristic
17#1434C2211yescharacteristic
18#1434C2211yescharacteristic
19C22⋊#95—21no—
20C22⋊#95—21no—
21#1280—21no—
22#1280—21no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4D411yescharacteristic
28C22×C22⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×#97—22no—
34C2×#97—22no—
35#380—42no—
36C4×C22⋊C4—22no—
37C4×C22⋊C4—22no—
38C2×#95—21no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—42no—
48C2×#95C2×C411yescharacteristic
49C2×#95C2×C411yescharacteristic
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95D411yescharacteristic
53C2×#95D411yescharacteristic
54#336—22no—
55#336—22no—
56#323—24no—
57#323—24no—
58C25C22⋊C411yescharacteristic
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—81no—
73C23×C4—81no—
74C23×C4—81no—
75C23×C4—81no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—42no—
83C22×C8—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4—81no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—81no—
99C2×C42—81no—
100C2×C42#4112yes—
101C2×C42#4112yes—
102C22⋊C8—24no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106#97—82no—
107#97—82no—
108#95—22no—
109#95—22no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—81no—
120#95—81no—
121#95—81no—
122#95—81no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—41no—
128C24—41no—
129C24—41no—
130C24—41no—
131C24#10112yes—
132C24#10412yes—
133C24#10412yes—
134C24C2×C22⋊C411yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
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—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—48no—
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—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4—84no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4—84no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4#10012yes—
205C22×C4#11711yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4#12312yes—
208C22×C4#12312yes—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C4⋊C4—84no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—84no—
219C42—42no—
220C42—42no—
221C42—81no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#33014yes—
250C23#38111yescharacteristic
251C23C2×#10011yescharacteristic
252C23C2×#10111yescharacteristic
253C23C2×#10411yescharacteristic
254C23C2×#10411yescharacteristic
255C23#42111yescharacteristic
256C23#42111yescharacteristic
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—816no—
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—
321C8—84no—
322C8—84no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—44no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22C2×#33011yescharacteristic
347C22#152711yescharacteristic
348C22#152711yescharacteristic
349C22#153411yescharacteristic
350C22#182111yescharacteristic
351C22#182611yescharacteristic
352C22#182611yescharacteristic
353C22#182711yescharacteristic
354C22#182711yescharacteristic
355C22#182811yescharacteristic
356C22#182911yescharacteristic
357C4—41no—
358C4—41no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4—161no—
367C4—161no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—21no—
372C2—44no—
373C2#871911yescharacteristic
374C2#874311yescharacteristic
375C2#875611yescharacteristic
376C2#877211yescharacteristic
377C2#877311yescharacteristic
378C2#877611yescharacteristic
379C2#877711yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7036C2semidirect
5. #7036C2semidirect

Maximal subgroups

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

subgroupindex
#139662
#81522
#70362