Finite Group Databasebeta

← #100372

Subgroups — #100372

Order 512.

705 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 705 conjugacy classes of subgroups — too many to draw. Only its 116 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#33106C211yesmaximal, characteristic
2#13821C211yesmaximal, characteristic, complemented
3#9712C211yesmaximal, characteristic
4#9680C211yesmaximal, characteristic
5#8706C211yesmaximal, characteristic, complemented
6#7039C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C2×#480—21no—
9#2856C2211yescharacteristic
10C2×#389—21no—
11C2×C4×C4⋊C4—21no—
12C2×C4×C4⋊C4—21no—
13C2×C4×C4⋊C4C2211yescharacteristic
14#1982C412yescomplemented
15#1947C412yescomplemented
16#1925—22no—
17#1874—21no—
18#1797C2211yescharacteristic
19#1797C2211yescharacteristic
20C4×C22⋊C8C2211yescharacteristic
21C2×#327—22no—
22C2×#327—21no—
23C2×#327—21no—
24#1436—22no—
25C4⋊#95C2211yescharacteristic
26#1434—21no—
27C4×#95C2211yescharacteristic
28#1279—22no—
29C2×#127C2×C411yescharacteristic
30C2×#121—21no—
31C2×#121—21no—
32C2×#121—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C42—21no—
38C22×C42—21no—
39C22×C42—22no—
40C22×C42C2311yescharacteristic
41#480—42no—
42#480—42no—
43C2×#107—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2×C412yes—
46C2×#102—22no—
47C2×#102—22no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C4×C8—21no—
52#390—21no—
53#390—21no—
54#390—21no—
55#389—28no—
56#384C2×C411yescharacteristic
57C4×C4⋊C4—21no—
58C4×C4⋊C4—21no—
59C4×C4⋊C4—42no—
60C4×C4⋊C4—42no—
61C4×C4⋊C4—42no—
62C4×C4⋊C4—42no—
63C4×C4⋊C4C2×C412yes—
64C4×C4⋊C4D412yes—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72C2×#95D412yes—
73#340—22no—
74#336—21no—
75#336—21no—
76#336—21no—
77#327—42no—
78#327—44no—
79#327—42no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—41no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95#127—42no—
96#121—42no—
97#121—42no—
98#121—41no—
99#121—41no—
100#121—41no—
101#121—41no—
102#121—41no—
103#121—41no—
104#121—41no—
105#121—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4D812yes—
124C2×C4⋊C4SD1612yes—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—24no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—44no—
140C2×C42—44no—
141C2×C42—82no—
142C2×C42C22×C411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146#107—44no—
147C4⋊C8—42no—
148C4⋊C8—42no—
149#102—44no—
150#102—44no—
151C22⋊C8—22no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C4×C8—42no—
158#95—28no—
159#95—22no—
160#95—22no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—42no—
165#95—42no—
166#95—44no—
167#95—41no—
168#95—41no—
169#95D812yes—
170C24—41no—
171C24#12011yescharacteristic
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—81no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×SD1611yescharacteristic
240C22×C4#13511yescharacteristic
241C22×C4AGL(1,ℤ/8)11yescharacteristic
242C22×C4AGL(1,ℤ/8)11yescharacteristic
243C22×C4AGL(1,ℤ/8)11yescharacteristic
244C22×C4#13711yescharacteristic
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—81no—
250C2×C8—81no—
251C2×C8—81no—
252C2×C8—81no—
253C2×C8—81no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—416no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4#10214yes—
268C4⋊C4#10414yes—
269C22⋊C4—82no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C42—22no—
275C42—22no—
276C42—41no—
277C42—41no—
278C42—44no—
279C42—44no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—88no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23#38611yescharacteristic
297C23#44711yescharacteristic
298C23#44711yescharacteristic
299C23#44811yescharacteristic
300C23#44911yescharacteristic
301C23#45311yescharacteristic
302C23#45411yescharacteristic
303D4—82no—
304D4—82no—
305D4—161no—
306D4—161no—
307D4—162no—
308D4—161no—
309D4—161no—
310D4—161no—
311D4—161no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4#41012yes—
371C2×C4C2×#10211yescharacteristic
372C2×C4#41711yescharacteristic
373C2×C4C2×#10412yes—
374C2×C4C4×D811yescharacteristic
375C2×C4#44211yescharacteristic
376C2×C4#45911yescharacteristic
377C2×C4#46311yescharacteristic
378C2×C4#48011yescharacteristic
379C2×C4#48111yescharacteristic
380C8—82no—
381C8—82no—
382C8—161no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22#161911yescharacteristic
403C22#162211yescharacteristic
404C22#162511yescharacteristic
405C22#162711yescharacteristic
406C22#187411yescharacteristic
407C22#187911yescharacteristic
408C22#188311yescharacteristic
409C22#189611yescharacteristic
410C22#189911yescharacteristic
411C22#190511yescharacteristic
412C22#190611yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—44no—
420C4—44no—
421C4—44no—
422C4—41no—
423C4—41no—
424C4—84no—
425C4—82no—
426C4—161no—
427C4#150212yes—
428C4#150912yes—
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—321no—
434C2#897711yescharacteristic
435C2#914811yescharacteristic
436C2#918011yescharacteristic
437C2#923311yescharacteristic
438C2#926011yescharacteristic
439C2#932611yescharacteristic
440C2#936611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ 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. #13821C2semidirect
5. #8706C2semidirect
6. #7039C2semidirect
7. #7028C2semidirect
14. #1982C4semidirect
15. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331062
#138212
#97122
#96802
#87062
#70392
#70282