Finite Group Databasebeta

← #115908

Subgroups — #115908

Order 512.

1020 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 150 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×#1508C212yesmaximal
2#14159C211yesmaximal, characteristic
3#13778C211yesmaximal, characteristic
4C2×#1297C212yesmaximal
5#7909C211yesmaximal, characteristic
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C2×C4×C22⋊C4C2212yes—
11C22×#95C2211yescharacteristic
12#1508—24no—
13#1508—24no—
14C2×#328—24no—
15#1446C412yescomplemented
16#1445C412yescomplemented
17#1436—22no—
18#1436—24no—
19C22⋊#95—21no—
20C22⋊#95—21no—
21C22⋊#95—21no—
22#1297—24no—
23#1280—22no—
24C23×Q8C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4C2311yescharacteristic
28C22×M16—22no—
29C2×C22⋊Q8—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8D412yes—
36C2×C22⋊Q8Q812yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—22no—
43C22×C42—22no—
44C22×C42—21no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407D414yes—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51#380—28no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4—22no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4—22no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95C2×C412yes—
68C2×#95C2×C412yes—
69C2×#95D414yes—
70C2×#95D412yes—
71#328—216no—
72C25C2×D411yescharacteristic
73C22×Q8—21no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—81no—
77C22×Q8—81no—
78C22×Q8C4⋊C412yescomplemented
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—24no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—24no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—81no—
92C23×C4—81no—
93C23×C4—81no—
94C23×C4—81no—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×Q811yescharacteristic
103C23×C4#4112yes—
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4112yes—
107C2×M16—24no—
108C2×M16—42no—
109C2×M16—82no—
110C2×M16—82no—
111C22×C8—42no—
112C22⋊Q8—24no—
113C22⋊Q8—24no—
114C22⋊Q8—24no—
115C22⋊Q8—24no—
116C22⋊Q8—42no—
117C22⋊Q8—42no—
118C22⋊Q8—42no—
119C22⋊Q8—42no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—82no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4C4⋊C412yescomplemented
131C2×C22⋊C4—24no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—42no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—42no—
148C2×C42—82no—
149C2×C42—81no—
150C2×C42—81no—
151C2×C42—82no—
152C22⋊C8—24no—
153C22⋊C8—24no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156#95—28no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—44no—
163#95—44no—
164#95—82no—
165#95—82no—
166#95—82no—
167#95—82no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—41no—
173C24—41no—
174C24C2×C22⋊C411yescharacteristic
175C24#12011yescharacteristic
176C24#12111yescharacteristic
177C24C22⋊Q811yescharacteristic
178C24#12311yescharacteristic
179C24#12411yescharacteristic
180C24#12711yescharacteristic
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—84no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—28no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—81no—
239C22×C4—82no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4—81no—
244C22×C4—88no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—88no—
250C22×C4#10414yes—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4C4×D412yes—
254C22×C4C4×D412yes—
255C22×C4#12012yes—
256C22×C4#12112yes—
257C22×C4#12112yes—
258C22×C4#12111yescharacteristic
259C22×C4#12112yes—
260C22×C4C22⋊Q812yes—
261C22×C4C22⋊Q812yes—
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12312yes—
264C22×C4#12312yes—
265C22×C4#12412yes—
266C22×C4#12612yes—
267M16—84no—
268M16—84no—
269C2×C8—44no—
270C2×C8—82no—
271C2×C8—82no—
272C4⋊C4—24no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—88no—
277C22⋊C4—48no—
278C22⋊C4—48no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C42—84no—
286C42—84no—
287C42—84no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—44no—
318C23#38011yescharacteristic
319C23#38611yescharacteristic
320C23#39011yescharacteristic
321C23#39211yescharacteristic
322C23#39311yescharacteristic
323C23#39711yescharacteristic
324C23#39911yescharacteristic
325C23C2×#10412yes—
326C23#42112yes—
327C23#45512yes—
328C23#45612yes—
329Q8—84no—
330Q8—84no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4#38512yes—
389C2×C4#38812yes—
390C2×C4#39412yes—
391C2×C4#39612yes—
392C8—84no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—44no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#143911yescharacteristic
424C22#180612yes—
425C22#188511yescharacteristic
426C22#188611yescharacteristic
427C22#189611yescharacteristic
428C22#189711yescharacteristic
429C22#191311yescharacteristic
430C22#191311yescharacteristic
431C22#198012yes—
432C22#200712yes—
433C22#202812yes—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—44no—
439C4—88no—
440C4—84no—
441C4—82no—
442C4—82no—
443C4—82no—
444C2—22no—
445C2—22no—
446C2—22no—
447C2—22no—
448C2—41no—
449C2—41no—
450C2#940711yescharacteristic
451C2#940811yescharacteristic
452C2#941111yescharacteristic
453C2#941211yescharacteristic
454C2#1104611yescharacteristic
455C2#1105511yescharacteristic
456C2#1107511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×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
15. #1446C4semidirect
16. #1445C4semidirect
78. C22×Q8C4⋊C4semidirect
130. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#15082
#141592
#137782
C2×#12972
#79092