Finite Group Databasebeta

← #125501

Subgroups — #125501

Order 512.

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

normal subgroups only

This group has 544 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#15112C211yesmaximal, characteristic
2#8560C211yesmaximal, characteristic, complemented
3#8558C211yesmaximal, characteristic, complemented
4#7122C212yesmaximal, complemented
5#7103C212yesmaximal
6C2×#397—21no—
7C2×#390C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—21no—
10#1797—21no—
11C2×#324—21no—
12C2×#324—21no—
13#1442C411yescharacteristic
14#1442C2211yescharacteristic
15#1439—21no—
16#1438C411yescharacteristic
17#1438C2211yescharacteristic
18#1437—21no—
19#1437—21no—
20#1434C411yescharacteristic
21#1280C2212yes—
22#1280C2212yes—
23C23×D4C2×C411yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40#397—42no—
41#397—42no—
42#390—41no—
43#390—41no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
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#324—42no—
59#324—42no—
60C25—21no—
61C22×D4—22no—
62C22×D4—41no—
63C22×D4—41no—
64C22×D4—42no—
65C22×D4—81no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4D812yes—
87C2×C4⋊C4SD1612yes—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—81no—
104C2×C22⋊C4—81no—
105C2×C22⋊C4—81no—
106C2×C22⋊C4—81no—
107C2×C22⋊C4—81no—
108C2×C22⋊C4—81no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—22no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—41no—
117C2×C42—81no—
118C2×C42—81no—
119C4⋊C8—82no—
120C4⋊C8—82no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#95—22no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—81no—
135#95—81no—
136C24—22no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24C2×C22⋊C411yescharacteristic
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—82no—
155C2×D4—84no—
156C2×D4—82no—
157C2×D4—84no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
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—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
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—41no—
209C22×C4—41no—
210C22×C4#9911yescharacteristic
211C22×C4#9911yescharacteristic
212C22×C4#10212yes—
213C22×C4#10212yes—
214C22×C4#13112yes—
215C22×C4C2×D811yescharacteristic
216C22×C4C2×SD1611yescharacteristic
217C22×C4AGL(1,ℤ/8)11yescharacteristic
218C22×C4#13711yescharacteristic
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—84no—
238C22⋊C4—82no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—81no—
244C42—81no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23#40811yescharacteristic
285C23C2×#9911yescharacteristic
286C23#41311yescharacteristic
287C23C2×#10211yescharacteristic
288C23#41711yescharacteristic
289C23#42111yescharacteristic
290C23#42111yescharacteristic
291D4—84no—
292D4—84no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
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—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#35512yes—
358C8—162no—
359C8—162no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—161no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22#146811yescharacteristic
388C22#148011yescharacteristic
389C22#149511yescharacteristic
390C22#151311yescharacteristic
391C22#151911yescharacteristic
392C22#153511yescharacteristic
393C22#154711yescharacteristic
394C22#212011yescharacteristic
395C22#212211yescharacteristic
396C22#212711yescharacteristic
397C22C2×#35511yescharacteristic
398C4—22no—
399C4—42no—
400C4—41no—
401C4—42no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—42no—
406C4—41no—
407C4—81no—
408C4—161no—
409C4—161no—
410C4—161no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—162no—
416C2#771411yescharacteristic
417C2#1154211yescharacteristic
418C2#1155311yescharacteristic
419C2#1182511yescharacteristic
420C2#1182911yescharacteristic
421C2#1187511yescharacteristic
422C2#1188211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8560C2semidirect
3. #8558C2semidirect
4. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#151122
#85602
#85582
#71222
#71032