Finite Group Databasebeta

← #191522

Subgroups — #191522

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21982C211yesmaximal, characteristic
4#21981C211yesmaximal, characteristic
5#20824C211yesmaximal, characteristic
6#20764C211yesmaximal, characteristic
7C2×#1453C211yesmaximal, characteristic
8#8821C211yesmaximal, characteristic, complemented
9#8821C211yesmaximal, characteristic, complemented
10#8821C211yesmaximal, characteristic, complemented
11#8821C211yesmaximal, characteristic, complemented
12#8820C211yesmaximal, characteristic, complemented
13#8819C211yesmaximal, characteristic, complemented
14#8817C211yesmaximal, characteristic, complemented
15#8814C211yesmaximal, characteristic, complemented
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#1949—22no—
24#1949—22no—
25#1948—22no—
26#1947—22no—
27#1854—22no—
28#1854—22no—
29#1853—22no—
30#1852—22no—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1845C2211yescharacteristic
34#1845C2211yescharacteristic
35#1845C2211yescharacteristic
36#1844C2211yescharacteristic
37#1844C2211yescharacteristic
38#1844C2211yescharacteristic
39#1844C2211yescharacteristic
40#1844C2211yescharacteristic
41#1836—22no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1809—22no—
57#1776—22no—
58#1736—22no—
59#1453C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C22×M16D412yes—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#431C2×C414yes—
73#431C2×C414yes—
74#431Q814yes—
75#431Q814yes—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×C4⋊C8D412yes—
87C2×C4⋊C8D412yes—
88C2×C22⋊C8D412yes—
89C2×C22⋊C8D412yes—
90C4×M16C2×C414yes—
91C4×M16Q814yes—
92C2×#97D412yes—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—21no—
111#382—21no—
112#382—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#322C2×C414yescomplemented
140#322C2×C414yescomplemented
141#322C2×C414yescomplemented
142C82C2×C414yescomplemented
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—42no—
148C2×M16—42no—
149C2×M16C4⋊C418yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156#117—44no—
157#117—82no—
158#117—82no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C22×C412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q812yes—
200C2×C42C2×Q812yes—
201C2×C42C2×Q812yes—
202C2×C42C2411yescharacteristic
203#107—44no—
204#107—82no—
205#107—82no—
206#107—82no—
207#107—82no—
208#106—44no—
209#106—82no—
210#106—82no—
211#106—82no—
212#106—82no—
213C4⋊C8C4⋊C418yes—
214C4⋊C8C4⋊C418yes—
215C22⋊C8C4⋊C418yes—
216C22⋊C8C4⋊C418yes—
217#97C4⋊C418yes—
218C4×C8C22×C412yes—
219C4×C8C22×C412yes—
220C4×C8C22×C412yes—
221C4×C8C22×C412yes—
222C4×C8C22×C412yes—
223C4×C8C22×C412yes—
224C4×C8C2×Q812yes—
225C4×C8C2×Q812yes—
226C4×C8C2×Q812yes—
227C4×C8C2×Q812yes—
228C4×C8C2×Q812yes—
229C4×C8C2×Q812yes—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236C24#12711yescharacteristic
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4C2×C4⋊C412yes—
255C22×C4C2×C4⋊C412yes—
256C22×C4C2×C4⋊C412yes—
257C22×C4#12711yescharacteristic
258C22×C4#12711yescharacteristic
259C22×C4#12711yescharacteristic
260C22×C4C4⋊Q812yes—
261C22×C4C4⋊Q812yes—
262C22×C4C4⋊Q812yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266M16—48no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8C2×C4⋊C414yes—
280C2×C8C2×C4⋊C414yes—
281C2×C8C2×C4⋊C414yes—
282C2×C8C2×C4⋊C414yes—
283C2×C8C2×C4⋊C414yes—
284C2×C8C2×C4⋊C414yes—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—82no—
316C42—82no—
317C42—82no—
318C42C2×C4⋊C412yes—
319C42C2×C4⋊C412yes—
320C42C2×C4⋊C412yes—
321C42C4⋊Q812yes—
322C42C4⋊Q812yes—
323C42C4⋊Q812yes—
324C42C23×C411yescharacteristic
325C42C22×Q811yescharacteristic
326C42C22×Q811yescharacteristic
327C42C22×Q811yescharacteristic
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#38412yes—
335C23C2×#12711yescharacteristic
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—164no—
373C2×C4#38412yes—
374C2×C4#38412yes—
375C2×C4#38412yes—
376C2×C4C22×C4⋊C411yescharacteristic
377C2×C4C22×C4⋊C411yescharacteristic
378C2×C4C22×C4⋊C411yescharacteristic
379C2×C4C2×C4⋊Q811yescharacteristic
380C2×C4C2×C4⋊Q811yescharacteristic
381C2×C4C2×C4⋊Q811yescharacteristic
382C2×C4#57512yes—
383C2×C4#57512yes—
384C2×C4#57612yes—
385C2×C4#57712yes—
386C2×C4#57712yes—
387C2×C4#57812yes—
388C8—24no—
389C8—24no—
390C8—24no—
391C8—24no—
392C8—24no—
393C8—24no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22C2×#38411yescharacteristic
402C22#315311yescharacteristic
403C22#315311yescharacteristic
404C22#315411yescharacteristic
405C22#315411yescharacteristic
406C22#315511yescharacteristic
407C22#315611yescharacteristic
408C4—22no—
409C4—22no—
410C4—22no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4#291112yes—
420C4#291212yes—
421C4#291312yes—
422C4#316312yes—
423C4#316412yes—
424C4#316612yes—
425C2—42no—
426C2#1971511yescharacteristic
427C2#1971611yescharacteristic
428C2#1972211yescharacteristic
429C2#1972311yescharacteristic
430C2#2080111yescharacteristic
431C2#2080211yescharacteristic
432C2#2080311yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#127
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
8. #8821C2semidirect
9. #8821C2semidirect
10. #8821C2semidirect
11. #8821C2semidirect
12. #8820C2semidirect
13. #8819C2semidirect
14. #8817C2semidirect
15. #8814C2semidirect
59. #1453C4semidirect
139. #322C2×C4semidirect
140. #322C2×C4semidirect
141. #322C2×C4semidirect
142. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219842
#219822
#219812
#208242
#207642
C2×#14532
#88212
#88202
#88192
#88172
#88142