Finite Group Databasebeta

← #196256

Subgroups — #196256

Order 512.

1165 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2982C211yesmaximal, characteristic, complemented
2#35246C211yesmaximal, characteristic
3#17480C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9102C214yesmaximal, complemented
6#9100C214yesmaximal, complemented
7#7857C212yesmaximal, complemented
8C2×#529C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2982—24no—
11#2982—22no—
12#2982C2214yes—
13C2×C8×Q8—24no—
14#2920C2212yes—
15C2×#431—22no—
16C2×#431C2212yes—
17#2853—22no—
18#2637—22no—
19#2599—22no—
20#2579—22no—
21#2565C412yes—
22#2545C412yes—
23#2504C411yescharacteristic
24#2502C411yescharacteristic
25#2436C411yescharacteristic
26#2434C411yescharacteristic
27#2327C2212yes—
28#2326C2212yes—
29#2283C2211yescharacteristic
30#1495—24no—
31#1494—24no—
32#1479C2214yes—
33#1478C2214yes—
34#1467—22no—
35#1463—22no—
36#1461C2214yes—
37#1274C2218yes—
38C23×Q8C2×C411yescharacteristic
39C2×C4⋊Q8C2×C412yes—
40#529—22no—
41#529—22no—
42#529—22no—
43#529—22no—
44C2×#126—21no—
45C2×#125C2×C411yescharacteristic
46C2×#124C2×C412yes—
47C2×#124C2×C411yescharacteristic
48C2×#123—21no—
49C2×C22⋊Q8C2×C411yescharacteristic
50C2×#121C2×C411yescharacteristic
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2312yes—
60C2×C4×D4—21no—
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67C8×Q8—216no—
68C8×Q8—216no—
69#432—22no—
70#432—22no—
71#431—28no—
72#431—28no—
73#431—22no—
74#431—22no—
75#431—22no—
76#431D414yes—
77#431C2314yes—
78#431C2314yes—
79#423—22no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82#407—22no—
83C2×C22⋊C8—22no—
84C4×M16—22no—
85C2×C4×C8—24no—
86C2×C4×C8D414yes—
87#398—24no—
88#398—22no—
89#397—22no—
90#396—24no—
91#395—22no—
92#393—22no—
93#391C2×C412yes—
94#391C2×C412yes—
95#391C2×C412yes—
96#390C2×C412yes—
97#388—24no—
98#388—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#386—22no—
103#386—22no—
104#385—22no—
105#385—22no—
106#385—24no—
107#384C2×C412yes—
108#383—21no—
109#382—21no—
110#382C2×C412yes—
111#381—21no—
112#380—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C22⋊C4C2×C411yescharacteristic
115C4×C22⋊C4C2×C411yescharacteristic
116#376C2312yes—
117Q8⋊C8—28no—
118D4⋊C8—28no—
119#324—28no—
120C22×Q8—28no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C22⋊C412yes—
124C22×Q8C22×C411yescharacteristic
125C22×Q8C22×C412yes—
126C22×D4—21no—
127C22×D4C22×C411yescharacteristic
128C23×C4—21no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C2×M16—42no—
135C22×C8—24no—
136C4⋊Q8—28no—
137#126—42no—
138#126—42no—
139#125—22no—
140#125—22no—
141#124—28no—
142#124—22no—
143#124—22no—
144#123—41no—
145#123—41no—
146#123—41no—
147#123—41no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150#121—22no—
151#121—22no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—42no—
165C4×D4—42no—
166#117—42no—
167#117—42no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C414yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42C22⋊C414yes—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D414yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—216no—
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—42no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—42no—
220#97—42no—
221C4×C8—216no—
222C4×C8—24no—
223C4×C8—24no—
224C4×C8C2×D418yes—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—22no—
234C24—21no—
235C24C2×C22⋊C411yescharacteristic
236C2×Q8—216no—
237C2×Q8—216no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—28no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8#13118yes—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—44no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C414yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12311yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—84no—
283C2×C8—28no—
284C2×C8—48no—
285C2×C8—44no—
286C4⋊C4—24no—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—48no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—48no—
312C22⋊C4—84no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—42no—
320C42C4×D412yes—
321C42C4×D412yes—
322C42#12012yes—
323C42#12112yes—
324C42#12112yes—
325C42#12312yes—
326C42C22×D412yes—
327C42C2×#4112yes—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23#38611yescharacteristic
336C23#38611yescharacteristic
337C23C22×C22⋊C411yescharacteristic
338Q8—216no—
339Q8—216no—
340Q8—44no—
341D4—44no—
342D4—82no—
343D4—82no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4#38611yescharacteristic
379C2×C4#38611yescharacteristic
380C2×C4#38612yes—
381C2×C4#38612yes—
382C2×C4#40818yes—
383C2×C4C2×C4×D412yes—
384C2×C4C2×#12011yescharacteristic
385C2×C4C2×#12112yes—
386C2×C4C2×#12311yescharacteristic
387C2×C4C2×#13114yes—
388C8—48no—
389C8—84no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—44no—
398C22—84no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22#191212yes—
403C22C2×#38611yescharacteristic
404C22C2×#40812yes—
405C4—22no—
406C4—22no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—28no—
413C4—44no—
414C4—82no—
415C4—82no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4#292914yes—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—161no—
424C2#2089511yescharacteristic
425C2#2100911yescharacteristic
426C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2982C2semidirect
4. #17365C2semidirect
5. #9102C2semidirect
6. #9100C2semidirect
7. #7857C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29822
#352462
#174802
#173652
#91022
#91002
#78572