Finite Group Databasebeta

← #196296

Subgroups — #196296

Order 512.

1097 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#35248C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4C2×#1479C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic
6#9104C214yesmaximal, complemented
7#9100C214yesmaximal, complemented
8#7858C211yesmaximal, characteristic
9#7857C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2986—22no—
13#2985—22no—
14#2984—22no—
15#2983C2212yes—
16#2982C2212yes—
17#2962—22no—
18#2960—22no—
19#2924—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2855—22no—
25#2634—22no—
26#2581—22no—
27#2568C412yes—
28#2546C412yes—
29#2503C412yes—
30#2484—22no—
31#2434C412yes—
32#2327C2214yes—
33#2283C2211yescharacteristic
34#1491—22no—
35#1487—22no—
36#1479—22no—
37#1479C2212yes—
38#1479C2212yes—
39#1479C2214yes—
40C2×Q8⋊C8—22no—
41#1472—22no—
42#1470—22no—
43#1461C2214yes—
44#1274C2212yes—
45#1274C2214yes—
46#1274C2212yes—
47C23×Q8C2×C411yescharacteristic
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125D412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×C22⋊Q8D412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8D412yes—
74C2×C4×Q8C2312yes—
75C2×C4×Q8C2312yes—
76C2×#117D412yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#446—24no—
82#446—28no—
83#446—24no—
84#446—24no—
85C8×Q8—24no—
86C8×Q8—24no—
87C8×Q8—24no—
88#433—28no—
89#432—21no—
90#432—21no—
91#432—24no—
92#432—21no—
93#432—21no—
94#431—21no—
95#431—21no—
96#431—24no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431D412yes—
102#431D412yes—
103#431C2312yes—
104#431C2312yes—
105#431C2311yescharacteristic
106#431C2311yescharacteristic
107#431C2312yes—
108#424—24no—
109#423—21no—
110#423—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113#407—21no—
114#407—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117#405—24no—
118C4×M16—21no—
119C4×M16D412yes—
120C2×C4×C8—21no—
121C2×C4×C8D412yes—
122#400—24no—
123#400—24no—
124#398—24no—
125#395—24no—
126#391—24no—
127#391C2×C414yes—
128#391C2×C414yes—
129#389—24no—
130#387—28no—
131#387—28no—
132#385—24no—
133#383—21no—
134#382—21no—
135#382C2×C414yes—
136#381—21no—
137#381—24no—
138#380—21no—
139C4×C4⋊C4C2×C412yes—
140C4×C22⋊C4C2×C412yes—
141#376C2312yes—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144Q8⋊C8—28no—
145#324—28no—
146C22×Q8—21no—
147C22×Q8—24no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—42no—
153C22×Q8C22⋊C412yes—
154C22×Q8C22×C412yes—
155C22×Q8C22×C412yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—22no—
162C2×M16—41no—
163C22×C8—22no—
164C22×C8—41no—
165C4⋊Q8—28no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—44no—
175#125—22no—
176#125—22no—
177#125—22no—
178#125—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4C22⋊C414yes—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42C22×C414yes—
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D414yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257C4⋊C8—28no—
258C4⋊C8—28no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—41no—
276C22⋊C8—41no—
277#97—28no—
278#97—22no—
279#97—41no—
280C4×C8—28no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—41no—
285C4×C8C2×D412yes—
286C4×C8C2×D412yes—
287C4×C8C2×D414yes—
288#95—22no—
289#95—22no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—22no—
294#95—22no—
295C24C2×C22⋊C411yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8#13114yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—44no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×C22⋊C414yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C414yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12311yescharacteristic
347C22×C4#12311yescharacteristic
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351M16—44no—
352M16—82no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—44no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—82no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—416no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—44no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42C4×D411yescharacteristic
396C42C4×D411yescharacteristic
397C42#12011yescharacteristic
398C42#12011yescharacteristic
399C42#12111yescharacteristic
400C42#12111yescharacteristic
401C42#12111yescharacteristic
402C42#12111yescharacteristic
403C42#12311yescharacteristic
404C42#12311yescharacteristic
405C42C22×D411yescharacteristic
406C42C22×D411yescharacteristic
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C23—22no—
410C23—22no—
411C23—41no—
412C23#38611yescharacteristic
413C23#38611yescharacteristic
414C23C22×C22⋊C411yescharacteristic
415Q8—24no—
416Q8—24no—
417Q8—44no—
418Q8—44no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—44no—
423Q8—44no—
424Q8—42no—
425Q8—82no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#38611yescharacteristic
462C2×C4#38611yescharacteristic
463C2×C4#38611yescharacteristic
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C2×C4#38611yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×#12011yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×#13112yes—
474C2×C4#56812yes—
475C8—42no—
476C8—42no—
477C8—44no—
478C8—82no—
479C8—82no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—44no—
488C22#191212yes—
489C22C2×#38611yescharacteristic
490C22#288012yes—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—44no—
502C4—44no—
503C4—84no—
504C4—82no—
505C4#190912yes—
506C4#292911yescharacteristic
507C4#293011yescharacteristic
508C4#293312yes—
509C2—22no—
510C2—22no—
511C2—41no—
512C2#2089911yescharacteristic
513C2C2×#190911yescharacteristic
514C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊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
6. #9104C2semidirect
7. #9100C2semidirect

Maximal subgroups

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

subgroupindex
#602502
#352482
#174492
C2×#14792
#173852
#91042
#91002
#78582
#78572