Finite Group Databasebeta

← #129337

Subgroups — #129337

Order 512.

1100 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1100 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×C4×#124C211yesmaximal, characteristic
2C2×#1324C212yesmaximal, complemented
3C2×#1318C212yesmaximal
4#8303C212yesmaximal, complemented
5C22×#124C411yescharacteristic
6C22×C4×Q8C2211yescharacteristic
7C22×C4×D4C411yescharacteristic
8C4×#124—24no—
9C4×#124—24no—
10C2×#391C411yescharacteristic
11C2×#390C2211yescharacteristic
12C2×#388—22no—
13C2×#382C2211yescharacteristic
14C2×C4×C22⋊C4—22no—
15C2×C43C411yescharacteristic
16#1817—22no—
17C2×#324—22no—
18C2×#324C2214yes—
19#1324—24no—
20#1318—24no—
21C23×Q8C2×C411yescharacteristic
22C23×D4C2×C411yescharacteristic
23C24×C4—21no—
24C2×#124—22no—
25C2×#124—22no—
26C2×#124—22no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124C814yescomplemented
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—42no—
33C2×C4×Q8—42no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—42no—
37C2×C4×D4—42no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4D412yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—22no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—22no—
44C22×C42—21no—
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42D412yes—
50C2×C22⋊C8—24no—
51#391—22no—
52#391—22no—
53#390—22no—
54#390D414yes—
55#388—216no—
56#382—22no—
57#382D414yes—
58C4×C22⋊C4—216no—
59C2×#95—22no—
60C2×#95—22no—
61C43—22no—
62C43—22no—
63#336—44no—
64#324—24no—
65#324—24no—
66#324—44no—
67C25—21no—
68C22×Q8—21no—
69C22×Q8—21no—
70C22×Q8—21no—
71C22×Q8—21no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2×C812yes—
77C22×Q8M1612yes—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4C22⋊C412yes—
86C22×D4C2×C812yes—
87C22×D4M1612yes—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—24no—
91C23×C4—21no—
92C23×C4—42no—
93C23×C4—48no—
94C23×C4—42no—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D412yes—
100C22×C8—44no—
101#124—28no—
102#124—28no—
103#124—28no—
104#124—28no—
105C4×Q8—48no—
106C4×Q8—48no—
107C4×D4—48no—
108C4×D4—48no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊C412yes—
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D412yes—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—48no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—24no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—21no—
139C2×C42—24no—
140C2×C42—42no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—44no—
144C2×C42—44no—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C2×C812yes—
148C2×C42M1612yes—
149C22⋊C8—44no—
150C22⋊C8—44no—
151#95—28no—
152#95—28no—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—44no—
157C24—44no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24C2×C22⋊C411yescharacteristic
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4C22⋊C814yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—48no—
219C22×C4—416no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—88no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4C22⋊C814yes—
234C22×C4#9912yes—
235C22×C4#9912yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4#12014yes—
239C22×C4C22×C811yescharacteristic
240C22×C4C2×M1611yescharacteristic
241C22×C4C2×M1611yescharacteristic
242C22×C4C2×M1611yescharacteristic
243C2×C8—84no—
244C2×C8—88no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—48no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C22⋊C4—232no—
251C22⋊C4—416no—
252C22⋊C4—48no—
253C42—24no—
254C42—24no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—48no—
259C42—42no—
260C42—48no—
261C42—42no—
262C42—48no—
263C42C22⋊C814yes—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—44no—
268C23—44no—
269C23—48no—
270C23—48no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—82no—
278C23—84no—
279C23—82no—
280C23—84no—
281C23—82no—
282C23—82no—
283C23#37911yescharacteristic
284C23C2×C22⋊C811yescharacteristic
285C23#40711yescharacteristic
286C23C2×#9911yescharacteristic
287C23C2×#9911yescharacteristic
288C23#41311yescharacteristic
289C23#41311yescharacteristic
290Q8—44no—
291Q8—44no—
292Q8—44no—
293Q8—44no—
294D4—44no—
295D4—44no—
296D4—44no—
297D4—44no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—816no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4#35514yes—
345C2×C4C2×C22⋊C811yescharacteristic
346C2×C4C2×C22⋊C811yescharacteristic
347C2×C4#40711yescharacteristic
348C2×C4#40711yescharacteristic
349C8—164no—
350C22—22no—
351C22—22no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—48no—
359C22—44no—
360C22—84no—
361C22—84no—
362C22—84no—
363C22—84no—
364C22—82no—
365C22—82no—
366C22—84no—
367C22#145911yescharacteristic
368C22#145911yescharacteristic
369C22#146511yescharacteristic
370C22#146511yescharacteristic
371C22#177911yescharacteristic
372C22#178111yescharacteristic
373C22#178511yescharacteristic
374C22#212612yes—
375C22C2×#35512yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—24no—
381C4—44no—
382C4—44no—
383C4—44no—
384C4—44no—
385C4—42no—
386C4—82no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—84no—
392C2#830111yescharacteristic
393C2#1147612yes—
394C2#1153012yes—
395C2#1269511yescharacteristic
396C2#1269911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1324C2semidirect
4. #8303C2semidirect
29. C2×#124C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#13242
C2×#13182
#83032