Finite Group Databasebeta

← #201179

Subgroups — #201179

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#37540C211yesmaximal, characteristic
3#17667C211yesmaximal, characteristic
4C2×#1534C211yesmaximal, characteristic
5#17404C211yesmaximal, characteristic
6#9450C218yesmaximal, complemented
7#7954C211yesmaximal, characteristic
8#7951C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×#125C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983C2214yes—
15#2962—22no—
16#2961—22no—
17#2924—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22#2822—22no—
23#2660C412yes—
24#2642C412yes—
25#2634—22no—
26#2627—22no—
27#2565C2214yes—
28#2503C412yes—
29#2502C412yes—
30#2484—22no—
31#2283C2211yescharacteristic
32#1534—22no—
33#1534C2214yes—
34#1534C2214yes—
35#1528—22no—
36#1528—22no—
37C2×#330—22no—
38#1472—22no—
39#1470—22no—
40#1467C2214yes—
41#1285C2214yes—
42#1285C2214yes—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#125—21no—
60C2×#125—21no—
61C2×#125—21no—
62C2×#125—21no—
63C2×#125C2×C412yes—
64C2×#125D412yes—
65C2×#125C2312yes—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8D412yes—
69C2×C4×Q8C2312yes—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42C2311yescharacteristic
76#446—28no—
77#446—24no—
78#446—28no—
79#446—28no—
80C8×Q8—24no—
81#433—28no—
82#432—21no—
83#432—21no—
84#432—24no—
85#432—21no—
86#432—21no—
87#432—21no—
88#432—21no—
89#432D412yes—
90#432D412yes—
91#432C2312yes—
92#432C2312yes—
93#432C2312yes—
94#432C2312yes—
95#431—21no—
96#431—21no—
97#431—24no—
98#431—21no—
99#431—21no—
100#424—24no—
101#423—21no—
102#423—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105#407—21no—
106#407—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109#405—24no—
110C4×M16—21no—
111C4×M16D412yes—
112C2×#97—21no—
113C2×#97D412yes—
114#400—24no—
115#400—24no—
116#400—24no—
117#398—24no—
118#398—24no—
119#398—28no—
120#395—24no—
121#391C2×C414yes—
122#391C2×C414yes—
123#387—28no—
124#385—24no—
125#384C2×C414yes—
126#383—21no—
127#382—21no—
128#382C2×C414yes—
129#381—21no—
130#381—24no—
131#380—21no—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4C2×C412yes—
134#376C2×C412yes—
135#330—28no—
136#330—28no—
137#324—28no—
138C22×Q8C22⋊C412yes—
139C22×Q8C22×C412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—22no—
146C2×M16—41no—
147C22×C8—22no—
148C22×C8—41no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161#125—22no—
162#125—22no—
163#125—22no—
164#125—22no—
165#125—22no—
166#125—22no—
167#125—22no—
168#125—22no—
169#125—41no—
170#125—41no—
171#125—41no—
172#125—41no—
173#125—41no—
174#125—41no—
175#125—41no—
176#125—41no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C414yes—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C22⋊C414yes—
231C2×C42C22×C414yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D414yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—28no—
246C4⋊C8—28no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263#97—28no—
264#97—22no—
265#97—22no—
266#97—22no—
267#97—41no—
268#97C2×D414yes—
269#97C2×D414yes—
270C4×C8—28no—
271C4×C8—22no—
272C4×C8—41no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—22no—
280C24C2×C22⋊C411yescharacteristic
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—44no—
286C2×Q8—82no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4C2×C22⋊C414yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C414yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319M16—44no—
320M16—82no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—44no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—82no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—82no—
356C4⋊C4—88no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—44no—
360C42—24no—
361C42—24no—
362C42—44no—
363C42C4×D411yescharacteristic
364C42C4×D411yescharacteristic
365C42C4×D411yescharacteristic
366C42C4×D411yescharacteristic
367C42#12011yescharacteristic
368C42#12011yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42#12111yescharacteristic
373C42#12311yescharacteristic
374C42#12311yescharacteristic
375C42C22×D411yescharacteristic
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C42C2×#4111yescharacteristic
379C23—22no—
380C23—22no—
381C23—41no—
382C23#38611yescharacteristic
383C23#38611yescharacteristic
384C23C22×C22⋊C411yescharacteristic
385Q8—44no—
386Q8—82no—
387Q8—82no—
388Q8—82no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×#12011yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C8—44no—
435C8—44no—
436C8—82no—
437C8—82no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22#191212yes—
447C22C2×#38611yescharacteristic
448C22#288912yes—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—44no—
462C4—44no—
463C4—84no—
464C4—82no—
465C4—82no—
466C2—22no—
467C2—22no—
468C2—41no—
469C2#2094711yescharacteristic
470C2#2099111yescharacteristic
471C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #9450C2semidirect

Maximal subgroups

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

subgroupindex
#602512
#375402
#176672
C2×#15342
#174042
#94502
#79542
#79512