Finite Group Databasebeta

← #124723

Subgroups — #124723

Order 512.

658 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 84 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#13844C211yesmaximal, characteristic
2#8559C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C22×#117C411yescharacteristic
7C2×C4×C4⋊C4—21no—
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11#1749—21no—
12C2×#323—21no—
13C2×#323—21no—
14#1444C411yescharacteristic
15#1444C2211yescharacteristic
16#1442C411yescharacteristic
17#1442C2211yescharacteristic
18#1439—21no—
19#1439—21no—
20C4×#95—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4C2×C411yescharacteristic
24C2×#117—44no—
25C2×#117—41no—
26C2×#117—41no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C2×C4⋊C8—41no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×#97—41no—
43C4×C4⋊C4—44no—
44C4×C22⋊C4—22no—
45C4×C22⋊C4—22no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4—41no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2311yescharacteristic
58#336—41no—
59#336—41no—
60#323—24no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C23×C4—22no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—42no—
81C22×C8—42no—
82#117—44no—
83#117—44no—
84#117—44no—
85#117—44no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C42—24no—
111C2×C42—24no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—42no—
122C2×C42—42no—
123C4⋊C8—82no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128#97—82no—
129#95—21no—
130#95—21no—
131#95—41no—
132#95—41no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95M1612yes—
140#95M1612yes—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24#10112yes—
152C24C2×C22⋊C411yescharacteristic
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—48no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
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—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4#9911yescharacteristic
218C22×C4#9911yescharacteristic
219C22×C4#10412yes—
220C22×C4C2×M1611yescharacteristic
221C22×C4C2×M1611yescharacteristic
222C22×C4#13512yes—
223C22×C4AGL(1,ℤ/8)11yescharacteristic
224C22×C4#13711yescharacteristic
225C2×C8—44no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—48no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C22⋊C4—44no—
239C22⋊C4—48no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—48no—
243C42—48no—
244C42—44no—
245C42—42no—
246C42—44no—
247C42—42no—
248C42—81no—
249C42—81no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23#40711yescharacteristic
278C23C2×#9911yescharacteristic
279C23C2×#10111yescharacteristic
280C23#41611yescharacteristic
281C23#41711yescharacteristic
282C23C2×#10411yescharacteristic
283C23#42111yescharacteristic
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
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—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4AΣL(1,𝔽16)12yes—
352C2×C4#35212yes—
353C8—84no—
354C8—162no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#146311yescharacteristic
379C22#148811yescharacteristic
380C22#148911yescharacteristic
381C22#150111yescharacteristic
382C22#151311yescharacteristic
383C22#153111yescharacteristic
384C22#155211yescharacteristic
385C22C2×AΣL(1,𝔽16)11yescharacteristic
386C22C2×#35211yescharacteristic
387C22#212611yescharacteristic
388C22#213211yescharacteristic
389C4—22no—
390C4—22no—
391C4—44no—
392C4—41no—
393C4—41no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—82no—
399C4—82no—
400C4—81no—
401C4—82no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—82no—
407C2#761311yescharacteristic
408C2#1151311yescharacteristic
409C2#1151711yescharacteristic
410C2#1172711yescharacteristic
411C2#1173311yescharacteristic
412C2#1182011yescharacteristic
413C2#1183011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×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. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#138442
#85592
#81522
#71232
#71222