Finite Group Databasebeta

← #128368

Subgroups — #128368

Order 512.

876 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C4×#390C211yesmaximal, characteristic
2C2×#1318C212yesmaximal
3#8153C212yesmaximal, complemented
4#7189C212yesmaximal, complemented
5C22×C4×D4C411yescharacteristic
6C2×#390C411yescharacteristic
7C2×#390C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×C4×C22⋊C4—22no—
10C2×C43C411yescharacteristic
11#1749—22no—
12C2×#324C2214yes—
13#1437—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16#1318C418yescomplemented
17#1280—22no—
18C23×D4C2×C411yescharacteristic
19C24×C4—21no—
20C2×C4×D4—22no—
21C2×C4×D4—22no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4—42no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2311yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—22no—
31C22×C42—21no—
32C22×C42—21no—
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42D412yes—
37C2×C22⋊C8—24no—
38C2×#97—42no—
39#390—22no—
40#390—22no—
41#390—22no—
42#390C2×C414yes—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C4×C22⋊C4—48no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95D412yes—
49C43—42no—
50#336—42no—
51#324—24no—
52#324C2×C418yescomplemented
53C25—21no—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—41no—
58C22×D4—41no—
59C22×D4—41no—
60C22×D4—41no—
61C22×D4C4212yes—
62C22×D4M1612yes—
63C22×D4M1612yes—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—24no—
67C23×C4—21no—
68C23×C4—42no—
69C23×C4—44no—
70C23×C4—44no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C22×C8—44no—
79C4×D4—44no—
80C4×D4—44no—
81C4×D4—44no—
82C4×D4—44no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4M1612yes—
93C2×C4⋊C4C22×C412yes—
94C2×C4⋊C4#4112yes—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—42no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—24no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—22no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—44no—
124C2×C42C4212yes—
125C2×C42M1612yes—
126C22⋊C8—28no—
127C22⋊C8—44no—
128#97—84no—
129#95—42no—
130#95—44no—
131#95—44no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24—42no—
136C24—42no—
137C24—42no—
138C24—42no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24C2×C22⋊C411yescharacteristic
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4#9714yescomplemented
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—416no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—88no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4#9911yescharacteristic
209C22×C4#9911yescharacteristic
210C22×C4#10012yes—
211C22×C4C2×C4211yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4C4×D414yes—
214C22×C4C2×M1611yescharacteristic
215C22×C4C2×M1611yescharacteristic
216C22×C4C2×M1611yescharacteristic
217C22×C4C2×M1611yescharacteristic
218C2×C8—48no—
219C2×C8—84no—
220C2×C8—84no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C22⋊C4—416no—
228C22⋊C4—416no—
229C22⋊C4—48no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—42no—
235C42—42no—
236C42—48no—
237C42—42no—
238C42—42no—
239C42—48no—
240C42—44no—
241C42—82no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—24no—
247C23—44no—
248C23—44no—
249C23—44no—
250C23—44no—
251C23—44no—
252C23—44no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—82no—
260C23—82no—
261C23—82no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23C4×C22⋊C411yescharacteristic
270C23#40711yescharacteristic
271C23#40711yescharacteristic
272C23C2×#9911yescharacteristic
273C23#41011yescharacteristic
274C23C2×#10011yescharacteristic
275C23#41311yescharacteristic
276D4—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—88no—
328C2×C4—88no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4#35312yes—
335C2×C4#35512yes—
336C2×C4C2×#9711yescharacteristic
337C2×C4C4×M1611yescharacteristic
338C2×C4#43211yescharacteristic
339C2×C4#43211yescharacteristic
340C8—88no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—48no—
354C22—44no—
355C22—84no—
356C22—84no—
357C22—84no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22#146211yescharacteristic
365C22#146411yescharacteristic
366C22#146511yescharacteristic
367C22#146611yescharacteristic
368C22#174911yescharacteristic
369C22C4×#9911yescharacteristic
370C22C4×#10011yescharacteristic
371C22C2×#35311yescharacteristic
372C22#212611yescharacteristic
373C22C2×#35511yescharacteristic
374C22#213211yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—22no—
380C4—22no—
381C4—44no—
382C4—44no—
383C4—44no—
384C4—42no—
385C4—82no—
386C4—82no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—84no—
392C2#815211yescharacteristic
393C2#1150511yescharacteristic
394C2#1150711yescharacteristic
395C2#1152911yescharacteristic
396C2#1153011yescharacteristic
397C2C4×#35311yescharacteristic
398C2C4×#35511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
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. #8153C2semidirect
4. #7189C2semidirect
16. #1318C4semidirect
52. #324C2×C4semidirect
159. C2×D4#97semidirect

Maximal subgroups

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

subgroupindex
C4×#3902
C2×#13182
#81532
#71892