Finite Group Databasebeta

← #197731

Subgroups — #197731

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×Q8×M16C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9224C214yesmaximal, complemented
7#9216C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic
9#7873C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#477—24no—
13C2×#477—21no—
14C2×#477—21no—
15C2×#477C418yescomplemented
16C2×#477C2212yes—
17C2×#477C2212yes—
18Q8×M16—24no—
19Q8×M16—24no—
20C2×#446—22no—
21C2×#446C2212yes—
22C2×C8×Q8C2212yes—
23C2×#431C2211yescharacteristic
24C22×#107—21no—
25C2×#423—21no—
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103—21no—
29C2×C4×M16—21no—
30C2×C4×M16C2211yescharacteristic
31#2285C2211yescharacteristic
32#1993—22no—
33#1990C2214yes—
34#1824—21no—
35#1803C2212yes—
36#1803C2212yes—
37#1787—21no—
38C4×#103—21no—
39C4×#103—21no—
40#1736—21no—
41#1736—21no—
42#1548—24no—
43C2×#333C2214yes—
44#1296C2214yes—
45#1277C2214yes—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C22×M16—21no—
49C22×M16D411yescharacteristic
50C22×M16D411yescharacteristic
51C23×C8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#477—28no—
78#477—28no—
79#477—24no—
80#477—24no—
81#477C2×C418yes—
82#477C2×C418yes—
83#446—216no—
84#446—24no—
85#446—24no—
86C8×Q8—24no—
87C8×Q8—24no—
88#431—22no—
89#431—22no—
90C2×#107—28no—
91C2×#107—22no—
92C2×#107—22no—
93#423—28no—
94#423—22no—
95#423D414yes—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2×C414yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×#103—28no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×C22⋊C8—21no—
111C4×M16—28no—
112C4×M16—22no—
113C4×M16—22no—
114C2×#97—22no—
115C2×#97D412yes—
116C2×C4×C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8C2312yes—
119#384C2312yes—
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#333—28no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—84no—
139C22×Q8C22⋊C414yes—
140C22×Q8C2×D411yescharacteristic
141C22×Q8C2×D411yescharacteristic
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4#4111yescharacteristic
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—24no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—42no—
155C2×M16—42no—
156C2×M16Q1614yes—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—44no—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8C22×C414yes—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—24no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8C22⋊C414yes—
185C4×Q8C22⋊C414yes—
186C4×Q8C22×C412yes—
187C4×Q8C22×C412yes—
188C4×Q8C2×D412yes—
189C4×Q8C2×D412yes—
190#117—42no—
191#117—42no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C22⋊C414yes—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—41no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—41no—
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42#4111yescharacteristic
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#107—28no—
231#107—28no—
232C4⋊C8—28no—
233C4⋊C8—28no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8C22×C414yes—
241C4⋊C8C2×D414yes—
242#103—28no—
243#103—28no—
244#103—44no—
245#103—44no—
246C22⋊C8—42no—
247#97—28no—
248#97—24no—
249C4×C8—28no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—42no—
253#95—41no—
254#95—41no—
255C24#12111yescharacteristic
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—28no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—48no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×Q8C2×C22⋊C412yes—
274C2×Q8C2×C22⋊C412yes—
275C2×Q8#13714yes—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—84no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12012yes—
295C22×C4#12112yes—
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12312yes—
300C22×C4C2×Q1612yes—
301C22×C4#13512yes—
302C22×C4#13712yes—
303C22×C4#13712yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307M16—28no—
308M16—44no—
309M16—48no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—48no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8C2×Q1614yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4C2×C22⋊C412yes—
346C4⋊C4C2×C22⋊C412yes—
347C22⋊C4—82no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—24no—
353C42—24no—
354C42—82no—
355C42C2×C22⋊C412yes—
356C42C4×D412yes—
357C42C4×D412yes—
358C42#12012yes—
359C42#12112yes—
360C42#12312yes—
361C42C23×C411yescharacteristic
362C42C22×D411yescharacteristic
363C42C22×D411yescharacteristic
364C42C2×#4111yescharacteristic
365C23—22no—
366C23—22no—
367C23#38612yes—
368C23#46212yes—
369C23#46412yes—
370C23C2×#12111yescharacteristic
371Q8—22no—
372Q8—24no—
373Q8—22no—
374Q8—22no—
375Q8—48no—
376Q8—42no—
377Q8—84no—
378Q8—88no—
379Q8—84no—
380Q8#41914yes—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—84no—
400C2×C4—88no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#38612yes—
406C2×C4#38612yes—
407C2×C4#38612yes—
408C2×C4C4×Q1614yes—
409C2×C4#44114yes—
410C2×C4#45114yes—
411C2×C4#45214yes—
412C2×C4#46212yes—
413C2×C4#46412yes—
414C2×C4C22×C22⋊C411yescharacteristic
415C2×C4C2×C4×D411yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×#12011yescharacteristic
418C2×C4C2×#12111yescharacteristic
419C2×C4C2×#12311yescharacteristic
420C2×C4C22×Q1611yescharacteristic
421C2×C4C2×#13511yescharacteristic
422C2×C4C2×#13712yes—
423C2×C4C2×#13711yescharacteristic
424C2×C4C2×#13711yescharacteristic
425C2×C4#57512yes—
426C8—24no—
427C8—44no—
428C8—42no—
429C8—44no—
430C8—82no—
431C22—22no—
432C22—24no—
433C22—24no—
434C22—22no—
435C22#187814yes—
436C22C2×#38611yescharacteristic
437C22C2×#46211yescharacteristic
438C22C2×#46411yescharacteristic
439C22#305612yes—
440C22#306812yes—
441C4—21no—
442C4—21no—
443C4—22no—
444C4—22no—
445C4—24no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4—162no—
451C4#187814yes—
452C4C2×#41911yescharacteristic
453C4#289611yescharacteristic
454C4C2×C4×Q1611yescharacteristic
455C4C2×#44111yescharacteristic
456C4C2×#45111yescharacteristic
457C4C2×#45211yescharacteristic
458C4#305612yes—
459C2—24no—
460C2#1917312yes—
461C2#1917612yes—
462C2C2×#187811yescharacteristic
463C2#2097211yescharacteristic
464C2#2100511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9224C2semidirect
7. #9216C2semidirect
15. C2×#477C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×Q8×M162
#213242
#207342
C2×#15482
#92242
#92162
#80452
#78732