Finite Group Databasebeta

← #97572

Subgroups — #97572

Order 512.

868 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2#8707C212yesmaximal, complemented
3#8557C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7573C212yesmaximal, complemented
6C22×#117C411yescharacteristic
7#2347—24no—
8#2347—24no—
9C2×#389C411yescharacteristic
10C2×#389C2211yescharacteristic
11C2×#388—21no—
12C2×#388—21no—
13C2×#383—21no—
14C2×#380C411yescharacteristic
15C2×C4×C4⋊C4—21no—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#323—21no—
22C2×#323—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1271—24no—
27C24×C4C2×C411yescharacteristic
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117C814yescomplemented
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4D411yescharacteristic
43C22×C22⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42—21no—
46C22×C42—21no—
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C4×C8—22no—
53#389—22no—
54#389—22no—
55#389—22no—
56#389D414yes—
57#388—28no—
58#388—28no—
59#383—28no—
60#380—22no—
61#380—22no—
62C4×C4⋊C4—28no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4D414yes—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95D411yescharacteristic
75C2×#95D411yescharacteristic
76#336—22no—
77#336—22no—
78#323—24no—
79#323—42no—
80C25C22⋊C411yescharacteristic
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C2×C812yes—
95C23×C4M1612yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C22×C8—42no—
100C22×C8—42no—
101#117—28no—
102#117—28no—
103#117—28no—
104#117—28no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C22⋊C412yes—
124C2×C4⋊C4C22⋊C412yes—
125C2×C4⋊C4C22⋊C412yes—
126C2×C4⋊C4C2×C812yes—
127C2×C4⋊C4C2×C812yes—
128C2×C4⋊C4M1612yes—
129C2×C4⋊C4M1612yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—82no—
142C2×C42—24no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—42no—
150C2×C42—42no—
151C2×C42—82no—
152C2×C42—82no—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C4⋊C8—44no—
156C22⋊C8—24no—
157C22⋊C8—42no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160C4×C8—44no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—22no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—41no—
175C24—41no—
176C24—41no—
177C24—41no—
178C24C22⋊C812yes—
179C24C2×C22⋊C411yescharacteristic
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4C22⋊C812yes—
236C22×C4#10412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4#12012yes—
240C22×C4#12012yes—
241C22×C4C22×C811yescharacteristic
242C22×C4C2×M1611yescharacteristic
243C22×C4C2×M1611yescharacteristic
244C22×C4C2×M1611yescharacteristic
245C22×C4#13512yes—
246C22×C4AGL(1,ℤ/8)11yescharacteristic
247C22×C4#13711yescharacteristic
248C2×C8—44no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—48no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4C22⋊C814yes—
268C4⋊C4C22⋊C814yes—
269C22⋊C4—216no—
270C22⋊C4—44no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—84no—
274C42—216no—
275C42—24no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—84no—
280C42—84no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23#37911yescharacteristic
309C23C2×C22⋊C811yescharacteristic
310C23#40711yescharacteristic
311C23#41611yescharacteristic
312C23#41711yescharacteristic
313C23C2×#10411yescharacteristic
314C23#42111yescharacteristic
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4C2×C22⋊C811yescharacteristic
373C2×C4C2×C22⋊C811yescharacteristic
374C2×C4#40711yescharacteristic
375C2×C4#40711yescharacteristic
376C2×C4#41012yes—
377C2×C4#41312yes—
378C2×C4#44912yes—
379C2×C4#45212yes—
380C8—84no—
381C8—162no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#147611yescharacteristic
406C22#147711yescharacteristic
407C22#148811yescharacteristic
408C22#148911yescharacteristic
409C22#177911yescharacteristic
410C22#178211yescharacteristic
411C22#178511yescharacteristic
412C22#178811yescharacteristic
413C22#178911yescharacteristic
414C22#179511yescharacteristic
415C22#179611yescharacteristic
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—44no—
422C4—44no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—82no—
427C4—81no—
428C4#145812yes—
429C4#146412yes—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—82no—
435C2#830011yescharacteristic
436C2#831211yescharacteristic
437C2#832411yescharacteristic
438C2#836511yescharacteristic
439C2#837311yescharacteristic
440C2#837811yescharacteristic
441C2#838911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊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
5. #7573C2semidirect
33. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
#87072
#85572
C4×#3232
#75732