Finite Group Databasebeta

← #125520

Subgroups — #125520

Order 512.

586 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 82 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#15001C211yesmaximal, characteristic
2#8559C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C2×#387—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1797—21no—
10#1797—21no—
11C2×#323—21no—
12C2×#323—21no—
13#1446C411yescharacteristic
14#1446C2211yescharacteristic
15#1444C411yescharacteristic
16#1442C2211yescharacteristic
17#1440—21no—
18#1439—21no—
19#1436—21no—
20C22⋊#95C411yescharacteristic
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C2×C22⋊Q8—44no—
26C2×C22⋊Q8—41no—
27C2×C22⋊Q8—41no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C4⋊C8—41no—
38C2×C4⋊C8—41no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41#387—44no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2311yescharacteristic
56#336—41no—
57#336—41no—
58#323—42no—
59#323—42no—
60C25C22⋊C411yescharacteristic
61C22×Q8—22no—
62C22×Q8—42no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—81no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—41no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—42no—
83C22×C8—42no—
84C22⋊Q8—44no—
85C22⋊Q8—44no—
86C22⋊Q8—44no—
87C22⋊Q8—44no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—81no—
109C2×C22⋊C4SD1612yes—
110C2×C22⋊C4Q1612yes—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—42no—
116C2×C42—81no—
117C4⋊C8—82no—
118C4⋊C8—82no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#95—22no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—81no—
135#95—81no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24#10312yes—
147C24C2×C22⋊C411yescharacteristic
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—82no—
151C2×Q8—84no—
152C2×Q8—84no—
153C2×Q8—82no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
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—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4#9911yescharacteristic
216C22×C4#9911yescharacteristic
217C22×C4#10312yes—
218C22×C4#13112yes—
219C22×C4C2×SD1611yescharacteristic
220C22×C4C2×Q1611yescharacteristic
221C22×C4AGL(1,ℤ/8)11yescharacteristic
222C22×C4#13711yescharacteristic
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C4⋊C4—48no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—48no—
239C22⋊C4—84no—
240C42—42no—
241C42—42no—
242C42—81no—
243C42—81no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23#40811yescharacteristic
272C23C2×#9911yescharacteristic
273C23#41311yescharacteristic
274C23C2×#10311yescharacteristic
275C23#41711yescharacteristic
276C23#42111yescharacteristic
277C23#42111yescharacteristic
278Q8—84no—
279Q8—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—84no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
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×C4AΣL(1,𝔽16)12yes—
347C8—162no—
348C8—162no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#146811yescharacteristic
373C22#148111yescharacteristic
374C22#149511yescharacteristic
375C22#151311yescharacteristic
376C22#152011yescharacteristic
377C22#153511yescharacteristic
378C22#155011yescharacteristic
379C22C2×AΣL(1,𝔽16)11yescharacteristic
380C22#212111yescharacteristic
381C22#212911yescharacteristic
382C22#213211yescharacteristic
383C4—22no—
384C4—41no—
385C4—42no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—42no—
391C4—82no—
392C4—82no—
393C4—81no—
394C4—82no—
395C4—161no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—82no—
401C2#771511yescharacteristic
402C2#1154111yescharacteristic
403C2#1155611yescharacteristic
404C2#1182611yescharacteristic
405C2#1182811yescharacteristic
406C2#1189811yescharacteristic
407C2#1190611yescharacteristic
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. #7131C2semidirect
5. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#150012
#85592
#85572
#71312
#71222