Finite Group Databasebeta

← #100630

Subgroups — #100630

Order 512.

540 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 96 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#29755C211yesmaximal, characteristic
2#14470C211yesmaximal, characteristic, complemented
3#9715C211yesmaximal, characteristic
4#9680C211yesmaximal, characteristic
5#7576C211yesmaximal, characteristic, complemented
6#7039C211yesmaximal, characteristic, complemented
7#7024C211yesmaximal, characteristic, complemented
8C2×#481—21no—
9#2856C2211yescharacteristic
10C2×#389C2211yescharacteristic
11#2084C412yescomplemented
12#2065—22no—
13#2041C412yescomplemented
14#1874—21no—
15#1797C2211yescharacteristic
16#1797C2211yescharacteristic
17C4×C22⋊C8C2211yescharacteristic
18C2×#327—21no—
19C2×#327—21no—
20C2×#327—21no—
21C2×#327—21no—
22#1445—21no—
23#1445—21no—
24#1442—21no—
25#1436—21no—
26#1436—21no—
27#1436—21no—
28C4⋊#95C2211yescharacteristic
29#1434C2211yescharacteristic
30#1271—22no—
31C2×#127C2×C411yescharacteristic
32C2×#121—21no—
33C2×#121—21no—
34C2×#121—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C42—21no—
40C22×C42C2311yescharacteristic
41#481—42no—
42#481—42no—
43C2×#106—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
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—21no—
56#389—21no—
57#389C2×C412yes—
58#389D412yes—
59#384C2×C411yescharacteristic
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95D411yescharacteristic
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74C2×#95D411yescharacteristic
75#340—22no—
76#336—21no—
77#336—21no—
78#336C2×C412yes—
79#327—42no—
80#327—42no—
81#327—42no—
82#327—42no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—41no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C22×C8—22no—
97C22×C8—41no—
98C22×C8—41no—
99#127—81no—
100#121—41no—
101#121—41no—
102#121—41no—
103#121—41no—
104#121—41no—
105#121—41no—
106#121—41no—
107#121—41no—
108#121—41no—
109#121—41no—
110#121—41no—
111#121—41no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4—81no—
128C2×C4⋊C4—81no—
129C2×C4⋊C4—81no—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C42—81no—
136C2×C42—81no—
137C2×C42—81no—
138C2×C42—81no—
139C2×C42C22×C411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143#106—44no—
144C4⋊C8—81no—
145C4⋊C8—81no—
146#102—44no—
147#102—82no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C4×C8—81no—
155#95—22no—
156#95—22no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—41no—
161#95—41no—
162#95—42no—
163#95—42no—
164#95—41no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—81no—
169#95—81no—
170#95—81no—
171#95—81no—
172#95—81no—
173#95—81no—
174#95SD1612yes—
175C24—41no—
176C24#12011yescharacteristic
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
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—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4C2×SD1611yescharacteristic
253C22×C4#13511yescharacteristic
254C22×C4#13511yescharacteristic
255C22×C4#13511yescharacteristic
256C22×C4AGL(1,ℤ/8)11yescharacteristic
257C22×C4AGL(1,ℤ/8)11yescharacteristic
258C22×C4AGL(1,ℤ/8)11yescharacteristic
259C22×C4#13711yescharacteristic
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—81no—
265C2×C8—81no—
266C2×C8—81no—
267C2×C8—81no—
268C2×C8—81no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C22⋊C4—81no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—84no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23#38611yescharacteristic
308C23#44811yescharacteristic
309C23#44911yescharacteristic
310C23#44911yescharacteristic
311C23#44911yescharacteristic
312C23#45311yescharacteristic
313C23#45411yescharacteristic
314D4—82no—
315D4—82no—
316D4—161no—
317D4—161no—
318D4—161no—
319D4—161no—
320D4—161no—
321D4—161no—
322D4—161no—
323D4—161no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4#41611yescharacteristic
394C2×C4#41811yescharacteristic
395C2×C4C4×SD1611yescharacteristic
396C2×C4#44011yescharacteristic
397C2×C4#46011yescharacteristic
398C2×C4#46411yescharacteristic
399C2×C4#48211yescharacteristic
400C2×C4#48211yescharacteristic
401C8—82no—
402C8—82no—
403C8—161no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22#162011yescharacteristic
424C22#162211yescharacteristic
425C22#162611yescharacteristic
426C22#162811yescharacteristic
427C22#187511yescharacteristic
428C22#188011yescharacteristic
429C22#188311yescharacteristic
430C22#190411yescharacteristic
431C22#190611yescharacteristic
432C22#191211yescharacteristic
433C22#191311yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—161no—
449C4—161no—
450C4—161no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—321no—
456C2#897811yescharacteristic
457C2#918111yescharacteristic
458C2#920611yescharacteristic
459C2#927811yescharacteristic
460C2#930211yescharacteristic
461C2#934311yescharacteristic
462C2#938211yescharacteristic
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. #14470C2semidirect
5. #7576C2semidirect
6. #7039C2semidirect
7. #7024C2semidirect
11. #2084C4semidirect
13. #2041C4semidirect

Maximal subgroups

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

subgroupindex
#297552
#144702
#97152
#96802
#75762
#70392
#70242