Finite Group Databasebeta

← #196103

Subgroups — #196103

Order 512.

1049 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#17449C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic
6#9092C218yesmaximal, complemented
7#7922C211yesmaximal, characteristic
8#7864C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3113—22no—
12#3113—22no—
13#3113—22no—
14#3113—22no—
15C2×#483C412yes—
16C2×#475C412yes—
17C2×#462C412yes—
18C2×#451C412yes—
19#2983—22no—
20#2983—22no—
21#2983—22no—
22#2983C2214yes—
23C2×#446—22no—
24C2×#446—22no—
25C2×#441C2214yes—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432—22no—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#1497—22no—
33#1493—22no—
34#1491—22no—
35#1487—22no—
36#1479C2214yes—
37#1479C2214yes—
38#1472—22no—
39#1470—22no—
40#1461C2214yes—
41#1281C2214yes—
42#1275C2214yes—
43C23×Q8C2×C411yescharacteristic
44C22×Q16C2×C414yes—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8D412yes—
67C2×C4×Q8D412yes—
68C2×C4×Q8D412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2312yes—
71C2×C4×Q8C2312yes—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#483—24no—
78#483—24no—
79#475—24no—
80#475—24no—
81#462—24no—
82#462—24no—
83#451—24no—
84#451—24no—
85#446—28no—
86#446—28no—
87#446—28no—
88#446—28no—
89#441—28no—
90#441—28no—
91#432—28no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432—28no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431D412yes—
108#431D412yes—
109#431C2312yes—
110#431C2312yes—
111#431C2312yes—
112#431C2312yes—
113C2×#106C2×C414yes—
114#423—21no—
115#423—21no—
116C2×C4⋊C8—24no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2×C412yes—
120C2×#103C2×C414yes—
121C2×#103C2×C414yes—
122#407—21no—
123#407—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8C2×C412yes—
126C2×C22⋊C8D412yes—
127C4×M16—21no—
128C4×M16—21no—
129C2×#97—24no—
130C2×#97C2312yes—
131C2×C4×C8—21no—
132C2×C4×C8D412yes—
133Q8⋊C8—28no—
134Q8⋊C8—28no—
135#324—28no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—44no—
141C22×Q8C22⋊C412yes—
142C22×Q8C22⋊C412yes—
143C22×Q8C22×C412yes—
144C22×Q8C22×C412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4111yescharacteristic
150C2×Q16—28no—
151C2×Q16—48no—
152C2×M16—41no—
153C2×M16—41no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C22×C414yes—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165#125—41no—
166#125—41no—
167#125—41no—
168#125—41no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8C22⋊C414yes—
198C4×Q8C22⋊C414yes—
199C4×Q8C2×D414yes—
200C4×Q8C2×D414yes—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243#106—28no—
244C4⋊C8—216no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255#103—28no—
256#103—28no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8C2×D414yes—
267#97—216no—
268#97—22no—
269#97—22no—
270#97—41no—
271#97—41no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8C2×D414yes—
278C24C4×D411yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—28no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8C2×C22⋊C414yes—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—82no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327Q16—416no—
328M16—82no—
329M16—82no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—28no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4C2×C22⋊C414yes—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—42no—
365C42—42no—
366C42C2×C22⋊C411yescharacteristic
367C42C2×C22⋊C411yescharacteristic
368C42C4×D411yescharacteristic
369C42C4×D411yescharacteristic
370C42#12011yescharacteristic
371C42#12011yescharacteristic
372C42#12111yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12311yescharacteristic
377C42#12311yescharacteristic
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C42C22×D411yescharacteristic
381C42C2×#4111yescharacteristic
382C23—22no—
383C23—22no—
384C23—41no—
385C23#38611yescharacteristic
386C23#38611yescharacteristic
387C23C2×C4×D411yescharacteristic
388Q8—24no—
389Q8—24no—
390Q8—44no—
391Q8—44no—
392Q8—44no—
393Q8—44no—
394Q8—44no—
395Q8—82no—
396Q8—82no—
397C2×C4—22no—
398C2×C4—22no—
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—24no—
408C2×C4—24no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38611yescharacteristic
432C2×C4C22×C22⋊C411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C2×#12011yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12311yescharacteristic
438C2×C4#56814yes—
439C8—44no—
440C8—44no—
441C8—44no—
442C8—82no—
443C8—82no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—44no—
452C22#190212yes—
453C22C2×#38611yescharacteristic
454C22#293012yes—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—44no—
466C4—82no—
467C4—82no—
468C4#190512yes—
469C4#287912yes—
470C4#293012yes—
471C2—22no—
472C2—22no—
473C2—41no—
474C2#2089611yescharacteristic
475C2C2×#190211yescharacteristic
476C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9092C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#29832
#174792
#174492
#173852
#90922
#79222
#78642