Finite Group Databasebeta

← #194038

Subgroups — #194038

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic
4#21400C211yesmaximal, characteristic
5C22×Q8⋊C8C211yesmaximal, characteristic
6#8982C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic
9#7897C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3078—24no—
13#3078C418yescomplemented
14C2×#479—22no—
15C2×#477C2212yes—
16C2×#475C2212yes—
17C2×#428—21no—
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#419—21no—
21C22×C4×C8C2211yescharacteristic
22#2356—22no—
23#2356—22no—
24#2355—22no—
25#2355C2214yes—
26#2313—22no—
27#2310—21no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30#2283—22no—
31#2274—21no—
32#1990C2212yes—
33#1987C2212yes—
34#1981C2212yes—
35#1956C2212yes—
36#1892—22no—
37#1892—22no—
38#1803—22no—
39C4×#103—22no—
40C2×Q8⋊C8—24no—
41C2×Q8⋊C8C2214yes—
42#1296C2214yes—
43#1279C2214yes—
44C23×Q8D411yescharacteristic
45C22×M16—21no—
46C23×C8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C414yes—
64C2×C4×Q8D414yes—
65C2×C4×Q8D412yes—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#479—216no—
75#477—24no—
76#477C2×C418yes—
77#475—24no—
78#475C2×C418yes—
79#428—28no—
80C2×#107—22no—
81C2×#106—22no—
82#423—22no—
83#423C2×C414yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90#419—28no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—24no—
94C2×#103—24no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8—21no—
98C2×C4×C8C2312yes—
99#391—24no—
100#391—24no—
101#387—28no—
102#387—28no—
103#384D412yes—
104#384C2312yes—
105#383—24no—
106#382—24no—
107#382—21no—
108#382—21no—
109#381—24no—
110#380—21no—
111#380—21no—
112#380D412yes—
113#380D412yes—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4C2312yes—
121C4×C22⋊C4—24no—
122#376—24no—
123#340—24no—
124#340—24no—
125Q8⋊C8—28no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8C22⋊C414yes—
135C22×Q8C2×D412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—44no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8—82no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8C22×C412yes—
158C4⋊Q8C22×C412yes—
159#125—24no—
160#125—42no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8C22⋊C418yes—
177C4×Q8C22×C414yes—
178C4×Q8C2×D414yes—
179#117—24no—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—42no—
184#117—42no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C22⋊C414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D414yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—41no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—28no—
215C2×C42—41no—
216C2×C42C22×C412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42C2411yescharacteristic
224#107—28no—
225#106—28no—
226C4⋊C8—24no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8C22×C414yes—
232#103—28no—
233#103—28no—
234#103—48no—
235#103—84no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—42no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246C24#12011yescharacteristic
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—28no—
256C2×Q8—28no—
257C2×Q8—44no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8C2×C22⋊C414yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12312yes—
290C22×C4#13512yes—
291C22×C4#13512yes—
292C22×C4#13712yes—
293C22×C4#13712yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—48no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—48no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—44no—
307C2×C8—84no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—28no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—44no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4C2×C22⋊C414yes—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C42—24no—
337C42—24no—
338C42—28no—
339C42—42no—
340C42—42no—
341C42—44no—
342C42—44no—
343C42—88no—
344C42C2×C22⋊C412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42#12112yes—
348C42#12112yes—
349C42#12312yes—
350C42C23×C411yescharacteristic
351C42C22×D411yescharacteristic
352C42C2×#4111yescharacteristic
353C42C2×#4111yescharacteristic
354C23—22no—
355C23—22no—
356C23#38612yes—
357C23#45212yes—
358C23#45212yes—
359C23C2×#12011yescharacteristic
360Q8—24no—
361Q8—24no—
362Q8—44no—
363Q8—42no—
364Q8—42no—
365Q8—48no—
366Q8—44no—
367Q8—82no—
368Q8—81no—
369Q8—81no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4#38612yes—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#41614yes—
399C2×C4#41914yes—
400C2×C4#45212yes—
401C2×C4#45212yes—
402C2×C4#48514yes—
403C2×C4#48514yes—
404C2×C4C22×C22⋊C411yescharacteristic
405C2×C4C2×C4×D411yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×#12111yescharacteristic
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×#12311yescharacteristic
410C2×C4C2×#13511yescharacteristic
411C2×C4C2×#13511yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4C2×#13711yescharacteristic
414C2×C4#57712yes—
415C2×C4#57812yes—
416C8—44no—
417C8—44no—
418C8—82no—
419C22—22no—
420C22—22no—
421C22—24no—
422C22—24no—
423C22#187314yes—
424C22C2×#38611yescharacteristic
425C22C2×#45211yescharacteristic
426C22C2×#45211yescharacteristic
427C22#301312yes—
428C22#301812yes—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—44no—
434C4—44no—
435C4—82no—
436C4—82no—
437C4—84no—
438C4—84no—
439C4#187314yes—
440C4C2×#41611yescharacteristic
441C4C2×#41911yescharacteristic
442C4#294712yes—
443C4#304311yescharacteristic
444C4#304411yescharacteristic
445C4C2×#48511yescharacteristic
446C4C2×#48511yescharacteristic
447C2—24no—
448C2#1866712yes—
449C2#1866912yes—
450C2C2×#187311yescharacteristic
451C2#2093011yescharacteristic
452C2#2094211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #8982C2semidirect
7. #8974C2semidirect
13. #3078C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#341872
#218852
#214002
C22×Q8⋊C82
#89822
#89742
#80452
#78972