Finite Group Databasebeta

← #196974

Subgroups — #196974

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35311C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3#20685C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5C2×#1508C211yesmaximal, characteristic
6#9152C214yesmaximal, complemented
7#9145C214yesmaximal, complemented
8#8016C211yesmaximal, characteristic
9#8012C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2855C2214yes—
15#2625C414yescomplemented
16#2503—21no—
17#2502—21no—
18#2499—22no—
19#2484—22no—
20#2483—22no—
21#2346—24no—
22#2346—24no—
23#2317C412yescomplemented
24#2310C412yescomplemented
25C2×#389—21no—
26C2×#388—22no—
27C2×#382C2212yes—
28#2280C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×#376C2212yes—
33#1818—22no—
34#1817C2214yes—
35#1729—22no—
36#1520—24no—
37#1508—24no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×Q8C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—21no—
47C23×C8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2312yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117C2×C412yes—
61C2×#117C2×C412yes—
62C2×#117C2×C412yes—
63C2×#117D412yes—
64C2×#117D412yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42—22no—
76C22×C42—21no—
77C22×C42C2312yes—
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#400—24no—
87#398—24no—
88#397—28no—
89#391—22no—
90#391—22no—
91#391C2×C418yes—
92#389—28no—
93#389C2×C418yes—
94#388—216no—
95#387—28no—
96#385—24no—
97#385—24no—
98#384—22no—
99#382—22no—
100#382—24no—
101#382—24no—
102#381—24no—
103#381—21no—
104#381—24no—
105#380—22no—
106#380C2×C412yes—
107#380D414yes—
108#380C2312yes—
109#379—21no—
110C4×C4⋊C4—28no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4D414yes—
115#376—24no—
116#376—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2311yescharacteristic
120C2×#95C2312yes—
121C2×#95C2311yescharacteristic
122#336—28no—
123#328—24no—
124#328—24no—
125#328—28no—
126C25C2×D411yescharacteristic
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—44no—
131C22×Q8C22×C412yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—44no—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C2×M16—44no—
151C2×M16—42no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—44no—
155C22×C8—42no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166#117—24no—
167#117—24no—
168#117—24no—
169#117—24no—
170#117—24no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C414yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4C22⋊C414yes—
211C2×C22⋊C4C22⋊C414yes—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—28no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—44no—
220C2×C42—48no—
221C2×C42C22×C414yes—
222C2×C42C2×D414yes—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229#95—28no—
230#95—28no—
231#95—22no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—21no—
236#95—22no—
237#95—21no—
238#95—22no—
239#95C2×D412yes—
240#95C2×D414yes—
241#95C2×D412yes—
242C24—22no—
243C24—22no—
244C24—42no—
245C24—42no—
246C24C2×C22⋊C411yescharacteristic
247C24C2×C22⋊C411yescharacteristic
248C24#12012yes—
249C24#12312yes—
250C24C22×D411yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—44no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—84no—
257C2×Q8—84no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—82no—
294C22×C4C2×C22⋊C414yes—
295C22×C4C2×C22⋊C414yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12012yes—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314M16—84no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—84no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—44no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—88no—
333C4⋊C4—88no—
334C22⋊C4—28no—
335C22⋊C4—28no—
336C22⋊C4—48no—
337C22⋊C4—44no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C22⋊C4—84no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—48no—
350C42—416no—
351C42—44no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—84no—
368C23—84no—
369C23#38612yes—
370C23#38612yes—
371C23C22×C22⋊C411yescharacteristic
372C23C2×#12011yescharacteristic
373C23C2×#12311yescharacteristic
374C23#57712yes—
375C23#57812yes—
376Q8—82no—
377Q8—82no—
378Q8—84no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—84no—
423C2×C4—88no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4#38612yes—
427C2×C4#38612yes—
428C2×C4#41014yes—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C8—82no—
434C8—82no—
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—44no—
448C22—44no—
449C22—82no—
450C22—82no—
451C22—84no—
452C22C2×#38611yescharacteristic
453C22C2×#41012yes—
454C22#289812yes—
455C22#290112yes—
456C22#300911yescharacteristic
457C22#301111yescharacteristic
458C22#309712yes—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—22no—
464C4—44no—
465C4—44no—
466C4—44no—
467C4—82no—
468C4—84no—
469C4—84no—
470C4—82no—
471C4—82no—
472C4#189914yes—
473C2—21no—
474C2—21no—
475C2—22no—
476C2—42no—
477C2—42no—
478C2#1760514yes—
479C2#2091311yescharacteristic
480C2#2098011yescharacteristic
481C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9152C2semidirect
7. #9145C2semidirect
15. #2625C4semidirect
23. #2317C4semidirect
24. #2310C4semidirect

Maximal subgroups

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

subgroupindex
#353112
C2×#23462
#206852
C2×#15202
C2×#15082
#91522
#91452
#80162
#80122