Finite Group Databasebeta

← #200545

Subgroups — #200545

Order 512.

1089 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic
3#21879C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#9404C214yesmaximal, complemented
7#9396C214yesmaximal, complemented
8#8097C211yesmaximal, characteristic
9#8092C211yesmaximal, 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×#423C2211yescharacteristic
24#2862—21no—
25#2862—21no—
26#2642C414yes—
27#2568—22no—
28#2568C2214yes—
29#2503—22no—
30#2503C414yes—
31#2310—22no—
32C4×#117C2211yescharacteristic
33#1997—21no—
34#1966—21no—
35#1901—22no—
36#1801C2214yes—
37#1556—22no—
38#1549—22no—
39#1549C2212yes—
40#1549C2212yes—
41#1549C2214yes—
42#1546—22no—
43C2×#332—22no—
44#1303C2212yes—
45#1303C2212yes—
46#1303C2214yes—
47C23×Q8—21no—
48C22×M16—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8C2×C412yes—
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8C2312yes—
71C2×#125C2×C412yes—
72C2×C22⋊Q8C2×C412yes—
73C2×C22⋊Q8D412yes—
74C2×C4×Q8D412yes—
75C2×C4×Q8C2312yes—
76C2×#117—21no—
77C2×#117D412yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2×C412yes—
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—24no—
91#432—21no—
92#432—21no—
93#432D412yes—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—24no—
98#431—21no—
99#431—21no—
100#431D412yes—
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103#431C2312yes—
104#424—24no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423D412yes—
109#423C2312yes—
110#423C2312yes—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8D412yes—
114C2×#101—24no—
115#407—21no—
116#407—21no—
117C2×C22⋊C8—21no—
118#405—24no—
119C4×M16—21no—
120C4×M16—21no—
121C2×C4×C8—21no—
122#400—28no—
123#400C2×C4116yes—
124#391—24no—
125#391—24no—
126#391—24no—
127#391—24no—
128#391—28no—
129#382—24no—
130#382—24no—
131#381—28no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—21no—
135C4×C22⋊C4C2×C412yes—
136#376C2×C412yes—
137C43—21no—
138#343—24no—
139#332—24no—
140#332—24no—
141#332—28no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—48no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8C22⋊C412yes—
150C22×Q8C2×D412yes—
151C23×C4—22no—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—41no—
159C2×M16—82no—
160C22×C8—22no—
161C22×C8—41no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
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—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186#125—22no—
187#125—22no—
188#125—22no—
189#125—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—44no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D414yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42C2411yescharacteristic
248C4⋊C8—28no—
249C4⋊C8—28no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8C2×D412yes—
259C4⋊C8C2×D412yes—
260C4⋊C8C2×D414yes—
261#101—84no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269#97—28no—
270#97—22no—
271#97—41no—
272C4×C8—28no—
273C4×C8—22no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—22no—
277C4×C8—41no—
278#95—22no—
279#95—216no—
280#95—216no—
281#95C22×C414yes—
282C24C2×C22⋊C411yescharacteristic
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—44no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C22⋊C414yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C414yes—
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329M16—44no—
330M16—42no—
331M16—44no—
332M16—42no—
333M16—82no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—22no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—82no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—44no—
365C42—24no—
366C42—24no—
367C42—44no—
368C42—44no—
369C42—42no—
370C42—42no—
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42#12011yescharacteristic
376C42#12011yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12311yescharacteristic
382C42#12311yescharacteristic
383C42C22×D411yescharacteristic
384C42C22×D411yescharacteristic
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C23—22no—
388C23—22no—
389C23—41no—
390C23#38611yescharacteristic
391C23#38611yescharacteristic
392C23C22×C22⋊C411yescharacteristic
393Q8—44no—
394Q8—42no—
395Q8—42no—
396Q8—44no—
397Q8—82no—
398Q8—84no—
399Q8—84no—
400Q8—84no—
401Q8—84no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—48no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4#45414yes—
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C2×#12011yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4#57712yes—
445C2×C4#57812yes—
446C8—42no—
447C8—42no—
448C8—44no—
449C8—82no—
450C8—82no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—44no—
459C22#188812yes—
460C22C2×#38611yescharacteristic
461C22#289812yes—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—44no—
471C4—84no—
472C4—84no—
473C4#188812yes—
474C4#191512yes—
475C4C2×#45412yes—
476C4#305711yescharacteristic
477C4#305911yescharacteristic
478C4#306312yes—
479C2—22no—
480C2—22no—
481C2—41no—
482C2C2×#188811yescharacteristic
483C2#2097411yescharacteristic
484C2C2×#191511yescharacteristic
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. #9404C2semidirect
7. #9396C2semidirect

Maximal subgroups

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

subgroupindex
#602502
#353202
#218792
#177422
C2×#15492
#94042
#93962
#80972
#80922