Finite Group Databasebeta

← #97617

Subgroups — #97617

Order 512.

828 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2489C211yesmaximal, characteristic
2#8706C212yesmaximal, complemented
3#8557C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7575C212yesmaximal, complemented
6C22×#125C411yescharacteristic
7C22×C22⋊Q8C411yescharacteristic
8#2489—24no—
9#2489—24no—
10C2×#400—21no—
11C2×#397—21no—
12C2×#391C411yescharacteristic
13C2×#391C2211yescharacteristic
14C2×#388—21no—
15C2×#382—21no—
16C2×#380C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#323—21no—
22C2×#323—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1272—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—44no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—44no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2311yescharacteristic
48C2×C4⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C4×C8—22no—
52#400—28no—
53#397—28no—
54#391—22no—
55#391—22no—
56#391—22no—
57#391D414yes—
58#388—28no—
59#382—28no—
60#380—22no—
61#380—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4D414yes—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#323—24no—
77#323—42no—
78C25C22⋊C411yescharacteristic
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—42no—
83C22×Q8—42no—
84C22×Q8C22⋊C412yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C22×C8—42no—
102C22×C8—42no—
103#125—48no—
104#125—44no—
105#125—44no—
106C22⋊Q8—48no—
107C22⋊Q8—44no—
108C22⋊Q8—44no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C22⋊C412yes—
124C2×C4⋊C4C22⋊C412yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—24no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42—42no—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C4⋊C8—44no—
148C22⋊C8—24no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152C4×C8—44no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—22no—
160#95—22no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24#10412yes—
172C24C2×C22⋊C411yescharacteristic
173C2×Q8—24no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—84no—
180C2×Q8—84no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4#12012yes—
237C22×C4#12012yes—
238C22×C4#13112yes—
239C22×C4#13112yes—
240C22×C4#13512yes—
241C22×C4AGL(1,ℤ/8)11yescharacteristic
242C22×C4#13711yescharacteristic
243C2×C8—44no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—48no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4#10014yes—
259C22⋊C4—44no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—24no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—44no—
269C42—44no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23#37911yescharacteristic
298C23#40811yescharacteristic
299C23#40811yescharacteristic
300C23#41611yescharacteristic
301C23#41711yescharacteristic
302C23C2×#10411yescharacteristic
303C23#42111yescharacteristic
304Q8—84no—
305Q8—84no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—84no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—84no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#40812yes—
360C2×C4#40812yes—
361C2×C4#41012yes—
362C2×C4C2×#10012yes—
363C2×C4#44912yes—
364C2×C4#45212yes—
365C8—84no—
366C8—162no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#148211yescharacteristic
391C22#148311yescharacteristic
392C22#149411yescharacteristic
393C22#149511yescharacteristic
394C22#178011yescharacteristic
395C22#178211yescharacteristic
396C22#178311yescharacteristic
397C22#178811yescharacteristic
398C22#178911yescharacteristic
399C22#179611yescharacteristic
400C22#179611yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—44no—
406C4—81no—
407C4—82no—
408C4—82no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4#146914yes—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—82no—
418C2#830511yescharacteristic
419C2#831511yescharacteristic
420C2#832711yescharacteristic
421C2#836311yescharacteristic
422C2#836911yescharacteristic
423C2#838211yescharacteristic
424C2#839311yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ 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. #8706C2semidirect
5. #7575C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24892
#87062
#85572
C4×#3232
#75752