Finite Group Databasebeta

← #100196

Subgroups — #100196

Order 512.

684 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 106 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#21902C211yesmaximal, characteristic
2#13992C211yesmaximal, characteristic
3#8700C211yesmaximal, characteristic
4#8698C212yesmaximal
5#7032C212yesmaximal
6C2×#385C411yescharacteristic
7C2×#381C411yescharacteristic
8C2×#381C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C2211yescharacteristic
11#1923—24no—
12#1797—21no—
13#1779C2212yes—
14#1779C2212yes—
15C4×C22⋊C8—21no—
16#1443—21no—
17#1442—21no—
18#1436—21no—
19#1434—21no—
20C22⋊#95C411yescharacteristic
21C22⋊#95C2211yescharacteristic
22#1280—24no—
23#1280—21no—
24#1280—21no—
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2311yescharacteristic
28C22×C4⋊C4—21no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8D414yes—
42C2×C4×C8—22no—
43#385—42no—
44#381—22no—
45#381—42no—
46#381C814yescomplemented
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95D411yescharacteristic
63C2×#95D411yescharacteristic
64#336—22no—
65#336—22no—
66C25C22×C411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—81no—
86C23×C4—81no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C2×C812yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4#4112yes—
96C22×C8—24no—
97C22×C8—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—81no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4C2×C812yes—
113C2×C22⋊C4C2×C812yes—
114C2×C22⋊C4M1612yes—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—42no—
120C2×C42—81no—
121C4⋊C8—82no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—44no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—44no—
128C22⋊C8—82no—
129C22⋊C8—82no—
130C4×C8—82no—
131#95—24no—
132#95—24no—
133#95—22no—
134#95—22no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—41no—
141#95—42no—
142#95—41no—
143#95—81no—
144#95—81no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24C2×C22⋊C411yescharacteristic
154C24C4×D412yes—
155C24C22×C811yescharacteristic
156C24C2×M1611yescharacteristic
157C24#13112yes—
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—22no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
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—42no—
202C22×C4—42no—
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—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4C22⋊C812yes—
236C22×C4C22⋊C812yes—
237C22×C4#10012yes—
238C22×C4#12011yescharacteristic
239C22×C4#12112yes—
240C22×C4#12311yescharacteristic
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C4⋊C4—84no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—82no—
254C42—42no—
255C42—42no—
256C42—81no—
257C42—81no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23#38611yescharacteristic
289C23C2×C22⋊C811yescharacteristic
290C23#40811yescharacteristic
291C23#41011yescharacteristic
292C23#41011yescharacteristic
293C23C2×#10011yescharacteristic
294C23#41311yescharacteristic
295C23C8×D412yes—
296C23#43512yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—22no—
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—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—84no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—84no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4#45812yes—
364C8—84no—
365C8—162no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—44no—
387C22—42no—
388C22—42no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22#145811yescharacteristic
395C22#146011yescharacteristic
396C22#146911yescharacteristic
397C22#147311yescharacteristic
398C22#186911yescharacteristic
399C22#189011yescharacteristic
400C22#189111yescharacteristic
401C22#189911yescharacteristic
402C22#190011yescharacteristic
403C22#191311yescharacteristic
404C22#191511yescharacteristic
405C4—22no—
406C4—42no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—42no—
411C4—82no—
412C4—81no—
413C4—82no—
414C4—82no—
415C4—161no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—42no—
421C2—42no—
422C2#896611yescharacteristic
423C2#908011yescharacteristic
424C2#910811yescharacteristic
425C2#950011yescharacteristic
426C2#950711yescharacteristic
427C2#951411yescharacteristic
428C2#951611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
46. #381C8semidirect

Maximal subgroups

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

subgroupindex
#219022
#139922
#87002
#86982
#70322