Finite Group Databasebeta

← #192975

Subgroups — #192975

Order 512.

802 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic
3#17796C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6#8909C214yesmaximal, complemented
7#8907C214yesmaximal, complemented
8#8076C211yesmaximal, characteristic
9#7883C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2986—21no—
13#2986—21no—
14#2985—22no—
15#2985—22no—
16Q8×M16—21no—
17Q8×M16—21no—
18Q8×M16C2212yes—
19Q8×M16C2212yes—
20#2921—22no—
21#2920C2211yescharacteristic
22#2920C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#1950C2214yes—
28#1855—22no—
29#1848—21no—
30#1837—21no—
31#1569—22no—
32#1567C2212yes—
33#1566C2212yes—
34#1565—22no—
35#1493—21no—
36#1493—21no—
37#1487C2214yes—
38#1485C411yescharacteristic
39#1485C411yescharacteristic
40#1485C411yescharacteristic
41#1485C411yescharacteristic
42#1485C414yes—
43#1481—22no—
44C2×Q8⋊C8C2214yes—
45#1301C2212yes—
46#1301C2212yes—
47#1278C2214yes—
48C22×M16—21no—
49C22×M16—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533C2×C411yescharacteristic
55#533C2×C411yescharacteristic
56#533C2×C412yes—
57#533D412yes—
58#533D411yescharacteristic
59#533D411yescharacteristic
60C2×C4⋊Q8Q811yescharacteristic
61C2×C4⋊Q8Q811yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8Q812yes—
65C2×C4×Q8—21no—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C22×C42C2311yescharacteristic
69#446—22no—
70#446—22no—
71#446—24no—
72#446—24no—
73#446—22no—
74#446—22no—
75#446—24no—
76#446—24no—
77C8×Q8—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81#433—24no—
82#433—24no—
83#433—24no—
84#433—24no—
85#432—21no—
86#432—21no—
87#432—21no—
88#432D412yes—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431C2×C411yescharacteristic
98#431C2×C411yescharacteristic
99#431C2×C412yes—
100#431D412yes—
101#431C2312yes—
102C2×#108—24no—
103#423—22no—
104#423—22no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423D411yescharacteristic
109#423D411yescharacteristic
110#423C2312yes—
111#423C2311yescharacteristic
112#423C2311yescharacteristic
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8C2312yes—
117#407—21no—
118#407D412yes—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C4×M16—22no—
122C4×M16—22no—
123C4×M16—21no—
124C4×M16—21no—
125C4×M16—21no—
126C4×M16C2×C411yescharacteristic
127C4×M16C2×C412yes—
128C4×M16C2×C411yescharacteristic
129C4×M16C2312yes—
130C2×#97—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8C2312yes—
134#341—24no—
135#335—24no—
136#334—24no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139Q8⋊C8C2×C414yes—
140Q8⋊C8C2×C414yes—
141Q8⋊C8C2×C414yes—
142Q8⋊C8C2×C414yes—
143C22×Q8—21no—
144C22×Q8C2×Q812yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×Q811yescharacteristic
147C23×C4#4111yescharacteristic
148C2×M16—21no—
149C2×M16—22no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—22no—
154C2×M16—44no—
155C2×M16—41no—
156C2×M16—82no—
157C2×M16C2×D412yes—
158C2×M16C2×D412yes—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—41no—
162C22×C8—41no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C4⋊C412yes—
168C4⋊Q8C4⋊C411yescharacteristic
169C4⋊Q8C4⋊C411yescharacteristic
170C4⋊Q8C4⋊C411yescharacteristic
171C4⋊Q8C4⋊C411yescharacteristic
172C4⋊Q8C4⋊C412yes—
173#125—22no—
174#125—21no—
175#125—21no—
176#125—21no—
177#125—21no—
178#125—22no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—22no—
187C22⋊Q8C4⋊C412yes—
188C22⋊Q8C4⋊C412yes—
189C22⋊Q8C4⋊C412yes—
190C22⋊Q8C4⋊C412yes—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—22no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8C22×C412yes—
200C4×Q8C22×C412yes—
201C4×Q8C2×D412yes—
202C4×Q8C2×D412yes—
203#117—22no—
204#117—21no—
205#117—21no—
206#117—22no—
207#117—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4C2×Q812yes—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×Q811yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#108—84no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249C4⋊C8C2×D412yes—
250C4⋊C8C2×D412yes—
251C22⋊C8—22no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—41no—
260#97—24no—
261#97—24no—
262#97—21no—
263#97—22no—
264#97—21no—
265#97—22no—
266#97—22no—
267C4×C8—24no—
268C4×C8—24no—
269C4×C8—22no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—41no—
277C4×C8C22×C412yes—
278C4×C8C22×C412yes—
279C24C22⋊Q811yescharacteristic
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—44no—
285C2×Q8—82no—
286C2×Q8C2×C4⋊C412yes—
287C2×Q8C2×C4⋊C412yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4C22⋊Q811yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×Q811yescharacteristic
311C22×C4C2×#4111yescharacteristic
312M16—24no—
313M16—24no—
314M16—24no—
315M16—44no—
316M16—42no—
317M16—42no—
318M16—42no—
319M16—44no—
320M16—82no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—44no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—82no—
335C2×C8—82no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4C2×C4⋊C412yes—
355C4⋊C4C2×C4⋊C412yes—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42C2×C4⋊C411yescharacteristic
365C42C2×C4⋊C411yescharacteristic
366C42C4×D411yescharacteristic
367C42C4×D411yescharacteristic
368C42C4×D411yescharacteristic
369C42C4×D411yescharacteristic
370C42#12011yescharacteristic
371C42#12011yescharacteristic
372C42C22⋊Q811yescharacteristic
373C42C22⋊Q811yescharacteristic
374C42#12311yescharacteristic
375C42#12311yescharacteristic
376C42C23×C411yescharacteristic
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C42C2×#4111yescharacteristic
380C23—22no—
381C23—41no—
382C23—41no—
383C23#38511yescharacteristic
384C23#38511yescharacteristic
385C23C2×C22⋊Q811yescharacteristic
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—42no—
390Q8—82no—
391Q8—82no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4#38511yescharacteristic
418C2×C4#38511yescharacteristic
419C2×C4#38511yescharacteristic
420C2×C4#38511yescharacteristic
421C2×C4#38511yescharacteristic
422C2×C4#38511yescharacteristic
423C2×C4C22×C4⋊C411yescharacteristic
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×#12011yescharacteristic
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×#12311yescharacteristic
429C2×C4#57712yes—
430C2×C4#57812yes—
431C8—24no—
432C8—42no—
433C8—42no—
434C8—42no—
435C8—42no—
436C8—42no—
437C8—42no—
438C8—42no—
439C8—82no—
440C8—82no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—82no—
450C22C2×#38511yescharacteristic
451C22#308212yes—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4#186112yes—
465C4#291612yes—
466C4#294712yes—
467C4#301011yescharacteristic
468C4#301111yescharacteristic
469C2—22no—
470C2—41no—
471C2—41no—
472C2#2085211yescharacteristic
473C2C2×#186111yescharacteristic
474C2#2087711yescharacteristic
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. #8909C2semidirect
7. #8907C2semidirect

Maximal subgroups

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

subgroupindex
#602492
#219932
#177962
#174862
C2×#14852
#89092
#89072
#80762
#78832