Finite Group Databasebeta

← #97566

Subgroups — #97566

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2367C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic, complemented
3#8311C212yesmaximal, complemented
4C4×#324C211yesmaximal, characteristic, complemented
5#7572C212yesmaximal
6C22×#124C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×C4×D4C411yescharacteristic
9#2367—24no—
10#2367—24no—
11C2×#391C411yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#388—21no—
14C2×#388—21no—
15C2×#385—21no—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4—21no—
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×Q8⋊C8—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1270—24no—
27C23×Q8C2×C411yescharacteristic
28C23×D4C2×C411yescharacteristic
29C24×C4—21no—
30C2×#124—22no—
31C2×#124—22no—
32C2×#124—22no—
33C2×#124—22no—
34C2×#124—22no—
35C2×#124C814yescomplemented
36C2×C4×Q8—22no—
37C2×C4×Q8—44no—
38C2×C4×Q8D414yes—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—44no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C4×C8—22no—
61#391—22no—
62#391—22no—
63#390—22no—
64#390D414yes—
65#388—28no—
66#388—28no—
67#385—28no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C22⋊C4—28no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95D411yescharacteristic
77C2×#95D411yescharacteristic
78#336—22no—
79#336—22no—
80Q8⋊C8—48no—
81#324—24no—
82#324—42no—
83C25—21no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8C22⋊C412yes—
90C22×Q8C2×C812yes—
91C22×Q8M1612yes—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4—21no—
95C22×D4—42no—
96C22×D4—42no—
97C22×D4C22⋊C412yes—
98C22×D4C2×C812yes—
99C22×D4M1612yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—24no—
104C23×C4—44no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—82no—
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C22×C8—42no—
116C22×C8—42no—
117#124—28no—
118#124—28no—
119#124—28no—
120#124—28no—
121C4×Q8—28no—
122C4×Q8—44no—
123C4×Q8—48no—
124C4×D4—48no—
125C4×D4—44no—
126C4×D4—44no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—82no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—42no—
165C2×C42—44no—
166C2×C42—42no—
167C2×C42—82no—
168C2×C42C2×C812yes—
169C2×C42M1612yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C4⋊C8—44no—
173C22⋊C8—24no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177C4×C8—44no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—22no—
182#95—22no—
183C24—22no—
184C24—22no—
185C24—22no—
186C24—42no—
187C24—42no—
188C24—42no—
189C24—42no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24—41no—
194C24C2×C22⋊C411yescharacteristic
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8C22⋊C814yes—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4C22⋊C814yes—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—84no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4C22⋊C812yes—
267C22×C4C22⋊C812yes—
268C22×C4#10412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4#12012yes—
272C22×C4#12012yes—
273C22×C4C22×C811yescharacteristic
274C22×C4C2×M1611yescharacteristic
275C22×C4C2×M1611yescharacteristic
276C22×C4C2×M1611yescharacteristic
277C22×C4#13512yes—
278C22×C4AGL(1,ℤ/8)11yescharacteristic
279C22×C4#13711yescharacteristic
280C2×C8—44no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8—82no—
285C2×C8—82no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4#9914yes—
295C22⋊C4—216no—
296C22⋊C4—216no—
297C22⋊C4—48no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—84no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—24no—
306C42—42no—
307C42—48no—
308C42—42no—
309C42—42no—
310C42—84no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—24no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—82no—
337C23—82no—
338C23#37911yescharacteristic
339C23C2×C22⋊C811yescharacteristic
340C23#40711yescharacteristic
341C23#41611yescharacteristic
342C23#41711yescharacteristic
343C23C2×#10411yescharacteristic
344C23#42111yescharacteristic
345Q8—28no—
346Q8—44no—
347Q8—44no—
348Q8—44no—
349D4—44no—
350D4—44no—
351D4—44no—
352D4—44no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—816no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4C2×C22⋊C811yescharacteristic
408C2×C4C2×C22⋊C811yescharacteristic
409C2×C4#40711yescharacteristic
410C2×C4#40711yescharacteristic
411C2×C4C2×#9912yes—
412C2×C4#41312yes—
413C2×C4#44912yes—
414C2×C4#45212yes—
415C8—84no—
416C8—162no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22—84no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22#147611yescharacteristic
441C22#147711yescharacteristic
442C22#148811yescharacteristic
443C22#148911yescharacteristic
444C22#177911yescharacteristic
445C22#178111yescharacteristic
446C22#178511yescharacteristic
447C22#178811yescharacteristic
448C22#178911yescharacteristic
449C22#179511yescharacteristic
450C22#179611yescharacteristic
451C4—21no—
452C4—21no—
453C4—22no—
454C4—22no—
455C4—44no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—44no—
460C4—42no—
461C4—81no—
462C4—82no—
463C4#145912yes—
464C4#146512yes—
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—84no—
470C2#830111yescharacteristic
471C2#831211yescharacteristic
472C2#832411yescharacteristic
473C2#835911yescharacteristic
474C2#836811yescharacteristic
475C2#837711yescharacteristic
476C2#838611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
2. #8558C2semidirect
3. #8311C2semidirect
4. C4×#324C2semidirect
35. C2×#124C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
#85582
#83112
C4×#3242
#75722