Finite Group Databasebeta

← #195626

Subgroups — #195626

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9060C218yesmaximal, complemented
7#7897C211yesmaximal, characteristic
8#7890C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#487—22no—
12C2×#487—24no—
13C2×#487C418yescomplemented
14C2×#487C2214yes—
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C22×#103—22no—
18C22×C4×C8C2211yescharacteristic
19#2369C2214yes—
20#2365—22no—
21#2345—22no—
22Q8×C22⋊C4—22no—
23#2331—22no—
24#2316—21no—
25C4×C4⋊Q8—22no—
26#2285C2211yescharacteristic
27#2274—21no—
28C4×#117C2211yescharacteristic
29#1988C2214yes—
30#1987—22no—
31#1981C2214yes—
32#1956—22no—
33#1787—22no—
34#1736—21no—
35#1736—21no—
36#1548—24no—
37C2×#333C2214yes—
38#1279C2214yes—
39#1279C2214yes—
40C23×Q8—21no—
41C23×Q8D412yes—
42C22×M16—21no—
43C23×C8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D412yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8C2312yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×Q8—24no—
61C2×#117—21no—
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#487—216no—
70#487—28no—
71#487C2×C4116yes—
72#431—22no—
73#431—22no—
74#423—22no—
75#423—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—216no—
80C2×#103—28no—
81C2×#103—28no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8C2×C414yes—
87C2×C4×C8C2312yes—
88#391—24no—
89#391—28no—
90#389—24no—
91#389—24no—
92#389—24no—
93#387—28no—
94#384D412yes—
95#382—21no—
96#380—24no—
97#380—21no—
98#380D412yes—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2312yes—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109#376—21no—
110C43—21no—
111#340—24no—
112#340—24no—
113#333—28no—
114C22×Q8—22no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—48no—
119C22×Q8—41no—
120C22×Q8—42no—
121C22×Q8—84no—
122C22×Q8C22⋊C418yes—
123C22×Q8C2×D414yes—
124C23×C4—22no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—24no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—44no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8C22⋊C418yes—
154C4⋊Q8C22×C414yes—
155C4⋊Q8C22×C414yes—
156C4⋊Q8C2×D414yes—
157C4×Q8—88no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—42no—
162#117—42no—
163#117Q1614yes—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—28no—
195C2×C42C22×C412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—42no—
208#103—232no—
209#103—88no—
210C22⋊C8—42no—
211C4×C8—24no—
212C4×C8—22no—
213C4×C8—42no—
214C4×C8C22×C414yes—
215#95—28no—
216#95—28no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221C24#12011yescharacteristic
222C2×Q8—24no—
223C2×Q8—28no—
224C2×Q8—24no—
225C2×Q8—48no—
226C2×Q8—416no—
227C2×Q8—42no—
228C2×Q8—44no—
229C2×Q8—48no—
230C2×Q8—48no—
231C2×Q8—88no—
232C2×Q8—88no—
233C2×Q8—84no—
234C2×Q8—88no—
235C2×Q8C2×C22⋊C418yes—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C4×D412yes—
249C22×C4C4×D412yes—
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12112yes—
254C22×C4#12112yes—
255C22×C4#12312yes—
256C22×C4C2×Q1612yes—
257C22×C4#13512yes—
258C22×C4#13712yes—
259C22×C4#13712yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263M16—84no—
264C2×C8—24no—
265C2×C8—44no—
266C2×C8—48no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—44no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—28no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—88no—
291C4⋊C4C2×Q1614yes—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—44no—
295C42—24no—
296C42—42no—
297C42—42no—
298C42—44no—
299C42—42no—
300C42—44no—
301C42—88no—
302C42C2×C22⋊C412yes—
303C42C4×D412yes—
304C42C4×D412yes—
305C42#12112yes—
306C42#12112yes—
307C42#12312yes—
308C42C23×C411yescharacteristic
309C42C22×D411yescharacteristic
310C42C2×#4111yescharacteristic
311C42C2×#4111yescharacteristic
312C23—22no—
313C23—22no—
314C23#38612yes—
315C23#45112yes—
316C23#45212yes—
317C23C2×#12011yescharacteristic
318Q8—48no—
319Q8—416no—
320Q8—48no—
321Q8—84no—
322Q8—88no—
323Q8—84no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—88no—
343C2×C4—88no—
344C2×C4—84no—
345C2×C4#38612yes—
346C2×C4#38612yes—
347C2×C4#38612yes—
348C2×C4C4×Q1614yes—
349C2×C4#44114yes—
350C2×C4#45112yes—
351C2×C4#45212yes—
352C2×C4#45514yes—
353C2×C4#46214yes—
354C2×C4#46414yes—
355C2×C4C22×C22⋊C411yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×#12311yescharacteristic
361C2×C4C22×Q1611yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C2×C4C2×#13711yescharacteristic
364C2×C4C2×#13711yescharacteristic
365C2×C4#57714yes—
366C8—44no—
367C8—82no—
368C8—82no—
369C22—22no—
370C22—24no—
371C22—24no—
372C22—22no—
373C22#187814yes—
374C22C2×#38611yescharacteristic
375C22C2×#45111yescharacteristic
376C22C2×#45211yescharacteristic
377C22#301012yes—
378C22C2×#45512yes—
379C4—22no—
380C4—21no—
381C4—21no—
382C4—44no—
383C4—48no—
384C4—84no—
385C4—82no—
386C4—84no—
387C4#187814yes—
388C4#188514yes—
389C4C2×C4×Q1611yescharacteristic
390C4C2×#44111yescharacteristic
391C4#294912yes—
392C4C2×#46211yescharacteristic
393C4C2×#46411yescharacteristic
394C4#304312yes—
395C2—24no—
396C2#1863312yes—
397C2#1863712yes—
398C2C2×#187811yescharacteristic
399C2#2093611yescharacteristic
400C2C2×#188511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9060C2semidirect
13. C2×#487C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
#342162
#218042
#213132
C2×#15482
#90602
#78972
#78902