Finite Group Databasebeta

← #146472

Subgroups — #146472

Order 512.

1842 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1842 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×#1785C212yesmaximal
3C2×#1394C214yesmaximal, complemented
4C23×#41C2211yescharacteristic
5C22×#117C411yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C22×#101C2214yes—
8C22×#100—22no—
9C2×#407—22no—
10#2311—28no—
11C2×#391C2212yes—
12C2×#390C412yes—
13C2×#386—22no—
14C2×#380—22no—
15#1785—24no—
16#1785C418yescomplemented
17#1394—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×Q8C2311yescharacteristic
28C23×D4C2×C411yescharacteristic
29C23×D4D412yes—
30C24×C4—21no—
31C24×C4C2×C411yescharacteristic
32C24×C4D412yes—
33C22×M16—24no—
34C2×#117—24no—
35C2×#117—42no—
36C2×#117—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4—42no—
41C22×C4⋊C4—42no—
42C22×C4⋊C4C2×C412yes—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—24no—
45C22×C42C2×C412yes—
46C2×#101—28no—
47C2×#101—28no—
48C2×#101—24no—
49C2×#101C2×C418yescomplemented
50C2×#100—28no—
51C2×#100—44no—
52C2×#100—44no—
53#407—28no—
54#407—44no—
55C2×C22⋊C8—44no—
56#391—24no—
57#391D418yes—
58#390—24no—
59#390—24no—
60#386—216no—
61#380—216no—
62C2×#95—24no—
63C25C22⋊C412yes—
64C25C22⋊C411yescharacteristic
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—41no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8C22×C412yes—
83C22×Q8C2×D411yescharacteristic
84C22×Q8C2×D414yes—
85C22×Q8C2×Q811yescharacteristic
86C22×Q8#4114yes—
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—24no—
92C22×D4—24no—
93C22×D4—41no—
94C22×D4—41no—
95C22×D4—41no—
96C22×D4—41no—
97C22×D4—42no—
98C22×D4—42no—
99C22×D4C4212yes—
100C22×D4C22⋊C414yes—
101C22×D4C4⋊C414yes—
102C22×D4C4⋊C412yes—
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—84no—
121C23×C4C4212yes—
122C23×C4C22⋊C414yes—
123C23×C4C22⋊C411yescharacteristic
124C23×C4C4⋊C412yes—
125C23×C4C4⋊C414yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D412yes—
128C2×M16—28no—
129C2×M16—44no—
130C2×M16—44no—
131C2×M16—44no—
132C2×M16—44no—
133C2×M16—44no—
134C22×C8—44no—
135#117—416no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—416no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C22⋊C414yes—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—88no—
152C2×C42—22no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42C22⋊C414yes—
156#101—216no—
157#101—48no—
158#100—416no—
159C22⋊C8—416no—
160#95—216no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24—42no—
172C24—42no—
173C24—44no—
174C24—44no—
175C24#9514yes—
176C24#9512yes—
177C24#9912yes—
178C24#9912yes—
179C24C2×C22⋊C411yescharacteristic
180#41—464no—
181#41—432no—
182#41—432no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8C4×D418yes—
194C2×Q8#12114yes—
195C2×Q8C22⋊Q814yes—
196C2×D4—28no—
197C2×D4—28no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—28no—
201C2×D4—28no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—48no—
209C2×D4—48no—
210C2×D4—48no—
211C2×D4—48no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—28no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—88no—
259C22×C4—88no—
260C22×C4—88no—
261C22×C4#9512yes—
262C22×C4#9912yes—
263C22×C4#9912yes—
264C22×C4C2×C4211yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C4⋊C411yescharacteristic
268C22×C4C2×C4⋊C412yes—
269C22×C4#11712yes—
270C22×C4#12014yes—
271C22×C4#12314yes—
272M16—48no—
273M16—48no—
274M16—84no—
275M16—84no—
276C2×C8—48no—
277C2×C8—84no—
278C2×C8—84no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C22⋊C4—48no—
284C22⋊C4—816no—
285C42—44no—
286C42—44no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—24no—
296C23—24no—
297C23—24no—
298C23—42no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—44no—
304C23—44no—
305C23—42no—
306C23—48no—
307C23—44no—
308C23—48no—
309C23—44no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—44no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—84no—
319C23—84no—
320C23—82no—
321C23—82no—
322C23#34214yes—
323C23C2×#9511yescharacteristic
324C23#37911yescharacteristic
325C23#38111yescharacteristic
326C23C2×#9911yescharacteristic
327C23C2×#9911yescharacteristic
328C23C2×#9911yescharacteristic
329C23C2×#9911yescharacteristic
330Q8—48no—
331Q8—44no—
332Q8—44no—
333D4—416no—
334D4—416no—
335D4—48no—
336D4—48no—
337C2×C4—216no—
338C2×C4—216no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—416no—
362C2×C4—416no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—816no—
372C2×C4—84no—
373C2×C4#34214yes—
374C2×C4C4×C22⋊C412yes—
375C2×C4#38012yes—
376C2×C4#38514yes—
377C2×C4#38614yes—
378C8—88no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—48no—
393C22—44no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—48no—
398C22—44no—
399C22—84no—
400C22—84no—
401C22—84no—
402C22—84no—
403C22—82no—
404C22—82no—
405C22C22⋊#9511yescharacteristic
406C22C2×#34211yescharacteristic
407C22C2×#34211yescharacteristic
408C22#178111yescharacteristic
409C22#178111yescharacteristic
410C22#181911yescharacteristic
411C22#181911yescharacteristic
412C22#213412yes—
413C22#213512yes—
414C4—24no—
415C4—24no—
416C4—22no—
417C4—22no—
418C4—24no—
419C4—84no—
420C4—84no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—21no—
425C2—44no—
426C2—84no—
427C2#1072511yescharacteristic
428C2#1258712yes—
429C2#1270311yescharacteristic
430C2#1270411yescharacteristic
431C2#1286312yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1394C2semidirect
16. #1785C4semidirect
49. C2×#101C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
C2×#17852
C2×#13942