Finite Group Databasebeta

← #198356

Subgroups — #198356

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#60232C211yesmaximal, characteristic
3#21829C211yesmaximal, characteristic, complemented
4#17746C211yesmaximal, characteristic, complemented
5#17490C211yesmaximal, characteristic
6#9247C218yesmaximal, complemented
7#8000C211yesmaximal, characteristic, complemented
8#7936C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3206—22no—
12#3206—22no—
13#3206—22no—
14#3206—22no—
15C2×#492C412yes—
16C2×#487C412yes—
17C2×#461C2214yes—
18C2×#448C414yes—
19#2981—22no—
20#2981—22no—
21#2980—22no—
22#2979—22no—
23D4×M16—22no—
24D4×M16C2214yes—
25#2920C2211yescharacteristic
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#431C2211yescharacteristic
30#2879—22no—
31#1997—22no—
32#1997C2214yes—
33#1821—22no—
34#1558—22no—
35#1548—22no—
36#1548C2214yes—
37#1496—22no—
38#1486C2214yes—
39#1484—22no—
40#1292C2214yes—
41#1283C2214yes—
42C22×#41—21no—
43C23×Q8C2×C412yes—
44C22×SD16C2×C418yes—
45C22×M16—24no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532D412yes—
51#532D412yes—
52#532D412yes—
53#532D412yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
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⋊Q8C2×C412yes—
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2312yes—
67C2×#127C2×C411yescharacteristic
68C2×#124—21no—
69C2×#121—21no—
70C2×#121C2×C412yes—
71C2×C4×D4—21no—
72C2×C4×D4C2312yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#492—24no—
77#492—24no—
78#487—24no—
79#487—24no—
80#461—28no—
81#461—28no—
82#448—28no—
83#448—28no—
84#436—24no—
85#436—24no—
86#435—28no—
87#435—28no—
88C8×D4—24no—
89C8×D4—24no—
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432D412yes—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431D412yes—
109#431C2312yes—
110#423—21no—
111#423D412yes—
112#423C2312yes—
113#423C2312yes—
114C2×C4⋊C8C2312yes—
115C2×#103C2×C418yes—
116#408—28no—
117#408—28no—
118#407—21no—
119#407—24no—
120#407—21no—
121#407—21no—
122#407D412yes—
123C2×C22⋊C8C2×C412yes—
124C4×M16—21no—
125C4×M16—21no—
126C4×M16—21no—
127C4×M16C2312yes—
128C2×C4×C8C2×C412yes—
129#343—28no—
130#333—28no—
131D4⋊C8—28no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—84no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—84no—
144C22×Q8C22×C414yes—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C2×SD16—216no—
155C2×SD16—88no—
156C2×M16—22no—
157C2×M16—28no—
158C2×M16—22no—
159C2×M16—41no—
160C2×M16—88no—
161C2×M16C2×D414yes—
162C22×C8—28no—
163C22×C8C22×C414yes—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8C22⋊C412yes—
183C4⋊Q8C22⋊C412yes—
184#127—21no—
185#127—21no—
186#127C22⋊C412yes—
187#127C22⋊C412yes—
188#124—22no—
189#124—22no—
190#124—22no—
191#124—22no—
192#121—22no—
193#121—22no—
194#121—22no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121C22⋊C414yes—
199#121C22⋊C414yes—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4C2×D414yes—
207C4×D4C2×D414yes—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C22×C412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42C2411yescharacteristic
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8C2×D414yes—
246#103—216no—
247C22⋊C8—28no—
248C22⋊C8—28no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—41no—
257C22⋊C8—41no—
258C22⋊C8—41no—
259#97—22no—
260#97—22no—
261#97—41no—
262#97—41no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—22no—
268C4×C8—41no—
269C24—22no—
270C24C4×D411yescharacteristic
271#41—82no—
272#41—84no—
273#41—82no—
274#41—84no—
275#41—84no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—48no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—82no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—44no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—84no—
300C2×D4C2×C22⋊C414yes—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—48no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—84no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12311yescharacteristic
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330SD16—816no—
331M16—28no—
332M16—44no—
333M16—44no—
334M16—88no—
335M16—82no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—44no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—88no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4C2×C22⋊C414yes—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C42C2×C22⋊C411yescharacteristic
360C42C2×C22⋊C411yescharacteristic
361C42C4×D411yescharacteristic
362C42C4×D411yescharacteristic
363C42#12011yescharacteristic
364C42#12011yescharacteristic
365C42#12111yescharacteristic
366C42#12111yescharacteristic
367C42#12111yescharacteristic
368C42#12111yescharacteristic
369C42#12311yescharacteristic
370C42#12311yescharacteristic
371C42C22×D411yescharacteristic
372C42C22×D411yescharacteristic
373C42C22×D411yescharacteristic
374C42C2×#4111yescharacteristic
375C23—22no—
376C23—24no—
377C23—41no—
378C23—41no—
379C23—44no—
380C23—82no—
381C23—82no—
382C23#38611yescharacteristic
383C23#38611yescharacteristic
384C23C2×C4×D411yescharacteristic
385Q8—48no—
386Q8—82no—
387Q8—82no—
388Q8—84no—
389Q8—84no—
390Q8—84no—
391D4—28no—
392D4—44no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4—82no—
397D4—84no—
398D4—84no—
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—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4C22×C22⋊C411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4#57714yes—
438C8—44no—
439C8—44no—
440C8—44no—
441C8—82no—
442C8—82no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—164no—
456C22C2×#38611yescharacteristic
457C22#294912yes—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—44no—
467C4—84no—
468C4#190212yes—
469C4#288812yes—
470C4#300912yes—
471C4#304212yes—
472C2—22no—
473C2—41no—
474C2—41no—
475C2—84no—
476C2#2093311yescharacteristic
477C2#2099011yescharacteristic
478C2C2×#190211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ 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
3. #21829C2semidirect
4. #17746C2semidirect
6. #9247C2semidirect
7. #8000C2semidirect
8. #7936C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
#602322
#218292
#177462
#174902
#92472
#80002
#79362