Finite Group Databasebeta

← #146468

Subgroups — #146468

Order 512.

1910 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2311C211yesmaximal, characteristic
2C2×#1782C212yesmaximal
3C2×#1393C214yesmaximal, complemented
4C23×#41C2211yescharacteristic
5C22×#117C411yescharacteristic, complemented
6C23×C4⋊C4C411yescharacteristic, complemented
7C22×#99—22no—
8C22×#99C2214yes—
9#2311—28no—
10C2×#391C412yescomplemented
11C2×#390C2212yes—
12C2×#386—22no—
13C2×#381—22no—
14C2×#380—22no—
15#1782—24no—
16#1782C418yescomplemented
17#1393—28no—
18C22×#41—24no—
19C22×#41—24no—
20C22×#41—42no—
21C22×#41—42no—
22C22×#41—42no—
23C22×#41—42no—
24C22×#41C2×C414yes—
25C22×#41D412yes—
26C22×#41Q812yes—
27C23×Q8C2×C411yescharacteristic
28C23×D4D412yes—
29C23×D4C2311yescharacteristic
30C24×C4—21no—
31C24×C4C2×C411yescharacteristic
32C24×C4D412yes—
33C2×#117—24no—
34C2×#117—42no—
35C2×#117—42no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—24no—
39C22×C4⋊C4—42no—
40C22×C4⋊C4—42no—
41C22×C4⋊C4C2×C412yes—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—24no—
44C22×C22⋊C4—24no—
45C22×C42C2×C412yes—
46C2×#99—28no—
47C2×#99—28no—
48C2×#99—28no—
49C2×#99—24no—
50C2×#99—44no—
51C2×#99—44no—
52C2×#99C2×C418yescomplemented
53#391—24no—
54#391—24no—
55#390—24no—
56#390D418yes—
57#386—216no—
58#381—28no—
59#381—44no—
60#380—216no—
61C2×#95—24no—
62C2×#95—44no—
63C25C22⋊C411yescharacteristic
64C25C2×D412yes—
65C2×#41—216no—
66C2×#41—28no—
67C2×#41—28no—
68C2×#41—48no—
69C2×#41—44no—
70C2×#41—44no—
71C2×#41—48no—
72C2×#41—44no—
73C2×#41—44no—
74C2×#41—416no—
75C2×#41—416no—
76C2×#41—416no—
77C2×#41—416no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8C4212yes—
85C22×Q8C4⋊C412yes—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—24no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4—41no—
93C22×D4—41no—
94C22×D4—41no—
95C22×D4—41no—
96C22×D4C22⋊C414yes—
97C22×D4C4⋊C414yes—
98C22×D4C22×C412yes—
99C22×D4C2×D414yes—
100C22×D4C2×D411yescharacteristic
101C22×D4C2×Q811yescharacteristic
102C22×D4#4114yes—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—44no—
121C23×C4—84no—
122C23×C4C4212yes—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C414yes—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C4⋊C412yes—
127C23×C4C4⋊C414yes—
128C23×C4C22×C411yescharacteristic
129#117—416no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—416no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C22⋊C414yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—216no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—88no—
152C2×C42—22no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42C22⋊C414yes—
156#99—216no—
157#99—416no—
158#99—48no—
159#95—216no—
160#95—416no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—42no—
170C24—42no—
171C24—44no—
172C24—44no—
173C24#9512yes—
174C24#9912yes—
175C24#9912yes—
176C24C2×C22⋊C411yescharacteristic
177C24C2×C22⋊C412yes—
178C24C2×C4⋊C412yes—
179C24#12014yes—
180C24#12314yes—
181#41—464no—
182#41—432no—
183#41—432no—
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×D4—28no—
195C2×D4—28no—
196C2×D4—28no—
197C2×D4—28no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—48no—
211C2×D4—48no—
212C2×D4—48no—
213C2×D4—48no—
214C2×D4C4×D418yes—
215C2×D4#12114yes—
216C2×D4C22⋊Q814yes—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—48no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4—88no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—88no—
268C22×C4—88no—
269C22×C4#9514yes—
270C22×C4#9512yes—
271C22×C4#9912yes—
272C22×C4#9912yes—
273C22×C4C2×C4211yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C4⋊C411yescharacteristic
276C22×C4#11712yes—
277C4⋊C4—44no—
278C4⋊C4—48no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—48no—
284C22⋊C4—816no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C42—44no—
288C42—44no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—24no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—24no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—44no—
307C23—44no—
308C23—42no—
309C23—42no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—48no—
318C23—42no—
319C23—44no—
320C23—44no—
321C23—42no—
322C23—84no—
323C23—84no—
324C23—82no—
325C23—82no—
326C23#34214yes—
327C23C2×#9511yescharacteristic
328C23#37911yescharacteristic
329C23#38111yescharacteristic
330C23#38514yes—
331C23#38614yes—
332C23C2×#9911yescharacteristic
333C23C2×#9911yescharacteristic
334C23C2×#9911yescharacteristic
335C23C2×#9911yescharacteristic
336Q8—48no—
337Q8—48no—
338D4—48no—
339D4—44no—
340D4—44no—
341D4—416no—
342D4—416no—
343C2×C4—216no—
344C2×C4—216no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—416no—
363C2×C4—416no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—816no—
377C2×C4—84no—
378C2×C4—88no—
379C2×C4—816no—
380C2×C4—168no—
381C2×C4—164no—
382C2×C4—164no—
383C2×C4#34214yes—
384C2×C4C4×C22⋊C412yes—
385C2×C4#38012yes—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—24no—
396C22—24no—
397C22—44no—
398C22—44no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—44no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—42no—
412C22—84no—
413C22—84no—
414C22—82no—
415C22—82no—
416C22C22⋊#9511yescharacteristic
417C22C2×#34211yescharacteristic
418C22C2×#34211yescharacteristic
419C22#178111yescharacteristic
420C22#178111yescharacteristic
421C22#181911yescharacteristic
422C22#181911yescharacteristic
423C22#212812yes—
424C22#212912yes—
425C4—24no—
426C4—22no—
427C4—22no—
428C4—44no—
429C4—84no—
430C4—84no—
431C4—168no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—44no—
437C2—44no—
438C2—44no—
439C2#1072511yescharacteristic
440C2#1258312yes—
441C2#1269711yescharacteristic
442C2#1269811yescharacteristic
443C2#1285912yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC23×D4C23
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1393C2semidirect
5. C22×#117C4semidirect
6. C23×C4⋊C4C4semidirect
10. C2×#391C4semidirect
16. #1782C4semidirect
52. C2×#99C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#17822
C2×#13932