Finite Group Databasebeta

← #110230

Subgroups — #110230

Order 512.

1106 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C2×#1511C212yesmaximal
2#14139C211yesmaximal, characteristic
3#13776C211yesmaximal, characteristic
4#10727C211yesmaximal, characteristic
5C2×#1284C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9C2×#391C412yescomplemented
10C2×#381C2212yes—
11C2×C4×C22⋊C4—21no—
12C2×C4×C22⋊C4C2211yescharacteristic
13C2×#343—22no—
14#1511—24no—
15#1511C418yescomplemented
16C2×#328—24no—
17#1441—22no—
18#1436—24no—
19C4⋊#95C412yescomplemented
20C22⋊#95—21no—
21C22⋊#95—21no—
22C22⋊#95—21no—
23#1284—24no—
24C23×Q8C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4C2311yescharacteristic
28C22×M16—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8C2×C414yes—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—44no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4D412yes—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—22no—
45C22×C42—22no—
46C22×C42C2×C412yes—
47#407—22no—
48#407—22no—
49#407—22no—
50#407C2×C414yes—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53#391—44no—
54#381—22no—
55#381—22no—
56#381C2×C414yes—
57#381D414yes—
58C4×C22⋊C4—22no—
59C4×C22⋊C4—22no—
60C4×C22⋊C4—44no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—21no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95D414yes—
71#343—44no—
72#328—216no—
73C25C2×D411yescharacteristic
74C22×Q8—21no—
75C22×Q8—22no—
76C22×Q8—22no—
77C22×Q8—42no—
78C22×Q8—42no—
79C22×Q8C4212yescomplemented
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—24no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—24no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—81no—
95C23×C4—81no—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4#4112yes—
104C23×C4#4112yes—
105C23×C4#4112yes—
106C2×M16—24no—
107C2×M16—42no—
108C2×M16—44no—
109C2×M16—44no—
110C22×C8—42no—
111C22⋊Q8—24no—
112C22⋊Q8—24no—
113C22⋊Q8—24no—
114C22⋊Q8—24no—
115C22⋊Q8—42no—
116C22⋊Q8—42no—
117C22⋊Q8—42no—
118C22⋊Q8—42no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C4212yescomplemented
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—41no—
147C2×C42—24no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42—44no—
151C2×C42—82no—
152C2×C42—82no—
153C22⋊C8—24no—
154C22⋊C8—24no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—44no—
161#95—44no—
162#95—44no—
163#95—41no—
164#95—41no—
165#95—82no—
166#95—82no—
167#95—82no—
168#95—82no—
169#95C22⋊C418yes—
170C24—22no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—41no—
175C24—41no—
176C24C2×C22⋊C411yescharacteristic
177C24C4×D412yes—
178C24#12011yescharacteristic
179C24#12112yes—
180C24#12311yescharacteristic
181C2×Q8—24no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—42no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—84no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—28no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—88no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—88no—
248C22×C4#9914yes—
249C22×C4C2×C4211yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4#11711yescharacteristic
252C22×C4#11712yes—
253C22×C4C4×D412yes—
254C22×C4C4×D412yes—
255C22×C4C4×D412yes—
256C22×C4C4×D412yes—
257C22×C4C4×D412yes—
258C22×C4#12112yes—
259C22×C4#12112yes—
260C22×C4#12312yes—
261C22×C4#12411yescharacteristic
262C22×C4#12412yes—
263C22×C4#12612yes—
264C22×C4#12711yescharacteristic
265M16—48no—
266M16—84no—
267C2×C8—44no—
268C2×C8—82no—
269C2×C8—82no—
270C4⋊C4—24no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C22⋊C4—48no—
277C22⋊C4—48no—
278C22⋊C4—48no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C42—84no—
282C42—88no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—24no—
292C23—24no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—24no—
300C23—24no—
301C23—42no—
302C23—42no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23C4×C22⋊C411yescharacteristic
308C23#38612yes—
309C23#38611yescharacteristic
310C23#38812yes—
311C23#39011yescharacteristic
312C23C2×#9912yes—
313C23#41012yes—
314C23#45512yes—
315C23#45612yes—
316Q8—84no—
317Q8—84no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4C4×C22⋊C412yes—
376C2×C4#38812yes—
377C2×C4#38812yes—
378C2×C4#39012yes—
379C8—84no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—48no—
398C22—48no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22#143711yescharacteristic
404C22C4×#9912yes—
405C22#188214yes—
406C22#188511yescharacteristic
407C22#188611yescharacteristic
408C22#189811yescharacteristic
409C22#189911yescharacteristic
410C22#191111yescharacteristic
411C22#191211yescharacteristic
412C22#196412yes—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—44no—
418C4—88no—
419C4—84no—
420C4—82no—
421C4—82no—
422C4—82no—
423C2—24no—
424C2—22no—
425C2—22no—
426C2—42no—
427C2#945911yescharacteristic
428C2#946011yescharacteristic
429C2#946111yescharacteristic
430C2#946211yescharacteristic
431C2#1094911yescharacteristic
432C2#1095711yescharacteristic
433C2#1096011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
9. C2×#391C4semidirect
15. #1511C4semidirect
19. C4⋊#95C4semidirect
79. C22×Q8C42semidirect
135. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#15112
#141392
#137762
#107272
C2×#12842