Finite Group Databasebeta

← #193363

Subgroups — #193363

Order 512.

823 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17718C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8933C214yesmaximal, complemented
7#8931C214yesmaximal, complemented
8#7885C211yesmaximal, characteristic
9#7872C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2986—21no—
12#2986—21no—
13#2985—22no—
14#2984—22no—
15#2983—21no—
16#2982—21no—
17#2962C2212yes—
18#2962C2212yes—
19#2960—21no—
20#2960—21no—
21#2924—22no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#1950C2214yes—
29#1855—22no—
30#1848—21no—
31#1837—21no—
32#1574—21no—
33#1574—21no—
34#1571C2212yes—
35#1570C2212yes—
36#1565—21no—
37#1565—21no—
38#1554C414yes—
39#1548C411yescharacteristic
40#1548C411yescharacteristic
41#1546C411yescharacteristic
42#1546C411yescharacteristic
43#1542C2214yes—
44#1493—22no—
45#1493C2214yes—
46C2×Q8⋊C8—22no—
47#1278C2214yes—
48#1277C2212yes—
49#1277C2212yes—
50C22×M16—21no—
51C23×C8—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yes—
57#533D412yes—
58#533Q812yes—
59#533C2312yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8Q812yes—
65C2×C4×Q8—21no—
66C2×C4×Q8D412yes—
67C2×#117—21no—
68C22×C42C2311yescharacteristic
69#446—22no—
70#446—22no—
71#446—22no—
72#446—22no—
73#446—24no—
74#446—24no—
75#446—22no—
76#446—22no—
77C8×Q8—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81C8×Q8—22no—
82C8×Q8—22no—
83#433—24no—
84#433—24no—
85#432—24no—
86#432—21no—
87#432—21no—
88#432—21no—
89#432C2×C412yes—
90#431—24no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431C2×C411yescharacteristic
99#431C2×C411yescharacteristic
100#431C2×C412yes—
101#431D411yescharacteristic
102#431D411yescharacteristic
103#431D411yescharacteristic
104#431D411yescharacteristic
105#431C2312yes—
106C2×#108—24no—
107#424—22no—
108#424—22no—
109#423—21no—
110#423—21no—
111#423—21no—
112#423C2×C411yescharacteristic
113#423C2×C411yescharacteristic
114#423D411yescharacteristic
115#423D411yescharacteristic
116#423C2312yes—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122#407—21no—
123#407D412yes—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126#405—22no—
127#405—22no—
128C4×M16—21no—
129C4×M16C2312yes—
130C2×C4×C8—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×C8C2312yes—
138#341—24no—
139#335—22no—
140#335—22no—
141#334—22no—
142#334—22no—
143#333C2×C414yes—
144#333C2×C414yes—
145#332C2×C414yes—
146#332C2×C414yes—
147Q8⋊C8—28no—
148C22×Q8—21no—
149C22×Q8C4⋊C412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×Q811yescharacteristic
152C23×C4#4111yescharacteristic
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—41no—
157C2×M16—82no—
158C2×M16C2×D412yes—
159C2×M16C2×D412yes—
160C22×C8—24no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—42no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C4⋊C412yes—
174C4⋊Q8C4⋊C412yes—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C22×C412yes—
177#125—22no—
178#125—22no—
179#125—22no—
180#125—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8C4⋊C414yes—
188C22⋊Q8C2×Q814yes—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8C4⋊C414yes—
196C4×Q8C2×D414yes—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4C4⋊C412yes—
207C2×C22⋊C4—22no—
208C2×C42—22no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q811yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#108—84no—
222C4⋊C8—24no—
223C4⋊C8—24no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8C22×C412yes—
240C4⋊C8C22×C412yes—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243C4⋊C8C2×D411yescharacteristic
244C4⋊C8C2×D411yescharacteristic
245C4⋊C8C2×D411yescharacteristic
246C4⋊C8C2×D411yescharacteristic
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258#97—24no—
259#97—24no—
260#97—22no—
261#97—22no—
262C4×C8—24no—
263C4×C8—24no—
264C4×C8—22no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—41no—
276C4×C8—41no—
277C24C22⋊Q811yescharacteristic
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—44no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8C2×C4⋊C414yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×Q811yescharacteristic
309C22×C4C2×#4111yescharacteristic
310M16—24no—
311M16—24no—
312M16—42no—
313M16—42no—
314M16—44no—
315M16—82no—
316C2×C8—24no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—82no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4C2×C4⋊C414yes—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C42—24no—
352C42—24no—
353C42C2×C4⋊C411yescharacteristic
354C42C2×C4⋊C411yescharacteristic
355C42C4×D411yescharacteristic
356C42C4×D411yescharacteristic
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42#12011yescharacteristic
360C42#12011yescharacteristic
361C42C22⋊Q811yescharacteristic
362C42C22⋊Q811yescharacteristic
363C42#12311yescharacteristic
364C42#12311yescharacteristic
365C42C23×C411yescharacteristic
366C42C22×D411yescharacteristic
367C42C22×D411yescharacteristic
368C42C2×#4111yescharacteristic
369C23—22no—
370C23—22no—
371C23—41no—
372C23#38511yescharacteristic
373C23#38511yescharacteristic
374C23C2×C22⋊Q811yescharacteristic
375Q8—44no—
376Q8—44no—
377Q8—44no—
378Q8—42no—
379Q8—42no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—44no—
403C2×C4—82no—
404C2×C4#38511yescharacteristic
405C2×C4#38511yescharacteristic
406C2×C4#38511yescharacteristic
407C2×C4#38511yescharacteristic
408C2×C4#38511yescharacteristic
409C2×C4#38511yescharacteristic
410C2×C4#44312yes—
411C2×C4#44412yes—
412C2×C4C22×C4⋊C411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×#12011yescharacteristic
416C2×C4C2×C22⋊Q811yescharacteristic
417C2×C4C2×#12311yescharacteristic
418C2×C4#57712yes—
419C2×C4#57812yes—
420C8—44no—
421C8—42no—
422C8—42no—
423C8—42no—
424C8—42no—
425C8—41no—
426C8—41no—
427C8—42no—
428C8—41no—
429C8—41no—
430C8—42no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—44no—
440C22#186612yes—
441C22C2×#38511yescharacteristic
442C22#308212yes—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—44no—
451C4—44no—
452C4#185812yes—
453C4#186612yes—
454C4#294712yes—
455C4C2×#44311yescharacteristic
456C4C2×#44411yescharacteristic
457C4#301011yescharacteristic
458C4#301111yescharacteristic
459C2—22no—
460C2—22no—
461C2—41no—
462C2C2×#185811yescharacteristic
463C2#2087711yescharacteristic
464C2C2×#186611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ 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. #8933C2semidirect
7. #8931C2semidirect

Maximal subgroups

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

subgroupindex
#602502
#219932
#178092
#177182
C2×#14932
#89332
#89312
#78852
#78722