Finite Group Databasebeta

← #192997

Subgroups — #192997

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic
3#17817C211yesmaximal, characteristic
4C2×#1487C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8909C214yesmaximal, complemented
7#8907C214yesmaximal, complemented
8#8076C211yesmaximal, characteristic
9#7884C211yesmaximal, 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×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#423C2211yescharacteristic
27#1950C2214yes—
28#1855—22no—
29#1848—21no—
30#1837—21no—
31#1574—22no—
32#1573—22no—
33#1567C2212yes—
34#1566C2212yes—
35#1487—22no—
36#1487C2214yes—
37#1479C412yes—
38#1479C414yes—
39#1479C412yes—
40C2×Q8⋊C8—21no—
41C2×Q8⋊C8—21no—
42C2×Q8⋊C8C2214yes—
43#1301C2212yes—
44#1301C2212yes—
45#1278C2214yes—
46C23×Q8Q811yescharacteristic
47C22×M16—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D412yes—
60C2×C4×Q8Q812yes—
61C2×C4×Q8C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4Q811yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66#446—24no—
67#446—24no—
68#446—22no—
69#446—22no—
70#446—22no—
71#446—22no—
72#446—24no—
73#446—24no—
74C8×Q8—22no—
75C8×Q8—22no—
76C8×Q8—22no—
77C8×Q8—22no—
78#432—24no—
79#432—24no—
80#432—21no—
81#432—21no—
82#432—21no—
83#432D412yes—
84#431—24no—
85#431—24no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431C2×C412yes—
97#431C2×C412yes—
98#431C2×C412yes—
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⋊C8—21no—
117C2×C4⋊C8C2312yes—
118#407—21no—
119#407D412yes—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C4×M16—22no—
124C4×M16—22no—
125C4×M16—21no—
126C4×M16C2312yes—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2312yes—
130#341—24no—
131#335—24no—
132#334—24no—
133Q8⋊C8—24no—
134Q8⋊C8—24no—
135Q8⋊C8C2×C418yes—
136Q8⋊C8C2×C418yes—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—24no—
141C22×Q8—42no—
142C22×Q8C4⋊C411yescharacteristic
143C22×Q8C4⋊C412yes—
144C22×Q8C4⋊C411yescharacteristic
145C22×Q8C2×Q812yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×Q811yescharacteristic
150C23×C4#4111yescharacteristic
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—22no—
154C2×M16—41no—
155C2×M16—82no—
156C2×M16C2×D412yes—
157C2×M16C2×D412yes—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—41no—
161C22×C8—41no—
162C4×Q8—21no—
163C4×Q8—22no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×Q8—22no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—22no—
170C4×Q8—21no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8C4⋊C412yes—
179C4×Q8C4⋊C412yes—
180C4×Q8C4⋊C412yes—
181C4×Q8C4⋊C412yes—
182C4×Q8C22×C412yes—
183C4×Q8C22×C412yes—
184C4×Q8C2×D412yes—
185C4×Q8C2×D412yes—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C4⋊C412yes—
198C2×C4⋊C4C4⋊C411yescharacteristic
199C2×C4⋊C4C4⋊C411yescharacteristic
200C2×C4⋊C4C2×Q812yes—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—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—22no—
227C4⋊C8—22no—
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—22no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8C22×C414yes—
242C4⋊C8C2×D412yes—
243C4⋊C8C2×D412yes—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—41no—
254#97—24no—
255#97—24no—
256#97—22no—
257#97—22no—
258C4×C8—24no—
259C4×C8—24no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—41no—
267C4×C8C22×C414yes—
268C24C22⋊Q811yescharacteristic
269C2×Q8—22no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—44no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8#10814yescomplemented
284C2×Q8C2×C4⋊C412yes—
285C2×Q8C2×C4⋊C412yes—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—44no—
299C22×C4—81no—
300C22×C4—82no—
301C22×C4C2×C4⋊C411yescharacteristic
302C22×C4C2×C4⋊C411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4C22⋊Q811yescharacteristic
310C22×C4C22⋊Q811yescharacteristic
311C22×C4C22⋊Q811yescharacteristic
312C22×C4C22⋊Q811yescharacteristic
313C22×C4C22⋊Q811yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q811yescharacteristic
318C22×C4C2×#4111yescharacteristic
319M16—24no—
320M16—24no—
321M16—42no—
322M16—42no—
323M16—44no—
324M16—82no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—82no—
336C2×C8—82no—
337C2×C8—82no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4C2×C4⋊C412yes—
351C4⋊C4C2×C4⋊C412yes—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42C2×C4⋊C411yescharacteristic
361C42C2×C4⋊C411yescharacteristic
362C42C4×D411yescharacteristic
363C42C4×D411yescharacteristic
364C42C4×D411yescharacteristic
365C42C4×D411yescharacteristic
366C42#12011yescharacteristic
367C42#12011yescharacteristic
368C42C22⋊Q811yescharacteristic
369C42C22⋊Q811yescharacteristic
370C42#12311yescharacteristic
371C42#12311yescharacteristic
372C42C23×C411yescharacteristic
373C42C22×D411yescharacteristic
374C42C22×D411yescharacteristic
375C42C2×#4111yescharacteristic
376C23—22no—
377C23—22no—
378C23—41no—
379C23#38511yescharacteristic
380C23#38511yescharacteristic
381C23C2×C22⋊Q811yescharacteristic
382Q8—24no—
383Q8—22no—
384Q8—22no—
385Q8—22no—
386Q8—44no—
387Q8—44no—
388Q8—42no—
389Q8C2×#10814yes—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
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—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4#38511yescharacteristic
423C2×C4#38511yescharacteristic
424C2×C4#38511yescharacteristic
425C2×C4#38511yescharacteristic
426C2×C4#38511yescharacteristic
427C2×C4#38511yescharacteristic
428C2×C4C2×#10812yes—
429C2×C4#43012yes—
430C2×C4C22×C4⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4#57712yes—
437C2×C4#57812yes—
438C8—44no—
439C8—44no—
440C8—42no—
441C8—42no—
442C8—42no—
443C8—42no—
444C8—82no—
445C8—82no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—44no—
454C22#185512yes—
455C22C2×#38511yescharacteristic
456C22#308212yes—
457C4—21no—
458C4—21no—
459C4—24no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4#185512yes—
470C4C22×#10811yescharacteristic
471C4C2×#43011yescharacteristic
472C4#294712yes—
473C4#301011yescharacteristic
474C4#301111yescharacteristic
475C2—22no—
476C2—22no—
477C2—41no—
478C2C2×#185511yescharacteristic
479C2#2086711yescharacteristic
480C2#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
283. C2×Q8#108semidirect

Maximal subgroups

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

subgroupindex
#602482
#219932
#178172
C2×#14872
C2×#14792
#89092
#89072
#80762
#78842