Finite Group Databasebeta

← #128424

Subgroups — #128424

Order 512.

640 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 112 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#13992C211yesmaximal, characteristic
2C2×#1317C212yesmaximal
3#8152C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7187C212yesmaximal
6C2×#385C411yescharacteristic
7C2×#381C411yescharacteristic
8C2×#381C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C411yescharacteristic
11#1749—21no—
12C4×C22⋊C8—21no—
13C2×#323C2212yes—
14C2×#323C2212yes—
15#1443—21no—
16#1442—21no—
17#1436—21no—
18#1434—21no—
19C22⋊#95C411yescharacteristic
20C22⋊#95C2211yescharacteristic
21#1317C418yescomplemented
22#1280—21no—
23#1280—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2311yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×#97—41no—
37C2×C4×C8—41no—
38#385—42no—
39#381—22no—
40#381—42no—
41#381C2×C414yes—
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×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—41no—
56C2×#95—41no—
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59#336—41no—
60#336—41no—
61#323—22no—
62#323—22no—
63#323C2×C414yes—
64#323C2×C414yes—
65C25C22×C411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—41no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—81no—
87C23×C4—81no—
88C23×C4C4212yes—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—42no—
96C22×C8—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—81no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4C4212yes—
112C2×C22⋊C4C22×C412yes—
113C2×C22⋊C4#4112yes—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—42no—
119C2×C42—81no—
120C22⋊C8—24no—
121C22⋊C8—24no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#97—82no—
125C4×C8—82no—
126#95—24no—
127#95—24no—
128#95—41no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—41no—
135#95—41no—
136#95—42no—
137#95—42no—
138#95—81no—
139#95—81no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24C2×C4211yescharacteristic
149C24C2×C22⋊C411yescharacteristic
150C24#11711yescharacteristic
151C24#13112yes—
152C24#13112yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
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—42no—
192C22×C4—42no—
193C22×C4—42no—
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—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4—84no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4#9911yescharacteristic
232C22×C4#9911yescharacteristic
233C22×C4#10012yes—
234C22×C4C4×D412yes—
235C22×C4C4×D412yes—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—82no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C4⋊C4—84no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—82no—
249C42—42no—
250C42—42no—
251C42—81no—
252C42—81no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—42no—
266C23—42no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C4×C22⋊C411yescharacteristic
284C23#40512yes—
285C23#40811yescharacteristic
286C23#40811yescharacteristic
287C23C2×#9911yescharacteristic
288C23#41011yescharacteristic
289C23C2×#10011yescharacteristic
290C23#41311yescharacteristic
291C23#43312yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
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—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—84no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—84no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—84no—
339C2×C4—84no—
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—84no—
350C2×C4—84no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4#35212yes—
359C8—84no—
360C8—84no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—44no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22#146811yescharacteristic
390C22#146911yescharacteristic
391C22#147111yescharacteristic
392C22#147311yescharacteristic
393C22#175011yescharacteristic
394C22C4×#9911yescharacteristic
395C22C4×#10011yescharacteristic
396C22C2×#35211yescharacteristic
397C22#212011yescharacteristic
398C22#212111yescharacteristic
399C22#212311yescharacteristic
400C4—22no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—82no—
409C4—82no—
410C4—161no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—42no—
417C2#815511yescharacteristic
418C2#1154311yescharacteristic
419C2#1154411yescharacteristic
420C2#1155811yescharacteristic
421C2#1155911yescharacteristic
422C2C4×#35211yescharacteristic
423C2#1262111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1317C4semidirect

Maximal subgroups

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

subgroupindex
#139922
C2×#13172
#81522
C4×#3232
#71872