Finite Group Databasebeta

← #98546

Subgroups — #98546

Order 512.

1204 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1204 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C2×#2501C211yesmaximal, characteristic
2#8723C212yesmaximal, complemented
3#8559C212yesmaximal, complemented
4#7640C212yesmaximal, complemented
5C22×#123—22no—
6C22×#121C411yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8#2501—24no—
9#2501—24no—
10C2×#399—22no—
11C2×#390C411yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#384C411yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C4×C22⋊C4C2211yescharacteristic
16#1797C2214yes—
17C2×#323—22no—
18#1296—24no—
19#1280—22no—
20#1270—24no—
21C23×D4—21no—
22C23×D4C2×C411yescharacteristic
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C2×#123—216no—
26C2×#123—48no—
27C2×#123—44no—
28C2×#123—44no—
29C2×#121—22no—
30C2×#121—22no—
31C2×#121—42no—
32C2×#121—42no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—42no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C22⋊C4—21no—
44C22×C22⋊C4—22no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4D412yes—
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—24no—
50C2×C22⋊C8—24no—
51#399—216no—
52#390—22no—
53#390—22no—
54#390—22no—
55#390D414yes—
56#384—22no—
57#384—22no—
58#384—22no—
59#384D414yes—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C2×#95D412yes—
63#336—24no—
64#323—44no—
65C25—21no—
66C25C22⋊C411yescharacteristic
67C22×D4—24no—
68C22×D4—21no—
69C22×D4—42no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—42no—
73C22×D4—42no—
74C22×D4—44no—
75C22×D4—42no—
76C22×D4—44no—
77C22×D4C22⋊C412yes—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—24no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—44no—
90C23×C4—42no—
91C23×C4—44no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C412yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D412yes—
96C22×C8—44no—
97#123—432no—
98#123—432no—
99#121—48no—
100#121—48no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—48no—
114C2×C4⋊C4—48no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C22⋊C412yes—
124C2×C4⋊C4C22⋊C412yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4D814yes—
138C2×C22⋊C4SD1614yes—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C4⋊C8—48no—
146C22⋊C8—44no—
147C22⋊C8—44no—
148#95—24no—
149C24—21no—
150C24—21no—
151C24—22no—
152C24—22no—
153C24—41no—
154C24—41no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—44no—
159C24—41no—
160C24—41no—
161C24—42no—
162C24—42no—
163C24#10012yes—
164C24#10214yes—
165C24C2×C22⋊C411yescharacteristic
166C2×D4—24no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—48no—
172C2×D4—416no—
173C2×D4—42no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4—88no—
177C2×D4—84no—
178C2×D4—84no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—48no—
205C22×C4—416no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—88no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4#10012yes—
222C22×C4#10214yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4#12014yes—
226C22×C4C2×D812yes—
227C22×C4C2×SD1612yes—
228C22×C4AGL(1,ℤ/8)14yes—
229C2×C8—84no—
230C2×C8—84no—
231C2×C8—84no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—48no—
238C4⋊C4—42no—
239C4⋊C4—416no—
240C4⋊C4—48no—
241C4⋊C4—48no—
242C22⋊C4—28no—
243C22⋊C4—28no—
244C22⋊C4—48no—
245C22⋊C4—416no—
246C22⋊C4—48no—
247C22⋊C4—48no—
248C42—24no—
249C42—42no—
250C42—42no—
251C42—42no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—24no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—42no—
260C23—42no—
261C23—44no—
262C23—44no—
263C23—42no—
264C23—42no—
265C23—44no—
266C23—44no—
267C23—42no—
268C23—41no—
269C23—41no—
270C23—44no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—44no—
275C23—82no—
276C23—82no—
277C23—82no—
278C23—82no—
279C23—84no—
280C23—84no—
281C23—84no—
282C23—84no—
283C23—81no—
284C23—82no—
285C23—81no—
286C23#33114yes—
287C23#37911yescharacteristic
288C23C2×#10011yescharacteristic
289C23C2×#10011yescharacteristic
290C23C2×#10212yes—
291C23#41812yes—
292D4—84no—
293D4—84no—
294D4—816no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—816no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4#41012yes—
335C2×C4#41312yes—
336C2×C4#44714yes—
337C2×C4#45014yes—
338C8—164no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—24no—
348C22—44no—
349C22—48no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—84no—
359C22—84no—
360C22—162no—
361C22—164no—
362C22—162no—
363C22C2×#33111yescharacteristic
364C22#154512yes—
365C22#155511yescharacteristic
366C22#178211yescharacteristic
367C22#178311yescharacteristic
368C22#178511yescharacteristic
369C22#178612yes—
370C22#179612yes—
371C4—22no—
372C4—22no—
373C4—24no—
374C4—24no—
375C4—42no—
376C4—44no—
377C4—82no—
378C4—82no—
379C4—82no—
380C2—21no—
381C2—22no—
382C2—21no—
383C2—44no—
384C2—162no—
385C2#833911yescharacteristic
386C2#837612yes—
387C2#838712yes—
388C2#852411yescharacteristic
389C2#855311yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#121C4
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
2. #8723C2semidirect
3. #8559C2semidirect
4. #7640C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
#87232
#85592
#76402