Finite Group Databasebeta

← #363871

Subgroups — #363871

Order 512.

2902 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#1873C214yesmaximal
3C2×#1817C211yesmaximal, characteristic
4#13756C211yesmaximal, characteristic
5#11384C212yesmaximal, complemented
6#11010C212yesmaximal, complemented
7#10789C214yesmaximal, complemented
8C2×#533—24no—
9C2×#533—216no—
10C2×#533C418yescomplemented
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125—21no—
14C22×C22⋊Q8—21no—
15C22×C22⋊Q8C2212yes—
16C22×C4×Q8—21no—
17C22×C4×Q8C2212yes—
18C22×#117—21no—
19C22×#117C2212yes—
20C23×C42C2211yescharacteristic
21C22×#103C2218yes—
22C22×C22⋊C8C2212yes—
23#1873—216no—
24#1817—24no—
25C2×#340C2218yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C4⋊#95C2212yes—
29C4⋊#95C2212yes—
30C22⋊#95—21no—
31C22⋊#95—21no—
32C4×#95—21no—
33C4×#95—21no—
34C23×Q8—21no—
35C23×Q8C2312yes—
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4C2311yescharacteristic
39C23×C8C2314yes—
40#533—264no—
41#533—264no—
42C2×C4⋊Q8—28no—
43C2×C4⋊Q8—28no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C414yes—
47C2×C4⋊Q8C2×C414yes—
48C2×#125—216no—
49C2×#125—28no—
50C2×C22⋊Q8—216no—
51C2×C22⋊Q8—28no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8C2×C418yes—
55C2×C22⋊Q8D4116yes—
56C2×C4×Q8—216no—
57C2×C4×Q8—28no—
58C2×C4×Q8—28no—
59C2×C4×Q8—28no—
60C2×C4×Q8—24no—
61C2×C4×Q8C2×C418yes—
62C2×#117—216no—
63C2×#117—28no—
64C2×#117—28no—
65C2×#117—28no—
66C2×#117—44no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—28no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42C2×C414yes—
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D414yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#103—216no—
86C2×#103—216no—
87C2×#103—48no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—28no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94#340—216no—
95#336—24no—
96#336—24no—
97C25C2×D411yescharacteristic
98C22×Q8—28no—
99C22×Q8—24no—
100C22×Q8—22no—
101C22×Q8—24no—
102C22×Q8—28no—
103C22×Q8—24no—
104C22×Q8—24no—
105C22×Q8—44no—
106C22×Q8—44no—
107C22×Q8C22×C414yes—
108C22×Q8C2×D418yes—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—42no—
121C23×C4C22⋊C414yes—
122C23×C4C22⋊C414yes—
123C23×C4C22×C412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D418yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2411yescharacteristic
132C22×C8—24no—
133C22×C8—28no—
134C22×C8—28no—
135C22×C8—24no—
136C22×C8—24no—
137C4⋊Q8—216no—
138C4⋊Q8—216no—
139C4⋊Q8—216no—
140C4⋊Q8—216no—
141#125—264no—
142C22⋊Q8—264no—
143C22⋊Q8—216no—
144C22⋊Q8—232no—
145C22⋊Q8C22⋊C4132yes—
146C4×Q8—232no—
147C4×Q8—264no—
148C4×Q8—216no—
149C4×Q8—216no—
150#117—264no—
151#117—432no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—216no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—216no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4C22×C414yes—
168C2×C4⋊C4C2×D418yes—
169C2×C4⋊C4#4118yes—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—216no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C42—216no—
176C2×C42—28no—
177C2×C42—216no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—216no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—28no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42C22⋊C414yes—
187C2×C42C22⋊C414yes—
188C2×C42C22×C412yes—
189C2×C42C22×C412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192#103—432no—
193C22⋊C8—48no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198C24—22no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24C2×C22⋊C412yes—
206C24#12014yes—
207C24C22×D411yescharacteristic
208C2×Q8—216no—
209C2×Q8—24no—
210C2×Q8—28no—
211C2×Q8—28no—
212C2×Q8—28no—
213C2×Q8—216no—
214C2×Q8—216no—
215C2×Q8—48no—
216C2×Q8—432no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—48no—
220C2×Q8—48no—
221C2×Q8—48no—
222C2×Q8C2×C22⋊C418yes—
223C22×C4—232no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—216no—
239C22×C4—216no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—44no—
253C22×C4—416no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C418yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4#12018yes—
272C22×C4#12014yes—
273C22×C4#13514yes—
274C22×C4#13714yes—
275C22×C4#13714yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D414yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4114yes—
280C22×C4H(5,𝔽2)12yes—
281C22×C42−1+412yes—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—48no—
286C2×C8—48no—
287C4⋊C4—216no—
288C4⋊C4—216no—
289C4⋊C4—232no—
290C4⋊C4—216no—
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—232no—
294C4⋊C4—216no—
295C4⋊C4—48no—
296C4⋊C4C2×C22⋊C418yes—
297C22⋊C4—232no—
298C22⋊C4—416no—
299C42—232no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—28no—
305C42—28no—
306C42—416no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—24no—
313C23—24no—
314C23—28no—
315C23—28no—
316C23—22no—
317C23—44no—
318C23—44no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23#37914yes—
324C23C22×C22⋊C411yescharacteristic
325C23C2×#12012yes—
326C23#52512yes—
327C23#53511yescharacteristic
328C23#53611yescharacteristic
329C23C2×#13512yes—
330C23C2×#13712yes—
331C23C2×#13712yes—
332C23#57512yes—
333Q8—216no—
334Q8—432no—
335Q8—48no—
336Q8—416no—
337C2×C4—232no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—216no—
352C2×C4—216no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—416no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—416no—
366C2×C4—44no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4#37914yes—
375C2×C4#41614yes—
376C2×C4#41914yes—
377C2×C4#41914yes—
378C2×C4C22×C22⋊C412yes—
379C2×C4C4×#4112yes—
380C2×C4#52012yes—
381C2×C4C2×#12012yes—
382C2×C4#53212yes—
383C2×C4#53712yes—
384C2×C4#54618yes—
385C2×C4#54718yes—
386C8—84no—
387C22—24no—
388C22—24no—
389C22—24no—
390C22—48no—
391C22—42no—
392C22—42no—
393C22—48no—
394C22—42no—
395C22—42no—
396C22C2×#37911yescharacteristic
397C22#229711yescharacteristic
398C22#231111yescharacteristic
399C22#240412yes—
400C22#240612yes—
401C22C2×#41611yescharacteristic
402C22C2×#41911yescharacteristic
403C22C2×#41911yescharacteristic
404C22#289611yescharacteristic
405C22#300214yes—
406C22#300414yes—
407C22#310714yes—
408C22#311614yes—
409C22#321012yes—
410C22#321612yes—
411C22#323112yes—
412C22#323712yes—
413C4—24no—
414C4—24no—
415C4—24no—
416C4—24no—
417C4—42no—
418C4—48no—
419C4—48no—
420C4—88no—
421C4#233914yes—
422C4#243014yes—
423C2—44no—
424C2#1395611yescharacteristic
425C2#2059711yescharacteristic
426C2#2059911yescharacteristic
427C2#2138911yescharacteristic
428C2#2139211yescharacteristic
429C2#2186411yescharacteristic
430C2#2187011yescharacteristic
431C2#2393514yes—
432C2#2406514yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11384C2semidirect
6. #11010C2semidirect
7. #10789C2semidirect
10. C2×#533C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#18732
C2×#18172
#137562
#113842
#110102
#107892