Finite Group Databasebeta

← #194823

Subgroups — #194823

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2#34380C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1472C211yesmaximal, characteristic
6#9027C214yesmaximal, complemented
7#9026C214yesmaximal, complemented
8#8019C211yesmaximal, characteristic
9#8012C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2502—24no—
15#2502C418yescomplemented
16C2×#398C2214yes—
17#2378—22no—
18#2360—22no—
19#2344—21no—
20#2342—22no—
21Q8×C22⋊C4—21no—
22#2329—22no—
23#2323—22no—
24#2316C2212yes—
25#2313C2212yes—
26C2×#391—21no—
27C2×#385—22no—
28C2×#384—21no—
29#2280C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1824—22no—
33#1823—22no—
34#1823C2212yes—
35#1823C2214yes—
36#1823C2212yes—
37#1736—22no—
38C2×#340—22no—
39#1472—24no—
40C2×#324C2214yes—
41#1294C2212yes—
42#1294C2214yes—
43#1294C2212yes—
44C23×Q8D411yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—21no—
48C23×C8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8C2×C414yes—
56C2×C22⋊Q8D414yes—
57C2×C4×Q8—22no—
58C2×#117—22no—
59C2×#117D412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4C2×C414yes—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D412yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D412yes—
77C22×C42—21no—
78#407—22no—
79#407—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86#398—28no—
87#398C2×C4116yescomplemented
88#391—22no—
89#391—24no—
90#391—28no—
91#389—22no—
92#389—24no—
93#389—24no—
94#388—24no—
95#388—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#385—24no—
100#385—216no—
101#384—28no—
102#381—22no—
103#381—21no—
104#380—22no—
105#380—21no—
106#380—21no—
107#380D412yes—
108#380D412yes—
109#380C2311yescharacteristic
110#380C2312yes—
111#380C2311yescharacteristic
112#379—21no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4C2×C414yes—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95C2312yes—
123#340—28no—
124#340—24no—
125#340—24no—
126#324—28no—
127C25C2×D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—44no—
132C22×Q8C22⋊C414yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—44no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—42no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C2×M16—44no—
156C2×M16—42no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—44no—
160C22×C8—42no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C4×Q8—84no—
170#117—44no—
171#117—84no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—216no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4—84no—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C22×C414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4C22⋊C418yes—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—44no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—42no—
233C22⋊C8—42no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—216no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95C22×C414yes—
246C24—22no—
247C24—22no—
248C24—24no—
249C24—42no—
250C24C2×C22⋊C412yes—
251C24#12011yescharacteristic
252C24#12011yescharacteristic
253C24#12312yes—
254C24C22×D411yescharacteristic
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—48no—
258C2×Q8—44no—
259C2×Q8—82no—
260C2×Q8—84no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—88no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—88no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4C2×C22⋊C414yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C414yes—
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4C4×D412yes—
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4#12312yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322M16—84no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—84no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—816no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C22⋊C4—216no—
348C22⋊C4—28no—
349C22⋊C4—28no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—48no—
357C42—44no—
358C42—84no—
359C42—84no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—21no—
370C23—21no—
371C23—22no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—48no—
376C23—48no—
377C23#38612yes—
378C23#38612yes—
379C23#45614yes—
380C23C22×C22⋊C411yescharacteristic
381C23C2×#12011yescharacteristic
382C23C2×#12311yescharacteristic
383C23#57612yes—
384C23#57712yes—
385Q8—48no—
386Q8—84no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—44no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—88no—
430C2×C4—88no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—164no—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#45814yes—
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C8—82no—
443C8—82no—
444C22—24no—
445C22—24no—
446C22—24no—
447C22—24no—
448C22—22no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—48no—
458C22—48no—
459C22—42no—
460C22—42no—
461C22#188614yes—
462C22C2×#38611yescharacteristic
463C22#289712yes—
464C22#300811yescharacteristic
465C22#301011yescharacteristic
466C22C2×#45612yes—
467C22C2×#45812yes—
468C22#309612yes—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—48no—
475C4—44no—
476C4—44no—
477C4—82no—
478C4—82no—
479C4—84no—
480C4—84no—
481C4#189014yes—
482C2—22no—
483C2—22no—
484C2—24no—
485C2—42no—
486C2#1826812yes—
487C2#1827012yes—
488C2#2091411yescharacteristic
489C2C2×#188611yescharacteristic
490C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9027C2semidirect
7. #9026C2semidirect
15. #2502C4semidirect
87. #398C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#343802
#207002
C2×#18232
C2×#14722
#90272
#90262
#80192
#80122