Finite Group Databasebeta

← #96936

Subgroups — #96936

Order 512.

1344 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 170 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×#2850C211yesmaximal, characteristic
2#10712C212yesmaximal, complemented
3C2×#1272C212yesmaximal, complemented
4C2×#1270C212yesmaximal, complemented
5C22×#124C411yescharacteristic
6C22×C22⋊Q8C411yescharacteristic
7C22×#121C411yescharacteristic
8C23×C42C2211yescharacteristic
9C22×C22⋊C8—22no—
10#2850—24no—
11#2850—24no—
12C2×#391C2211yescharacteristic
13C2×#390C2211yescharacteristic
14C2×#388—22no—
15C2×#386—22no—
16C2×#382C411yescharacteristic
17C2×#336—22no—
18C2×#336C2214yes—
19#1272—24no—
20#1270—24no—
21C23×Q8C2×C411yescharacteristic
22C23×D4—21no—
23C23×D4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4D412yes—
26C23×C8—24no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124—44no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—42no—
33C2×C22⋊Q8—42no—
34C2×#121—22no—
35C2×#121—22no—
36C2×#121—42no—
37C2×#121—42no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—22no—
45C22×C42—22no—
46C22×C42—22no—
47C22×C42—44no—
48C22×C42C2×C411yescharacteristic
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51C2×C22⋊C8—28no—
52C2×C22⋊C8—44no—
53C2×C22⋊C8—44no—
54#391—22no—
55#391D414yes—
56#390—22no—
57#390D414yes—
58#388—216no—
59#386—216no—
60#382—22no—
61#382—22no—
62C2×#95—22no—
63C2×#95—22no—
64#336—24no—
65#336—24no—
66#336—44no—
67C25—21no—
68C25C22⋊C411yescharacteristic
69C22×Q8—21no—
70C22×Q8—24no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—42no—
74C22×Q8C22⋊C412yes—
75C22×D4—24no—
76C22×D4—21no—
77C22×D4—24no—
78C22×D4—41no—
79C22×D4—41no—
80C22×D4—42no—
81C22×D4—42no—
82C22×D4—44no—
83C22×D4—42no—
84C22×D4—44no—
85C22×D4C22⋊C412yes—
86C23×C4—22no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4C22⋊C412yes—
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D412yes—
107C22×C8—28no—
108C22×C8—44no—
109C22×C8—44no—
110C22×C8—44no—
111C22×C8—44no—
112C22×C8—44no—
113C22×C8—44no—
114#124—48no—
115#124—48no—
116C22⋊Q8—48no—
117C22⋊Q8—48no—
118#121—48no—
119#121—48no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C42—24no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—48no—
150C2×C42—48no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C22⋊C8—416no—
156#95—28no—
157#95—28no—
158C24—21no—
159C24—22no—
160C24—22no—
161C24—21no—
162C24—22no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—42no—
167C24—42no—
168C24—41no—
169C24—41no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24—42no—
174C24#10012yes—
175C24#10414yes—
176C24C2×C22⋊C411yescharacteristic
177C2×Q8—24no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—84no—
184C2×Q8—84no—
185C2×D4—24no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—48no—
191C2×D4—416no—
192C2×D4—42no—
193C2×D4—84no—
194C2×D4—84no—
195C2×D4—88no—
196C2×D4—84no—
197C2×D4—84no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—48no—
225C22×C4—48no—
226C22×C4—48no—
227C22×C4—48no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4#9914yes—
249C22×C4#10012yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#12014yes—
253C22×C4#13114yes—
254C22×C4#13514yes—
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4#13712yes—
257C2×C8—48no—
258C2×C8—48no—
259C2×C8—48no—
260C2×C8—84no—
261C2×C8—84no—
262C2×C8—84no—
263C2×C8—84no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—416no—
271C22⋊C4—48no—
272C22⋊C4—48no—
273C22⋊C4—48no—
274C42—24no—
275C42—24no—
276C42—42no—
277C42—42no—
278C42—48no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—24no—
286C23—22no—
287C23—22no—
288C23—24no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—41no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23—81no—
305C23—82no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—81no—
318C23—82no—
319C23—81no—
320C23#32914yes—
321C23#37911yescharacteristic
322C23#40812yes—
323C23C2×#9912yes—
324C23C2×#10011yescharacteristic
325C23C2×#10011yescharacteristic
326C23#41612yes—
327C23#41712yes—
328C23C2×#10412yes—
329C23#42112yes—
330Q8—84no—
331Q8—84no—
332D4—84no—
333D4—84no—
334D4—816no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—416no—
357C2×C4—416no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4#40812yes—
381C2×C4#40812yes—
382C2×C4#44914yes—
383C2×C4#45214yes—
384C8—88no—
385C22—28no—
386C22—24no—
387C22—44no—
388C22—44no—
389C22—42no—
390C22—44no—
391C22—44no—
392C22—42no—
393C22—44no—
394C22—82no—
395C22—84no—
396C22—84no—
397C22—82no—
398C22—82no—
399C22—162no—
400C22—162no—
401C22—162no—
402C22—162no—
403C22#147112yes—
404C22#147112yes—
405C22#148212yes—
406C22#148312yes—
407C22#149412yes—
408C22#149512yes—
409C22#150112yes—
410C22#150412yes—
411C22#151312yes—
412C22#151612yes—
413C22C2×#32911yescharacteristic
414C22#152712yes—
415C22#153311yescharacteristic
416C22#154311yescharacteristic
417C22#155112yes—
418C22#156011yescharacteristic
419C22#178011yescharacteristic
420C22#178111yescharacteristic
421C22#178311yescharacteristic
422C22#178812yes—
423C22#178912yes—
424C4—22no—
425C4—22no—
426C4—24no—
427C4—24no—
428C4—42no—
429C4—44no—
430C4—82no—
431C4—82no—
432C4—82no—
433C2—44no—
434C2—162no—
435C2#767012yes—
436C2#769012yes—
437C2#771012yes—
438C2#773012yes—
439C2#830711yescharacteristic
440C2#831512yes—
441C2#833112yes—
442C2#833411yescharacteristic
443C2#834411yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#121C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10712C2semidirect
3. C2×#1272C2semidirect
4. C2×#1270C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#107122
C2×#12722
C2×#12702