Finite Group Databasebeta

← #101855

Subgroups — #101855

Order 512.

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

normal subgroups only

This group has 635 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#25811C211yesmaximal, characteristic
2#14420C211yesmaximal, characteristic
3#9912C211yesmaximal, characteristic
4#9714C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7024C211yesmaximal, characteristic
7#7024C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#498C412yescomplemented
10#2857C2211yescharacteristic
11C2×#391—21no—
12C2×#387—21no—
13#2058—22no—
14#1990C412yescomplemented
15#1853—21no—
16#1853—21no—
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#332—22no—
21C2×#328—21no—
22C2×#328—21no—
23#1446—21no—
24#1445—21no—
25#1441—21no—
26#1436—21no—
27#1434C2211yescharacteristic
28#1434C2211yescharacteristic
29#1294—21no—
30#1294—21no—
31C23×Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—44no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4⋊Q8C2×C412yes—
37C2×C4⋊Q8D412yes—
38C2×C22⋊Q8—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45#498—44no—
46C2×#106—22no—
47C2×#106—22no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C4×C8C2×C412yes—
54#391—21no—
55#391—42no—
56#391—42no—
57#389—21no—
58#389—21no—
59#387—44no—
60#384C2×C411yescharacteristic
61#380—21no—
62#380—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×#95—21no—
72C2×#95—21no—
73C2×#95D411yescharacteristic
74C2×#95D411yescharacteristic
75C2×#95D411yescharacteristic
76C2×#95D411yescharacteristic
77#340—22no—
78#340—22no—
79#336—21no—
80#336—21no—
81#336—21no—
82#332—44no—
83#328—42no—
84#328—42no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—21no—
88C22×Q8—42no—
89C22×Q8—42no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—81no—
95C22×Q8—81no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C22×C8—22no—
106C22×C8—41no—
107C22×C8—41no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C22⋊Q8—41no—
116C22⋊Q8—41no—
117C22⋊Q8—41no—
118C22⋊Q8—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
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—81no—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C22⋊C412yes—
138C2×C22⋊C4—41no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42—81no—
143C2×C42C22×C411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147#106—44no—
148#106—44no—
149C4⋊C8—42no—
150C4⋊C8—42no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C4×C8—42no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95—81no—
177#95—81no—
178#95SD1612yes—
179#95SD1612yes—
180C24#12011yescharacteristic
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
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×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4C2×SD1611yescharacteristic
260C22×C4C2×SD1611yescharacteristic
261C22×C4#13511yescharacteristic
262C22×C4#13511yescharacteristic
263C22×C4AGL(1,ℤ/8)11yescharacteristic
264C22×C4AGL(1,ℤ/8)11yescharacteristic
265C22×C4#13711yescharacteristic
266C22×C4#13711yescharacteristic
267C2×C8—24no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—81no—
271C2×C8—81no—
272C2×C8—81no—
273C2×C8—81no—
274C2×C8—81no—
275C2×C8—81no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—84no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4#9914yes—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—22no—
296C42—22no—
297C42—41no—
298C42—41no—
299C42—84no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23#38611yescharacteristic
305C23#44811yescharacteristic
306C23#44911yescharacteristic
307C23#45011yescharacteristic
308C23#45211yescharacteristic
309C23#45611yescharacteristic
310C23#45611yescharacteristic
311Q8—84no—
312Q8—84no—
313Q8—84no—
314Q8—84no—
315Q8—161no—
316Q8—161no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
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—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—81no—
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×C4C2×#9912yes—
389C2×C4C4×SD1611yescharacteristic
390C2×C4C4×SD1611yescharacteristic
391C2×C4#44011yescharacteristic
392C2×C4#44011yescharacteristic
393C2×C4#46011yescharacteristic
394C2×C4#46111yescharacteristic
395C2×C4#46311yescharacteristic
396C2×C4#46411yescharacteristic
397C8—44no—
398C8—161no—
399C8—161no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22#165111yescharacteristic
413C22#165211yescharacteristic
414C22#165311yescharacteristic
415C22#165411yescharacteristic
416C22#187711yescharacteristic
417C22#188011yescharacteristic
418C22#188611yescharacteristic
419C22#189811yescharacteristic
420C22#190411yescharacteristic
421C22#190511yescharacteristic
422C22#191211yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—44no—
433C4—41no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2#905911yescharacteristic
446C2#925511yescharacteristic
447C2#926211yescharacteristic
448C2#929611yescharacteristic
449C2#930211yescharacteristic
450C2#942711yescharacteristic
451C2#946211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
9. C2×#498C4semidirect
14. #1990C4semidirect

Maximal subgroups

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

subgroupindex
#258112
#144202
#99122
#97142
#83402
#70242