Finite Group Databasebeta

← #195273

Subgroups — #195273

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2#34398C211yesmaximal, characteristic
3#21841C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic
5#14274C211yesmaximal, characteristic, complemented
6#9045C218yesmaximal, complemented
7#8013C211yesmaximal, characteristic, complemented
8#7914C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3227—24no—
12#3227C418yescomplemented
13C2×#498—21no—
14C2×#486—21no—
15C2×#481C2214yes—
16C2×#476—22no—
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2379—22no—
20#2361—22no—
21#2357—21no—
22#2349—21no—
23#2339—22no—
24#2331—21no—
25#2328—21no—
26#2325C2214yes—
27#2318—22no—
28C2×#391—22no—
29#2283C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1993—22no—
33#1988—21no—
34#1988—21no—
35#1824C2214yes—
36#1505—24no—
37C2×#327C2214yes—
38#1445—22no—
39#1445C2214yes—
40#1294C2214yes—
41#1280C2214yes—
42C22×#41—21no—
43C23×Q8—21no—
44#532—22no—
45#532—22no—
46#532C2×C414yes—
47#532D414yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yescomplemented
56C2×C4⋊Q8D411yescharacteristic
57C2×#127D411yescharacteristic
58C2×#124—21no—
59C2×#121—21no—
60C2×#121C2312yes—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—21no—
64C2×C4×D4D412yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#498—28no—
71#486—28no—
72#481—28no—
73#481C2×C4116yescomplemented
74#476—216no—
75#432—22no—
76#432—22no—
77#431—22no—
78#431C2×C414yescomplemented
79C2×#106—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×#102—24no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85C2×#97—21no—
86C2×C4×C8—21no—
87#391—24no—
88#391—48no—
89#390—22no—
90#389—22no—
91#388—24no—
92#388—24no—
93#387—24no—
94#386—24no—
95#386—24no—
96#386—24no—
97#385—24no—
98#385—24no—
99#383—21no—
100#382—21no—
101#382—21no—
102#382—21no—
103#382—21no—
104#382D412yes—
105#381—22no—
106#381—21no—
107#380—22no—
108#380C2312yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C22⋊C4—22no—
113C4×C22⋊C4—21no—
114#376—21no—
115#376—22no—
116#376—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—28no—
120C2×#95D412yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123#340—24no—
124#340—24no—
125#327—28no—
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—84no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—44no—
132C22×Q8—44no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—42no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C2×D412yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D414yes—
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8C22⋊C414yes—
160#127—22no—
161#127C22⋊C414yes—
162#124—24no—
163#124—24no—
164#121—24no—
165#121—24no—
166#121—22no—
167#121—22no—
168#121C22×C414yes—
169#121C2×D414yes—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—84no—
174C4×D4—88no—
175C4×D4C22⋊C418yes—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C42—24no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—24no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4112yes—
213C2×C42#4112yes—
214#106—216no—
215C4⋊C8—24no—
216C4⋊C8—42no—
217#102—216no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8C22×C414yes—
222#97—42no—
223C4×C8—24no—
224#95—24no—
225#95—24no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—416no—
238#95—42no—
239C24—22no—
240C24C22×D411yescharacteristic
241#41—88no—
242#41—88no—
243C2×Q8—24no—
244C2×Q8—416no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—82no—
250C2×Q8—88no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—44no—
259C2×D4—48no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—82no—
263C2×D4C2×C22⋊C414yes—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—28no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C2×C8—24no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—42no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4C2×C22⋊C414yes—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—28no—
325C22⋊C4—44no—
326C22⋊C4—84no—
327C22⋊C4—84no—
328C22⋊C4—84no—
329C42—48no—
330C42—42no—
331C42—42no—
332C42—84no—
333C42C2×C22⋊C412yes—
334C42C2×C22⋊C412yes—
335C42C4×D412yes—
336C42C4×D412yes—
337C42#12112yes—
338C42#12112yes—
339C42#12312yes—
340C42#12312yes—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—24no—
345C23—44no—
346C23—42no—
347C23—42no—
348C23—44no—
349C23C2×C4×D411yescharacteristic
350C23C2×#12011yescharacteristic
351C23C2×#12111yescharacteristic
352C23#57714yes—
353Q8—84no—
354Q8—88no—
355Q8—88no—
356D4—44no—
357D4—48no—
358D4—48no—
359D4—48no—
360D4—82no—
361D4—84no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—88no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—88no—
399C2×C4—168no—
400C2×C4#38612yes—
401C2×C4#38612yes—
402C2×C4#38612yes—
403C2×C4#38612yes—
404C2×C4#45312yes—
405C2×C4#45412yes—
406C2×C4C22×C22⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×#12311yescharacteristic
410C8—44no—
411C8—82no—
412C22—24no—
413C22—24no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—84no—
422C22—84no—
423C22C2×#38611yescharacteristic
424C22#294912yes—
425C22#300912yes—
426C22C2×#45311yescharacteristic
427C22C2×#45411yescharacteristic
428C22#302812yes—
429C22#304212yes—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—44no—
438C4—84no—
439C4—84no—
440C4—82no—
441C4—82no—
442C4—84no—
443C4#188314yes—
444C2—22no—
445C2—22no—
446C2—84no—
447C2#1843712yes—
448C2#1844112yes—
449C2#2093311yescharacteristic
450C2C2×#188311yescharacteristic
451C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21841C2semidirect
5. #14274C2semidirect
6. #9045C2semidirect
7. #8013C2semidirect
8. #7914C2semidirect
12. #3227C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
73. #481C2×C4semidirect
78. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
#343982
#218412
C2×#15052
#142742
#90452
#80132
#79142