Finite Group Databasebeta

← #192963

Subgroups — #192963

Order 512.

867 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#21994C211yesmaximal, characteristic
3#17796C211yesmaximal, characteristic
4C2×#1491C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8909C214yesmaximal, complemented
7#8907C214yesmaximal, complemented
8#8080C211yesmaximal, characteristic
9#7883C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11#2984—22no—
12#2984—22no—
13#2983—22no—
14#2983—21no—
15#2982—21no—
16Q8×M16—21no—
17Q8×M16—21no—
18Q8×M16C2212yes—
19Q8×M16C2212yes—
20#2920—22no—
21#2920C2211yescharacteristic
22#2920C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#1950C2214yes—
28#1855—22no—
29#1849—21no—
30#1838—21no—
31#1569—22no—
32#1567C2212yes—
33#1566C2212yes—
34#1565—22no—
35#1491C412yes—
36#1491C412yes—
37#1491C412yes—
38#1491C412yes—
39#1487—21no—
40#1487—21no—
41#1487C2214yes—
42C2×Q8⋊C8—22no—
43C2×Q8⋊C8C2214yes—
44#1301C2212yes—
45#1301C2212yes—
46#1278C2214yes—
47C23×Q8Q811yescharacteristic
48C22×M16—21no—
49C22×M16—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D412yes—
62C2×C4×Q8Q812yes—
63C2×C4×Q8C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4Q811yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#446—24no—
69#446—24no—
70#446—22no—
71#446—22no—
72#446—22no—
73#446—22no—
74#446—24no—
75#446—24no—
76C8×Q8—22no—
77C8×Q8—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80#432—24no—
81#432—24no—
82#432—21no—
83#432—21no—
84#432—21no—
85#432—21no—
86#432—21no—
87#432C2×C412yes—
88#432C2×C412yes—
89#432D412yes—
90#431—24no—
91#431—24no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431D412yes—
99#431C2312yes—
100C2×#108—24no—
101#423—22no—
102#423—22no—
103#423—21no—
104#423—21no—
105#423—21no—
106#423D411yescharacteristic
107#423D411yescharacteristic
108#423C2312yes—
109#423C2311yescharacteristic
110#423C2311yescharacteristic
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2312yes—
115#407—21no—
116#407D412yes—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C4×M16—22no—
120C4×M16—22no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16C2×C411yescharacteristic
125C4×M16C2×C412yes—
126C4×M16C2×C411yescharacteristic
127C4×M16C2312yes—
128C2×#97—21no—
129C2×#97—21no—
130C2×C4×C8—21no—
131C2×C4×C8C2312yes—
132#341—24no—
133#335—24no—
134#334—24no—
135Q8⋊C8—24no—
136Q8⋊C8—24no—
137Q8⋊C8C2×C414yes—
138Q8⋊C8C2×C414yes—
139Q8⋊C8C2×C414yes—
140Q8⋊C8C2×C414yes—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—44no—
145C22×Q8C4⋊C411yescharacteristic
146C22×Q8C4⋊C412yes—
147C22×Q8C4⋊C411yescharacteristic
148C22×Q8C2×Q812yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×Q811yescharacteristic
153C23×C4#4111yescharacteristic
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—22no—
160C2×M16—44no—
161C2×M16—41no—
162C2×M16—82no—
163C2×M16C2×D412yes—
164C2×M16C2×D412yes—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—41no—
168C22×C8—41no—
169C4×Q8—21no—
170C4×Q8—22no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—22no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—22no—
177C4×Q8—21no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8C4⋊C412yes—
186C4×Q8C4⋊C412yes—
187C4×Q8C4⋊C412yes—
188C4×Q8C4⋊C412yes—
189C4×Q8C22×C412yes—
190C4×Q8C22×C412yes—
191C4×Q8C2×D412yes—
192C4×Q8C2×D412yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C4⋊C412yes—
205C2×C4⋊C4C4⋊C411yescharacteristic
206C2×C4⋊C4C4⋊C411yescharacteristic
207C2×C4⋊C4C2×Q812yes—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×Q811yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42C2411yescharacteristic
228#108—84no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8C22×C414yes—
247C4⋊C8C2×D412yes—
248C4⋊C8C2×D412yes—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—41no—
257#97—24no—
258#97—24no—
259#97—21no—
260#97—22no—
261#97—21no—
262#97—22no—
263#97—22no—
264#97—22no—
265#97—22no—
266C4×C8—24no—
267C4×C8—24no—
268C4×C8—22no—
269C4×C8—22no—
270C4×C8—22no—
271C4×C8—22no—
272C4×C8—41no—
273C4×C8C22×C412yes—
274C4×C8C22×C412yes—
275C24C22⋊Q811yescharacteristic
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8C2×C4⋊C412yes—
290C2×Q8C2×C4⋊C412yes—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—82no—
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4C22⋊Q811yescharacteristic
314C22×C4C22⋊Q811yescharacteristic
315C22×C4C22⋊Q811yescharacteristic
316C22×C4C22⋊Q811yescharacteristic
317C22×C4C22⋊Q811yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323M16—24no—
324M16—24no—
325M16—24no—
326M16—44no—
327M16—42no—
328M16—42no—
329M16—42no—
330M16—44no—
331M16—82no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—44no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—82no—
345C2×C8—82no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4C2×C4⋊C412yes—
359C4⋊C4C2×C4⋊C412yes—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42C2×C4⋊C411yescharacteristic
369C42C2×C4⋊C411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42#12011yescharacteristic
375C42#12011yescharacteristic
376C42C22⋊Q811yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42#12311yescharacteristic
379C42#12311yescharacteristic
380C42C23×C411yescharacteristic
381C42C22×D411yescharacteristic
382C42C22×D411yescharacteristic
383C42C2×#4111yescharacteristic
384C23—22no—
385C23—22no—
386C23—41no—
387C23#38511yescharacteristic
388C23#38511yescharacteristic
389C23C2×C22⋊Q811yescharacteristic
390Q8—24no—
391Q8—22no—
392Q8—22no—
393Q8—22no—
394Q8—44no—
395Q8—44no—
396Q8—42no—
397Q8#43014yes—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#38511yescharacteristic
430C2×C4#38511yescharacteristic
431C2×C4#38511yescharacteristic
432C2×C4#38511yescharacteristic
433C2×C4#38511yescharacteristic
434C2×C4#38511yescharacteristic
435C2×C4C22×C4⋊C411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×C4×D411yescharacteristic
438C2×C4C2×#12011yescharacteristic
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4#57712yes—
442C2×C4#57812yes—
443C8—24no—
444C8—42no—
445C8—44no—
446C8—42no—
447C8—42no—
448C8—42no—
449C8—42no—
450C8—82no—
451C8—82no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—44no—
460C22#186112yes—
461C22C2×#38511yescharacteristic
462C22#308212yes—
463C4—24no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4C2×#43012yes—
477C4#294712yes—
478C4#301011yescharacteristic
479C4#301111yescharacteristic
480C2—22no—
481C2—22no—
482C2—41no—
483C2#2085111yescharacteristic
484C2C2×#186111yescharacteristic
485C2#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
#602482
#219942
#177962
C2×#14912
C2×#14872
#89092
#89072
#80802
#78832