Finite Group Databasebeta

← #97565

Subgroups — #97565

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2#8707C212yesmaximal, complemented
3#8558C211yesmaximal, characteristic, complemented
4C4×#324C211yesmaximal, characteristic, complemented
5#7572C212yesmaximal
6C22×C4×D4C411yescharacteristic
7#2357—24no—
8#2357—24no—
9C2×#390C411yescharacteristic
10C2×#390C2211yescharacteristic
11C2×#388—21no—
12C2×#388—21no—
13C2×#382C411yescharacteristic
14C2×#381—21no—
15C2×C4×C4⋊C4C411yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4—21no—
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1270—24no—
27C23×D4C2×C411yescharacteristic
28C24×C4—21no—
29C2×C4×D4—22no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×C4×D4—22no—
33C2×C4×D4—22no—
34C2×C4×D4C814yescomplemented
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C42—21no—
46C22×C42—21no—
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42C2311yescharacteristic
50C2×C4⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C4×C8—22no—
54#390—22no—
55#390—22no—
56#390—22no—
57#390D414yes—
58#388—28no—
59#388—28no—
60#382—22no—
61#382—22no—
62#381—28no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4D414yes—
69C4×C22⋊C4—28no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95D411yescharacteristic
75C2×#95D411yescharacteristic
76#336—22no—
77#336—22no—
78#324—24no—
79#324—42no—
80C25—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—42no—
85C22×D4—42no—
86C22×D4C22⋊C412yes—
87C22×D4C2×C812yes—
88C22×D4M1612yes—
89C23×C4—21no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—82no—
96C23×C4—82no—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C22×C8—42no—
105C22×C8—42no—
106C4×D4—28no—
107C4×D4—28no—
108C4×D4—28no—
109C4×D4—28no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C22⋊C412yes—
125C2×C4⋊C4C22⋊C412yes—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C2×C812yes—
128C2×C4⋊C4M1612yes—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—82no—
142C2×C22⋊C4—82no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42—82no—
154C2×C42—82no—
155C2×C42C2×C812yes—
156C2×C42M1612yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C4⋊C8—44no—
160C22⋊C8—24no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163C22⋊C8—42no—
164C4×C8—44no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—22no—
170#95—22no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24C2×C22⋊C411yescharacteristic
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4C22⋊C814yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—216no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—88no—
239C22×C4—88no—
240C22×C4—84no—
241C22×C4—84no—
242C22×C4C22⋊C812yes—
243C22×C4C22⋊C812yes—
244C22×C4#10412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4#12012yes—
248C22×C4#12012yes—
249C22×C4C22×C811yescharacteristic
250C22×C4C2×M1611yescharacteristic
251C22×C4C2×M1611yescharacteristic
252C22×C4C2×M1611yescharacteristic
253C22×C4#13512yes—
254C22×C4AGL(1,ℤ/8)11yescharacteristic
255C22×C4#13711yescharacteristic
256C2×C8—44no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4C22⋊C814yes—
273C22⋊C4—216no—
274C22⋊C4—48no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—84no—
278C22⋊C4—84no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—24no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—84no—
288C42—84no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—24no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23#37911yescharacteristic
317C23C2×C22⋊C811yescharacteristic
318C23#40711yescharacteristic
319C23#41611yescharacteristic
320C23#41711yescharacteristic
321C23C2×#10411yescharacteristic
322C23#42111yescharacteristic
323D4—44no—
324D4—44no—
325D4—44no—
326D4—44no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—816no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4C2×C22⋊C811yescharacteristic
382C2×C4C2×C22⋊C811yescharacteristic
383C2×C4#40711yescharacteristic
384C2×C4#40711yescharacteristic
385C2×C4#41012yes—
386C2×C4#41312yes—
387C2×C4#44912yes—
388C2×C4#45212yes—
389C8—84no—
390C8—162no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—84no—
407C22—84no—
408C22—84no—
409C22—84no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22#147611yescharacteristic
415C22#147711yescharacteristic
416C22#148811yescharacteristic
417C22#148911yescharacteristic
418C22#177911yescharacteristic
419C22#178211yescharacteristic
420C22#178511yescharacteristic
421C22#178811yescharacteristic
422C22#178911yescharacteristic
423C22#179511yescharacteristic
424C22#179511yescharacteristic
425C4—21no—
426C4—21no—
427C4—22no—
428C4—22no—
429C4—42no—
430C4—44no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—82no—
437C4#145812yes—
438C4#146412yes—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—84no—
444C2#830011yescharacteristic
445C2#831211yescharacteristic
446C2#832211yescharacteristic
447C2#836211yescharacteristic
448C2#837011yescharacteristic
449C2#837811yescharacteristic
450C2#838611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8707C2semidirect
3. #8558C2semidirect
4. C4×#324C2semidirect
34. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
#87072
#85582
C4×#3242
#75722