Finite Group Databasebeta

← #191801

Subgroups — #191801

Order 512.

863 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#60250C211yesmaximal, characteristic
2#21425C211yesmaximal, characteristic
3#21300C211yesmaximal, characteristic
4#17809C211yesmaximal, characteristic
5#17796C211yesmaximal, characteristic
6#8838C214yesmaximal, complemented
7#8836C214yesmaximal, complemented
8#7885C211yesmaximal, characteristic
9C2×#1278C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2986—21no—
12#2986—21no—
13#2985—22no—
14#2984—22no—
15#2983C2212yes—
16#2982C2212yes—
17#2962—21no—
18#2962—21no—
19#2960—21no—
20#2960—21no—
21#2924—22no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#1997C411yescharacteristic
29#1997C411yescharacteristic
30#1984—22no—
31#1984C2214yes—
32#1966C411yescharacteristic
33#1966C411yescharacteristic
34#1901C414yes—
35C4×#101C2214yes—
36#1729—22no—
37#1574—21no—
38#1574—21no—
39#1571—21no—
40#1570—21no—
41#1569—22no—
42#1567—21no—
43#1566—21no—
44#1565C2212yes—
45#1565C2214yes—
46#1565C2212yes—
47#1278C2212yes—
48#1278C2214yes—
49#1278C2212yes—
50C22×M16—21no—
51C22×M16C2×C412yes—
52C23×C8—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×#125Q812yes—
64C2×C22⋊Q8C2×C412yes—
65C2×C22⋊Q8Q812yes—
66C2×C4×Q8D412yes—
67C2×C4×Q8C2312yes—
68C2×#117D412yes—
69C22×C42C2311yescharacteristic
70#446—22no—
71#446—22no—
72#446—24no—
73#446—24no—
74#446—22no—
75#446—22no—
76#446—22no—
77#446—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81C8×Q8—22no—
82C8×Q8—22no—
83C8×Q8—22no—
84#433—24no—
85#433—24no—
86#432—21no—
87#432—24no—
88#432—21no—
89#432—21no—
90#432D412yes—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—24no—
97#431D411yescharacteristic
98#431D411yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2312yes—
102#424—22no—
103#424—22no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423C2×C411yescharacteristic
109#423C2×C411yescharacteristic
110#423D411yescharacteristic
111#423D411yescharacteristic
112C2×C4⋊C8—21no—
113C2×C4⋊C8D411yescharacteristic
114C2×C4⋊C8D411yescharacteristic
115C2×C4⋊C8C2312yes—
116C2×#101C2×C414yes—
117C2×#101C2×C414yes—
118#407—21no—
119#407C2×C412yes—
120C2×C22⋊C8—21no—
121#405—22no—
122#405—22no—
123C4×M16—21no—
124C4×M16—21no—
125C4×M16—21no—
126C4×M16C2×C411yescharacteristic
127C4×M16C2×C411yescharacteristic
128C4×M16C2312yes—
129C4×M16C2312yes—
130C2×#97—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8C2312yes—
139#343C2×C414yes—
140#343C2×C414yes—
141#336—28no—
142#335—24no—
143#335—22no—
144#335—22no—
145#334—24no—
146#334—22no—
147#334—22no—
148C22×Q8C4⋊C412yes—
149C22×Q8C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q811yescharacteristic
153C2×M16—22no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—24no—
157C2×M16—42no—
158C2×M16—44no—
159C2×M16—41no—
160C2×M16—41no—
161C2×M16C22×C412yes—
162C2×M16C22×C412yes—
163C2×M16C22×C412yes—
164C2×M16C22×C412yes—
165C22×C8—24no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—41no—
173C22×C8—41no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182#125—22no—
183#125—22no—
184#125—22no—
185#125—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊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—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—22no—
206C2×C4⋊C4C4⋊C412yes—
207C2×C4⋊C4C4⋊C412yes—
208C2×C4⋊C4C4⋊C414yes—
209C2×C4⋊C4C4⋊C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C2×Q814yes—
212C2×C22⋊C4C4⋊C414yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D414yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q811yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8C2×D411yescharacteristic
243C4⋊C8C2×D411yescharacteristic
244C4⋊C8C2×D412yes—
245C4⋊C8C2×D411yescharacteristic
246C4⋊C8C2×D411yescharacteristic
247C4⋊C8C2×D412yes—
248#101—28no—
249#101—44no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258#97—24no—
259#97—24no—
260#97—21no—
261#97—21no—
262#97—22no—
263#97—22no—
264#97—22no—
265C4×C8—24no—
266C4×C8—24no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—41no—
279C4×C8—41no—
280C24C2×C4⋊C411yescharacteristic
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—44no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4C2×C4⋊C414yes—
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C2×C4⋊C411yescharacteristic
297C22×C4C2×C4⋊C414yes—
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4C22⋊Q811yescharacteristic
307C22×C4C22⋊Q811yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×Q811yescharacteristic
313M16—22no—
314M16—24no—
315M16—22no—
316M16—22no—
317M16—24no—
318M16—44no—
319M16—42no—
320M16—42no—
321M16—82no—
322M16—82no—
323M16#13114yes—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—44no—
343C2×C8—42no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C42—24no—
359C42—24no—
360C42C4×D411yescharacteristic
361C42C4×D411yescharacteristic
362C42C4×D411yescharacteristic
363C42C4×D411yescharacteristic
364C42#12011yescharacteristic
365C42#12011yescharacteristic
366C42C22⋊Q811yescharacteristic
367C42C22⋊Q811yescharacteristic
368C42C22⋊Q811yescharacteristic
369C42C22⋊Q811yescharacteristic
370C42#12311yescharacteristic
371C42#12311yescharacteristic
372C42C22×D411yescharacteristic
373C42C22×D411yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C23—22no—
377C23—22no—
378C23—41no—
379C23#38511yescharacteristic
380C23#38511yescharacteristic
381C23C22×C4⋊C411yescharacteristic
382Q8—44no—
383Q8—42no—
384Q8—42no—
385Q8—44no—
386Q8—82no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4#38511yescharacteristic
412C2×C4#38511yescharacteristic
413C2×C4#38511yescharacteristic
414C2×C4#38511yescharacteristic
415C2×C4#38511yescharacteristic
416C2×C4#38511yescharacteristic
417C2×C4#45412yes—
418C2×C4#45612yes—
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C2×C4×D411yescharacteristic
421C2×C4C2×#12011yescharacteristic
422C2×C4C2×C22⋊Q811yescharacteristic
423C2×C4C2×C22⋊Q811yescharacteristic
424C2×C4C2×#12311yescharacteristic
425C2×C4C2×#13112yes—
426C2×C4#56812yes—
427C8—24no—
428C8—44no—
429C8—42no—
430C8—42no—
431C8—42no—
432C8—41no—
433C8—41no—
434C8—42no—
435C8—41no—
436C8—41no—
437C8—42no—
438C8—82no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—44no—
447C22#185712yes—
448C22C2×#38511yescharacteristic
449C22#290612yes—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—44no—
458C4—44no—
459C4#185712yes—
460C4#185812yes—
461C4#293111yescharacteristic
462C4#293211yescharacteristic
463C4#293312yes—
464C4C2×#45411yescharacteristic
465C4C2×#45611yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2#2083911yescharacteristic
470C2C2×#185711yescharacteristic
471C2C2×#185811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C2×C4⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8838C2semidirect
7. #8836C2semidirect

Maximal subgroups

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

subgroupindex
#602502
#214252
#213002
#178092
#177962
#88382
#88362
#78852
C2×#12782