Finite Group Databasebeta

← #188100

Subgroups — #188100

Order 512.

2510 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2#20727C212yesmaximal, complemented
3#17394C212yesmaximal, complemented
4#17364C212yesmaximal
5#8555C214yesmaximal, complemented
6#8550C214yesmaximal, complemented
7C2×#554—28no—
8C2×#554—24no—
9C2×#554—24no—
10C2×#554—24no—
11C2×#554—24no—
12C22×C4⋊Q8C411yescharacteristic
13C22×#125C411yescharacteristic
14C22×#125C2211yescharacteristic
15C22×C22⋊Q8C412yes—
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C411yescharacteristic
20C22×C4×D4C411yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C412yes—
23#2920C2214yes—
24#1821C2214yes—
25#1801—24no—
26#1784C2214yes—
27C4×#101—24no—
28#1470—28no—
29#1466—24no—
30#1463C2218yes—
31#1462C2218yes—
32#1460—24no—
33C23×Q8C2×C412yes—
34C23×Q8C2311yescharacteristic
35C23×D4C2×C411yescharacteristic
36C24×C4C2×C412yes—
37C24×C4C2×C412yes—
38C24×C4C2×C412yes—
39#554—232no—
40#554—232no—
41#554—232no—
42#554—232no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—24no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—24no—
50C2×#125—24no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125D418yes—
56C2×C22⋊Q8—28no—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—28no—
66C2×C4×Q8—24no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8D414yes—
70C2×C4×Q8D414yes—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4D414yes—
83C2×C4×D4D414yes—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C412yes—
90C22×C4⋊C4C2×C412yes—
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C4⋊C4C2×C412yes—
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C4⋊C4C2×C411yescharacteristic
95C22×C4⋊C4C2×C411yescharacteristic
96C22×C4⋊C4C2×C411yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2×C412yes—
100C22×C22⋊C4C2×C412yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2×C411yescharacteristic
103C22×C42C2×C411yescharacteristic
104C22×C42C2311yescharacteristic
105#432—24no—
106#432—24no—
107#431—24no—
108#431—24no—
109#423—24no—
110C2×#101—28no—
111#407C2318yes—
112C4×M16—24no—
113#343—28no—
114#324—216no—
115#323—216no—
116C25C22×C412yes—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—84no—
124C22×Q8C2×D414yes—
125C22×D4—28no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4C22⋊C412yes—
129C22×D4C22⋊C412yes—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—48no—
140C23×C4—84no—
141C23×C4—84no—
142C23×C4C22⋊C418yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2411yescharacteristic
153C2×M16—28no—
154C4⋊Q8—216no—
155C4⋊Q8—216no—
156#125—216no—
157#125—216no—
158#125—28no—
159#125—28no—
160#125—28no—
161#125—28no—
162C22⋊Q8—232no—
163C22⋊Q8—232no—
164C22⋊Q8—264no—
165C22⋊Q8—264no—
166C4×Q8—28no—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×D4—216no—
171C4×D4—216no—
172C4×D4—216no—
173C4×D4—216no—
174C4×D4—28no—
175C4×D4—28no—
176C4×D4—28no—
177C4×D4—28no—
178C2×C4⋊C4—232no—
179C2×C4⋊C4—232no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C418yes—
211C2×C4⋊C4C22⋊C414yes—
212C2×C4⋊C4C22⋊C414yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D418yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4C2×D418yes—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42C22⋊C414yes—
235C2×C42C22⋊C414yes—
236C2×C42C2×D414yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C4⋊C8—44no—
242C4⋊C8—44no—
243#101—416no—
244C22⋊C8—28no—
245C22⋊C8—28no—
246#97—44no—
247C4×C8—44no—
248C24—22no—
249C24—22no—
250C24—28no—
251C24—44no—
252C24—44no—
253C24C2×C22⋊C414yes—
254C24C23×C411yescharacteristic
255C2×Q8—216no—
256C2×Q8—28no—
257C2×Q8—28no—
258C2×Q8—44no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8—88no—
264C2×D4—216no—
265C2×D4—216no—
266C2×D4—28no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4#10118yescomplemented
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—216no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—216no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—416no—
296C22×C4—416no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—88no—
308C22×C4—88no—
309C22×C4—88no—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C414yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C414yes—
320C22×C4C2×C22⋊C414yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328M16—416no—
329C2×C8—48no—
330C4⋊C4—216no—
331C4⋊C4—216no—
332C4⋊C4—216no—
333C4⋊C4—28no—
334C4⋊C4—216no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C22⋊C4—216no—
348C22⋊C4—216no—
349C22⋊C4—216no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42#12014yes—
355C42#12014yes—
356C42#12014yes—
357C42#12014yes—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—28no—
362C23—28no—
363C23—24no—
364C23—24no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—432no—
370C23—44no—
371C23—44no—
372C23C2×#10118yes—
373C23C22×C22⋊C411yescharacteristic
374C23C22×C22⋊C411yescharacteristic
375C23C22×C22⋊C411yescharacteristic
376Q8—416no—
377Q8—816no—
378D4—216no—
379D4—216no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—432no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—88no—
420C2×C4—88no—
421C2×C4—816no—
422C2×C4#37912yes—
423C2×C4#37912yes—
424C2×C4#37912yes—
425C2×C4#37912yes—
426C2×C4#41014yes—
427C2×C4C2×#10114yes—
428C2×C4C2×#12014yes—
429C8—88no—
430C22—28no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—28no—
438C22—48no—
439C22—416no—
440C22—48no—
441C22—416no—
442C22—44no—
443C22—44no—
444C22C2×#37911yescharacteristic
445C22C2×#41012yes—
446C22C22×#10112yes—
447C22#288812yes—
448C4—24no—
449C4—24no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—42no—
454C4—44no—
455C4—44no—
456C4—42no—
457C4—44no—
458C4—42no—
459C4#178414yes—
460C2—22no—
461C2—22no—
462C2—22no—
463C2—48no—
464C2#2058411yescharacteristic
465C2C2×#178411yescharacteristic
466C2#2059211yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #17394C2semidirect
5. #8555C2semidirect
6. #8550C2semidirect
270. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#207272
#173942
#173642
#85552
#85502