Finite Group Databasebeta

← #124173

Subgroups — #124173

Order 512.

600 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 108 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#14313C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic
3#7907C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7104C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#397C411yescharacteristic
8C2×#389C411yescharacteristic
9C2×#385—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#323—21no—
13#1445C2211yescharacteristic
14#1444C2211yescharacteristic
15#1442—21no—
16#1440—21no—
17#1439C411yescharacteristic
18#1436—21no—
19#1434C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C23×C8—41no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—24no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—41no—
39C2×C22⋊C8—42no—
40C2×C4×C8—41no—
41#397—22no—
42#397C814yescomplemented
43#389—41no—
44#389—41no—
45#385—42no—
46#385—42no—
47C4×C22⋊C4—41no—
48C4×C22⋊C4—41no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95C2311yescharacteristic
61#336—41no—
62#323—24no—
63C25C22⋊C411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—81no—
72C23×C4—81no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4D812yes—
77C23×C4SD1612yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—42no—
82C22×C8—44no—
83C22×C8—42no—
84C22×C8—82no—
85C22×C8—82no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4C2×C812yescomplemented
100C2×C22⋊C4—24no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—81no—
110C2×C22⋊C4—81no—
111C2×C22⋊C4C2×C812yes—
112C2×C22⋊C4M1612yes—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—81no—
118C2×C42—81no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8—44no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C4×C8—82no—
125#95—24no—
126#95—21no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—81no—
136#95—81no—
137#95—81no—
138#95C2×C812yescomplemented
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24C22⋊C812yes—
148C24#10212yes—
149C24#10412yes—
150C24C2×C22⋊C411yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
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—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—84no—
213C22×C4C22⋊C812yes—
214C22×C4#9911yescharacteristic
215C22×C4#9911yescharacteristic
216C22×C4#10212yes—
217C22×C4#10412yes—
218C22×C4C22×C811yescharacteristic
219C22×C4C2×M1611yescharacteristic
220C22×C4C2×D811yescharacteristic
221C22×C4C2×SD1611yescharacteristic
222C22×C4#13512yes—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—88no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—48no—
241C22⋊C4—84no—
242C42—42no—
243C42—81no—
244C42—81no—
245C42—81no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23D4⋊C814yes—
273C23C2×C22⋊C811yescharacteristic
274C23C2×#9911yescharacteristic
275C23#41311yescharacteristic
276C23C2×#10211yescharacteristic
277C23#41611yescharacteristic
278C23C2×#10411yescharacteristic
279C23C2×#10411yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4#32712yes—
348C2×C4#32712yes—
349C8—84no—
350C8—84no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22#145911yescharacteristic
375C22C2×D4⋊C811yescharacteristic
376C22#147611yescharacteristic
377C22C2×#32711yescharacteristic
378C22#150111yescharacteristic
379C22#152411yescharacteristic
380C22#154011yescharacteristic
381C22#212111yescharacteristic
382C22#212311yescharacteristic
383C22#212711yescharacteristic
384C22#213511yescharacteristic
385C4—22no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—42no—
392C4—42no—
393C4—82no—
394C4—82no—
395C4—81no—
396C4—161no—
397C4—161no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—44no—
403C2#756511yescharacteristic
404C2#1147411yescharacteristic
405C2#1148111yescharacteristic
406C2#1169211yescharacteristic
407C2#1170011yescharacteristic
408C2#1173211yescharacteristic
409C2#1173911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect
5. #7104C2semidirect
42. #397C8semidirect
99. C2×C4⋊C4C2×C8semidirect
138. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#143132
C4×#3232
#79072
#71232
#71042