Finite Group Databasebeta

← #100628

Subgroups — #100628

Order 512.

590 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 104 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#29754C211yesmaximal, characteristic
2#14439C211yesmaximal, characteristic, complemented
3#9715C211yesmaximal, characteristic
4#9680C211yesmaximal, characteristic
5#7573C211yesmaximal, characteristic, complemented
6#7039C211yesmaximal, characteristic, complemented
7#7024C211yesmaximal, characteristic, complemented
8C2×#480—21no—
9#2856C2211yescharacteristic
10C2×#389—21no—
11C2×#389C2211yescharacteristic
12C2×#387—21no—
13#2083C412yescomplemented
14#2067—22no—
15#2041C412yescomplemented
16#1874—21no—
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#327—21no—
21C2×#327—21no—
22C2×#327—21no—
23C2×#327—21no—
24#1442—21no—
25#1441—21no—
26#1441—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⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42C2311yescharacteristic
43#480—42no—
44#480—42no—
45C2×#107—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#102—22no—
49C2×#102—22no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C4×C8—21no—
54#390—21no—
55#390—21no—
56#390—21no—
57#389—21no—
58#389—21no—
59#389—42no—
60#389—42no—
61#389C2×C412yes—
62#389D412yes—
63#387—42no—
64#387—42no—
65#384C2×C411yescharacteristic
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95D411yescharacteristic
76C2×#95D411yescharacteristic
77C2×#95D411yescharacteristic
78C2×#95D411yescharacteristic
79#340—22no—
80#336—21no—
81#336—21no—
82#336C2×C412yes—
83#327—42no—
84#327—42no—
85#327—42no—
86#327—42no—
87C22×D4—21no—
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—41no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C22×C8—22no—
101C22×C8—41no—
102C22×C8—41no—
103#127—81no—
104#121—41no—
105#121—41no—
106#121—41no—
107#121—41no—
108#121—41no—
109#121—41no—
110#121—41no—
111#121—41no—
112#121—41no—
113#121—41no—
114#121—41no—
115#121—41no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—81no—
137C2×C4⋊C4—81no—
138C2×C4⋊C4—81no—
139C2×C4⋊C4—81no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4D812yes—
143C2×C4⋊C4SD1612yes—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—81no—
150C2×C42—81no—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155#107—44no—
156C4⋊C8—81no—
157C4⋊C8—81no—
158#102—44no—
159#102—82no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166C4×C8—81no—
167#95—22no—
168#95—22no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—41no—
173#95—41no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—81no—
183#95—81no—
184#95D812yes—
185C24—41no—
186C24#12011yescharacteristic
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4—81no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
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×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×SD1611yescharacteristic
265C22×C4#13511yescharacteristic
266C22×C4AGL(1,ℤ/8)11yescharacteristic
267C22×C4AGL(1,ℤ/8)11yescharacteristic
268C22×C4AGL(1,ℤ/8)11yescharacteristic
269C22×C4#13711yescharacteristic
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—81no—
275C2×C8—81no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—81no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—81no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4#10214yes—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—84no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23#38611yescharacteristic
322C23#44711yescharacteristic
323C23#44711yescharacteristic
324C23#44811yescharacteristic
325C23#44911yescharacteristic
326C23#45311yescharacteristic
327C23#45411yescharacteristic
328D4—82no—
329D4—82no—
330D4—161no—
331D4—161no—
332D4—161no—
333D4—161no—
334D4—161no—
335D4—161no—
336D4—161no—
337D4—161no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
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—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
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—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4C2×#10211yescharacteristic
408C2×C4#41711yescharacteristic
409C2×C4C4×D811yescharacteristic
410C2×C4#44211yescharacteristic
411C2×C4#45911yescharacteristic
412C2×C4#46311yescharacteristic
413C2×C4#48011yescharacteristic
414C2×C4#48111yescharacteristic
415C8—82no—
416C8—82no—
417C8—161no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22#161911yescharacteristic
438C22#162211yescharacteristic
439C22#162511yescharacteristic
440C22#162711yescharacteristic
441C22#187411yescharacteristic
442C22#187911yescharacteristic
443C22#188311yescharacteristic
444C22#190511yescharacteristic
445C22#190611yescharacteristic
446C22#191211yescharacteristic
447C22#191311yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—44no—
457C4—41no—
458C4—41no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—161no—
463C4—161no—
464C4—161no—
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—321no—
470C2#897711yescharacteristic
471C2#918011yescharacteristic
472C2#920411yescharacteristic
473C2#927711yescharacteristic
474C2#930011yescharacteristic
475C2#934411yescharacteristic
476C2#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. #14439C2semidirect
5. #7573C2semidirect
6. #7039C2semidirect
7. #7024C2semidirect
13. #2083C4semidirect
15. #2041C4semidirect

Maximal subgroups

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

subgroupindex
#297542
#144392
#97152
#96802
#75732
#70392
#70242