Finite Group Databasebeta

← #134232

Subgroups — #134232

Order 512.

760 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 122 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
1#13785C211yesmaximal, characteristic
2C2×#1318C212yesmaximal
3#8558C212yesmaximal, complemented
4#7189C212yesmaximal, complemented
5C22×C4×D4C411yescharacteristic
6C2×#390C411yescharacteristic
7C2×#390C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×#376C411yescharacteristic
10#1797—22no—
11C2×#324C2214yes—
12#1437—22no—
13C4⋊#95C411yescharacteristic
14C4⋊#95C2211yescharacteristic
15C22⋊#95—22no—
16#1318—24no—
17#1280—22no—
18C23×D4C2×C411yescharacteristic
19C24×C4—21no—
20C2×C4×D4—22no—
21C2×C4×D4—42no—
22C2×C4×D4—42no—
23C2×C4×D4C814yescomplemented
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2311yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—22no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D412yes—
34C2×C4⋊C8—42no—
35C2×C22⋊C8—24no—
36#390—22no—
37#390—22no—
38#390—22no—
39#390D412yes—
40#390Q812yes—
41C4×C4⋊C4—42no—
42#376—42no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95D412yes—
48#336—42no—
49#324—24no—
50#324—24no—
51C25—21no—
52C22×D4—21no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4—41no—
56C22×D4—41no—
57C22×D4—41no—
58C22×D4—41no—
59C22×D4C4⋊C412yes—
60C22×D4C2×C812yes—
61C22×D4M1612yes—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—24no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4—82no—
69C23×C4—82no—
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D412yes—
75C22×C8—44no—
76C4×D4—48no—
77C4×D4—48no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2×C812yes—
88C2×C4⋊C4C2×D411yescharacteristic
89C2×C4⋊C4C2×Q811yescharacteristic
90C2×C4⋊C4#4112yes—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—28no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—82no—
98C2×C22⋊C4—82no—
99C2×C22⋊C4—82no—
100C2×C22⋊C4—82no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—82no—
112C2×C42C4⋊C412yes—
113C2×C42C2×C812yes—
114C2×C42M1612yes—
115C2×C42M1612yes—
116C4⋊C8—84no—
117C22⋊C8—44no—
118C22⋊C8—44no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—82no—
123#95—82no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—42no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24—41no—
135C24C2×C22⋊C411yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4C4⋊C814yescomplemented
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—48no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—88no—
192C22×C4—88no—
193C22×C4—84no—
194C22×C4—84no—
195C22×C4C22⋊C812yes—
196C22×C4C22⋊C812yes—
197C22×C4#9912yes—
198C22×C4#10012yes—
199C22×C4C2×C4⋊C411yescharacteristic
200C22×C4#11711yescharacteristic
201C22×C4#12112yes—
202C22×C4C22⋊Q812yes—
203C22×C4C22×C811yescharacteristic
204C22×C4C2×M1611yescharacteristic
205C22×C4C2×M1611yescharacteristic
206C22×C4C2×M1611yescharacteristic
207C2×C8—84no—
208C2×C8—84no—
209C2×C8—84no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C22⋊C4—48no—
218C22⋊C4—84no—
219C22⋊C4—84no—
220C22⋊C4—88no—
221C42—24no—
222C42—24no—
223C42—42no—
224C42—42no—
225C42—44no—
226C42—42no—
227C42—42no—
228C42—82no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—24no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—44no—
239C23—44no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—82no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23—82no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23#38011yescharacteristic
257C23C2×C22⋊C811yescharacteristic
258C23#40711yescharacteristic
259C23C2×#9911yescharacteristic
260C23#41011yescharacteristic
261C23C2×#10011yescharacteristic
262C23#41311yescharacteristic
263D4—44no—
264D4—44no—
265D4—44no—
266D4—44no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—816no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4#32314yes—
321C2×C4C2×C4⋊C811yescharacteristic
322C2×C4#42311yescharacteristic
323C2×C4#43111yescharacteristic
324C2×C4#43211yescharacteristic
325C8—164no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—48no—
339C22—44no—
340C22—84no—
341C22—84no—
342C22—84no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22C2×#32311yescharacteristic
350C22#145811yescharacteristic
351C22#146511yescharacteristic
352C22#146611yescharacteristic
353C22#179711yescharacteristic
354C22#179911yescharacteristic
355C22#180011yescharacteristic
356C22#212711yescharacteristic
357C22#212811yescharacteristic
358C22#213311yescharacteristic
359C22#213411yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—24no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4—82no—
371C4—82no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—84no—
377C2#855711yescharacteristic
378C2#1143211yescharacteristic
379C2#1143311yescharacteristic
380C2#1153311yescharacteristic
381C2#1153411yescharacteristic
382C2#1279911yescharacteristic
383C2#1280311yescharacteristic
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 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8558C2semidirect
4. #7189C2semidirect
23. C2×C4×D4C8semidirect
151. C2×D4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#137852
C2×#13182
#85582
#71892