Finite Group Databasebeta

← #239186

Subgroups — #239186

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21974C211yesmaximal, characteristic
3#21389C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1561C211yesmaximal, characteristic
6C2×#1546C211yesmaximal, characteristic
7#10542C214yesmaximal, complemented
8#10538C214yesmaximal, complemented
9#7880C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—21no—
17C2×#462—21no—
18C2×#451—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1988C2212yes—
25#1988C2214yes—
26#1987—21no—
27#1987—21no—
28#1981C2212yes—
29#1939C2214yes—
30#1873—22no—
31#1851—22no—
32#1842—21no—
33#1831—21no—
34C4×#103—21no—
35C4×#103—21no—
36#1736—22no—
37#1561—24no—
38#1546—24no—
39C2×#332C2214yes—
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1277C2214yes—
43C23×Q8—21no—
44C22×Q16—22no—
45C22×Q16—22no—
46C22×M16—21no—
47C22×M16—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533D414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C2×#117D412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#494—24no—
73#494—24no—
74#487—24no—
75#487—24no—
76#462—28no—
77#462—28no—
78#451—216no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#423—22no—
84#423—22no—
85#423—22no—
86#423—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8D412yes—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103C4×C4⋊C4D411yescharacteristic
104C4×C4⋊C4D411yescharacteristic
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108#376—21no—
109C43—21no—
110#340—22no—
111#340—22no—
112#340—24no—
113#340—24no—
114#336—24no—
115#332—28no—
116#332—24no—
117#332—24no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C2×Q16—28no—
131C2×Q16—28no—
132C2×Q16—84no—
133C2×Q16—84no—
134C2×M16—44no—
135C2×M16—42no—
136C2×M16—42no—
137C2×M16—82no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—21no—
147C4⋊Q8—22no—
148C4⋊Q8—21no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8C2×D414yes—
159C4⋊Q8C2×D412yes—
160C4⋊Q8C2×D412yes—
161#125—24no—
162#125—42no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—42no—
170C4×Q8—42no—
171#117—24no—
172#117—24no—
173#117—42no—
174#117—42no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—41no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—41no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42#4112yes—
203C2×C42#4112yes—
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8—84no—
217#103—28no—
218#103—28no—
219#103—44no—
220#103—44no—
221C22⋊C8—24no—
222C22⋊C8—42no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—42no—
226#95—21no—
227#95—21no—
228#95—22no—
229C24#12111yescharacteristic
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—88no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—84no—
260C22×C4#12012yes—
261C22×C4#12112yes—
262C22×C4#12112yes—
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12312yes—
267C22×C4#12312yes—
268C22×C4#12412yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4111yescharacteristic
272Q16—88no—
273Q16—88no—
274M16—84no—
275M16—84no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—44no—
286C2×C8—44no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4#13714yes—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C42—44no—
310C42—42no—
311C42—42no—
312C42—44no—
313C42—42no—
314C42—42no—
315C42#12012yes—
316C42#12112yes—
317C42#12112yes—
318C42#12312yes—
319C42#12312yes—
320C42#12412yes—
321C42C22×D411yescharacteristic
322C42C22×D411yescharacteristic
323C42C2×#4111yescharacteristic
324C42C2×#4111yescharacteristic
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23#39412yes—
330C23C2×#12111yescharacteristic
331Q8—42no—
332Q8—44no—
333Q8—42no—
334Q8—44no—
335Q8—42no—
336Q8—42no—
337Q8—88no—
338Q8—82no—
339Q8—84no—
340Q8—82no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—88no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4#39412yes—
367C2×C4#39412yes—
368C2×C4#39412yes—
369C2×C4#47312yes—
370C2×C4#47312yes—
371C2×C4C2×#12011yescharacteristic
372C2×C4C2×#12111yescharacteristic
373C2×C4C2×#12111yescharacteristic
374C2×C4C2×#12311yescharacteristic
375C2×C4C2×#12311yescharacteristic
376C2×C4C2×#12411yescharacteristic
377C2×C4C2×#13712yes—
378C2×C4#57512yes—
379C2×C4#57612yes—
380C2×C4#57712yes—
381C8—42no—
382C8—42no—
383C8—82no—
384C8—82no—
385C8—82no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22C2×#39411yescharacteristic
392C22#304512yes—
393C22#305612yes—
394C22C2×#47311yescharacteristic
395C22C2×#47311yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—44no—
401C4—82no—
402C4—84no—
403C4—82no—
404C4—82no—
405C4#204714yes—
406C4#300811yescharacteristic
407C4#300911yescharacteristic
408C4#303712yes—
409C4#303911yescharacteristic
410C4#304011yescharacteristic
411C4#313612yes—
412C2—42no—
413C2#1948512yes—
414C2#1949712yes—
415C2#2350911yescharacteristic
416C2C2×#204711yescharacteristic
417C2#2354711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
7. #10542C2semidirect
8. #10538C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#219742
#213892
#213132
C2×#15612
C2×#15462
#105422
#105382
#78802