Finite Group Databasebeta

← #97593

Subgroups — #97593

Order 512.

1094 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3#8311C212yesmaximal, complemented
4C4×#323C211yesmaximal, characteristic
5#7576C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic
9#2360—24no—
10#2360—24no—
11C2×#391C411yescharacteristic
12C2×#389—21no—
13C2×#389C2211yescharacteristic
14C2×#387—21no—
15C2×#385—21no—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C22⋊C4—21no—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×Q8⋊C8—24no—
22C2×#323—21no—
23C2×#323—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4—21no—
29C24×C4C2×C411yescharacteristic
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8C814yescomplemented
36C2×C4×Q8—22no—
37C2×C4×Q8—44no—
38C2×C4×Q8D414yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—44no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4D411yescharacteristic
51C22×C22⋊C4D411yescharacteristic
52C22×C42—21no—
53C22×C42—21no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C4×C8—22no—
60#391—22no—
61#391—22no—
62#389—28no—
63#389—22no—
64#389D414yes—
65#387—28no—
66#385—28no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C22⋊C4—28no—
70C4×C22⋊C4—22no—
71C4×C22⋊C4—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95D411yescharacteristic
77C2×#95D411yescharacteristic
78#336—22no—
79#336—22no—
80Q8⋊C8—48no—
81#323—24no—
82#323—42no—
83C25C22⋊C411yescharacteristic
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8C22⋊C412yes—
90C22×Q8C2×C812yes—
91C22×Q8M1612yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—24no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—44no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—82no—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C2×C812yes—
111C23×C4M1612yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C22×C8—42no—
116C22×C8—42no—
117C22⋊Q8—28no—
118C22⋊Q8—28no—
119C22⋊Q8—28no—
120C22⋊Q8—28no—
121C4×Q8—28no—
122C4×Q8—44no—
123C4×Q8—48no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—48no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C2×C812yes—
147C2×C4⋊C4M1612yes—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4SD1612yes—
157C2×C22⋊C4Q1612yes—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—42no—
163C2×C42—44no—
164C2×C42—42no—
165C2×C42—82no—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C4⋊C8—44no—
169C22⋊C8—24no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173C4×C8—44no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—22no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—41no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24C22⋊C812yes—
189C24#10312yes—
190C24#10412yes—
191C24C2×C22⋊C411yescharacteristic
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8C22⋊C814yes—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—48no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—84no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4C22⋊C812yes—
263C22×C4#10312yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4#12012yes—
267C22×C4#12012yes—
268C22×C4C22×C811yescharacteristic
269C22×C4C2×M1611yescharacteristic
270C22×C4C2×M1611yescharacteristic
271C22×C4C2×M1611yescharacteristic
272C22×C4C2×SD1611yescharacteristic
273C22×C4C2×Q1611yescharacteristic
274C22×C4#13712yes—
275C2×C8—44no—
276C2×C8—82no—
277C2×C8—82no—
278C2×C8—82no—
279C2×C8—82no—
280C2×C8—82no—
281C4⋊C4—24no—
282C4⋊C4—216no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—48no—
288C4⋊C4—42no—
289C4⋊C4—48no—
290C4⋊C4—42no—
291C4⋊C4—84no—
292C4⋊C4C22⋊C814yes—
293C4⋊C4#9914yes—
294C22⋊C4—216no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—44no—
298C42—24no—
299C42—42no—
300C42—48no—
301C42—42no—
302C42—42no—
303C42—84no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23Q8⋊C814yescomplemented
331C23#37911yescharacteristic
332C23C2×C22⋊C811yescharacteristic
333C23#40711yescharacteristic
334C23C2×#10311yescharacteristic
335C23#41911yescharacteristic
336C23C2×#10411yescharacteristic
337C23#42111yescharacteristic
338Q8—28no—
339Q8—44no—
340Q8—44no—
341Q8—44no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—816no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4C2×C22⋊C811yescharacteristic
399C2×C4C2×C22⋊C811yescharacteristic
400C2×C4#40711yescharacteristic
401C2×C4#40711yescharacteristic
402C2×C4C2×#9912yes—
403C2×C4#41312yes—
404C2×C4#44812yes—
405C2×C4#45112yes—
406C8—84no—
407C8—162no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—24no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—44no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22C2×Q8⋊C811yescharacteristic
432C22#147911yescharacteristic
433C22#148711yescharacteristic
434C22#149111yescharacteristic
435C22#177911yescharacteristic
436C22#178111yescharacteristic
437C22#178511yescharacteristic
438C22#178711yescharacteristic
439C22#178911yescharacteristic
440C22#179511yescharacteristic
441C22#179611yescharacteristic
442C4—21no—
443C4—21no—
444C4—22no—
445C4—22no—
446C4—42no—
447C4—44no—
448C4—44no—
449C4—42no—
450C4—44no—
451C4—42no—
452C4—82no—
453C4—81no—
454C4#145912yes—
455C4#146512yes—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—44no—
461C2#830111yescharacteristic
462C2#831111yescharacteristic
463C2#832411yescharacteristic
464C2#835911yescharacteristic
465C2#837111yescharacteristic
466C2#837911yescharacteristic
467C2#839511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
3. #8311C2semidirect
5. #7576C2semidirect
35. C2×C22⋊Q8C8semidirect
330. C23Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
#85572
#83112
C4×#3232
#75762