Finite Group Databasebeta

← #97077

Subgroups — #97077

Order 512.

1428 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×#389C211yesmaximal, characteristic
2#8698C212yesmaximal
3#8299C212yesmaximal
4#8299C212yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C23×C42C411yescharacteristic
10C2×#389—24no—
11C2×#389—24no—
12C2×#389—24no—
13C22×#95C411yescharacteristic
14C22×#95C2211yescharacteristic
15#1779C2214yes—
16C2×#324—24no—
17C2×#324—24no—
18#1280—24no—
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4—44no—
41C22×C4⋊C4—44no—
42C22×C4⋊C4C814yescomplemented
43C22×C4⋊C4D414yes—
44C22×C4⋊C4D414yes—
45C22×C42—22no—
46C22×C42—22no—
47C22×C42—22no—
48C22×C42—44no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—24no—
52#389—232no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95D414yes—
59#324—48no—
60#324—48no—
61C25C22×C411yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—24no—
67C23×C4—24no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C412yes—
104C23×C4C2×C812yes—
105C23×C4C2×C812yes—
106C23×C4C2×C812yes—
107C23×C4M1612yes—
108C23×C4M1612yes—
109C23×C4M1612yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C22×C8—44no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—48no—
129C2×C4⋊C4—48no—
130C2×C4⋊C4—48no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—48no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C42—28no—
137C2×C42—48no—
138C2×C42—48no—
139C2×C42—44no—
140C2×C42—44no—
141C22⋊C8—48no—
142C22⋊C8—48no—
143C22⋊C8—48no—
144#95—28no—
145#95—28no—
146C24—22no—
147C24—22no—
148C24—22no—
149C24—22no—
150C24—22no—
151C24—22no—
152C24C2×C22⋊C411yescharacteristic
153C24C2×C22⋊C411yescharacteristic
154C24C2×C22⋊C411yescharacteristic
155C24C22×C811yescharacteristic
156C24C2×M1611yescharacteristic
157C24C2×M1611yescharacteristic
158C24C2×M1611yescharacteristic
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—28no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—416no—
216C22×C4—416no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—88no—
249C22×C4—88no—
250C22×C4C22⋊C814yes—
251C22×C4C22⋊C814yes—
252C22×C4C22⋊C814yes—
253C22×C4#9914yes—
254C22×C4#9914yes—
255C22×C4#10114yes—
256C22×C4#10114yes—
257C22×C4#12014yes—
258C2×C8—84no—
259C2×C8—84no—
260C2×C8—84no—
261C4⋊C4—232no—
262C4⋊C4—416no—
263C4⋊C4—416no—
264C4⋊C4—416no—
265C42—416no—
266C23—28no—
267C23—28no—
268C23—28no—
269C23—28no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#37911yescharacteristic
295C23C2×C22⋊C811yescharacteristic
296C23C2×C22⋊C811yescharacteristic
297C23C2×C22⋊C811yescharacteristic
298C23#40711yescharacteristic
299C23#40711yescharacteristic
300C23#40711yescharacteristic
301C23C2×#9912yes—
302C23C2×#9912yes—
303C23#41012yes—
304C23C2×#10112yes—
305C23C2×#10112yes—
306C23#41312yes—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—28no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4#32418yescomplemented
362C8—164no—
363C22—28no—
364C22—28no—
365C22—28no—
366C22—28no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22C2×#32412yes—
392C22#145912yes—
393C22#146112yes—
394C22#146312yes—
395C22#146512yes—
396C22#146712yes—
397C22#177911yescharacteristic
398C22#178111yescharacteristic
399C22#178111yescharacteristic
400C22#178211yescharacteristic
401C22#178411yescharacteristic
402C22#178411yescharacteristic
403C22#178511yescharacteristic
404C4—28no—
405C4—44no—
406C4—44no—
407C4—44no—
408C4—44no—
409C4—44no—
410C4—82no—
411C2—22no—
412C2—22no—
413C2—22no—
414C2—22no—
415C2—22no—
416C2—22no—
417C2#829911yescharacteristic
418C2#830111yescharacteristic
419C2#830311yescharacteristic
420C2#854211yescharacteristic
421C2#855011yescharacteristic
422C2#855111yescharacteristic
423C2#855411yescharacteristic
subgroupG/H
center, socleC23#379
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
42. C22×C4⋊C4C8semidirect
361. C2×C4#324semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#86982
#82992