Finite Group Databasebeta

← #97396

Subgroups — #97396

Order 512.

954 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 116 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×#2492C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3#8713C212yesmaximal
4#8700C211yesmaximal, characteristic
5#8329C212yesmaximal
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×C22⋊Q8C2211yescharacteristic
8#2492—24no—
9#2492—24no—
10C2×#400—21no—
11C2×#393—21no—
12C2×#391—21no—
13C2×#391C411yescharacteristic, complemented
14C2×#391C411yescharacteristic, complemented
15C2×#388—21no—
16C2×#380C2211yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic, complemented
18C2×C4×C22⋊C4C2211yescharacteristic
19#1779C2212yes—
20C2×#342—21no—
21C2×#342—21no—
22C2×#328—24no—
23C22⋊#95C2212yes—
24#1294—24no—
25#1280—21no—
26#1280—21no—
27C23×Q8—21no—
28C23×Q8C2×C411yescharacteristic
29C24×C4C2311yescharacteristic
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—44no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—44no—
35C2×C22⋊Q8D414yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4D411yescharacteristic
45C22×C22⋊C4D411yescharacteristic
46C22×C42—21no—
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52#400—28no—
53#393—28no—
54#391—28no—
55#391—22no—
56#391—22no—
57#391—22no—
58#391—22no—
59#388—28no—
60#380—22no—
61#380D414yes—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95D411yescharacteristic
75C2×#95D411yescharacteristic
76#342—24no—
77#342—42no—
78#328—48no—
79C25C2×D411yescharacteristic
80C22×Q8—24no—
81C22×Q8—21no—
82C22×Q8—42no—
83C22×Q8—44no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—42no—
89C22×Q8—82no—
90C22×Q8C22⋊C412yes—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—42no—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C22×C8—42no—
109C4⋊Q8—48no—
110C4⋊Q8—44no—
111C4⋊Q8—44no—
112C22⋊Q8—28no—
113C22⋊Q8—44no—
114C22⋊Q8—48no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C412yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—82no—
144C2×C22⋊C4SD1612yes—
145C2×C22⋊C4Q1612yes—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—24no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42—82no—
154C22⋊C8—44no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C22⋊C8—44no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—22no—
165#95—22no—
166#95—44no—
167#95—44no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—41no—
173C24—41no—
174C24—41no—
175C24—41no—
176C24C2×C22⋊C411yescharacteristic
177C24#12012yes—
178C24C2×SD1611yescharacteristic
179C24C2×Q1611yescharacteristic
180C24#13712yes—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8—48no—
184C2×Q8—42no—
185C2×Q8—48no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—84no—
190C2×Q8—84no—
191C2×Q8—84no—
192C2×Q8—88no—
193C2×Q8—88no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
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—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
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—44no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4#10312yes—
255C22×C4#10312yes—
256C22×C4#10412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4#12012yes—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—48no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4#9914yes—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C22⋊C4—48no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—84no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—84no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23#37911yescharacteristic
317C23#41011yescharacteristic
318C23#41011yescharacteristic
319C23C2×#10311yescharacteristic
320C23#41911yescharacteristic
321C23C2×#10411yescharacteristic
322C23#42111yescharacteristic
323C23#44812yes—
324C23#45112yes—
325Q8—84no—
326Q8—84no—
327Q8—816no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—162no—
385C2×C4—162no—
386C2×C4—162no—
387C2×C4—162no—
388C2×C4C2×#9912yes—
389C2×C4#41012yes—
390C8—162no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—81no—
415C22—82no—
416C22—82no—
417C22—81no—
418C22—81no—
419C22#150311yescharacteristic
420C22#150611yescharacteristic
421C22#151511yescharacteristic
422C22#151811yescharacteristic
423C22#178111yescharacteristic
424C22#178211yescharacteristic
425C22#178211yescharacteristic
426C22#178711yescharacteristic
427C22#178911yescharacteristic
428C22#179511yescharacteristic
429C22#179611yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—44no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4—81no—
441C4—164no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—42no—
447C2—42no—
448C2#833311yescharacteristic
449C2#835911yescharacteristic
450C2#836511yescharacteristic
451C2#837911yescharacteristic
452C2#838011yescharacteristic
453C2#851911yescharacteristic
454C2#854411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Q8C4semidirect
13. C2×#391C4semidirect
14. C2×#391C4semidirect
17. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24922
#108012
#87132
#87002
#83292