Finite Group Databasebeta

← #242437

Subgroups — #242437

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21962C211yesmaximal, characteristic, complemented
2#21888C211yesmaximal, characteristic
3#21401C211yesmaximal, characteristic
4#21351C211yesmaximal, characteristic, complemented
5C22×#334C211yesmaximal, characteristic, complemented
6C2×#1551C211yesmaximal, characteristic
7#10597C212yesmaximal, complemented
8#10595C212yesmaximal, complemented
9#10589C212yesmaximal, complemented
10#10587C212yesmaximal, complemented
11#8044C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2285C2211yescharacteristic
18#2283C2211yescharacteristic
19#1990C2212yes—
20#1987C2212yes—
21#1982C2212yes—
22#1957C2212yes—
23#1949—22no—
24#1948C2212yes—
25#1947C2212yes—
26#1939C2212yes—
27#1939C2212yes—
28#1893—21no—
29#1893—21no—
30#1893—21no—
31#1893—21no—
32#1850—21no—
33#1850—21no—
34#1808—21no—
35#1807—21no—
36#1804—22no—
37#1768—22no—
38C4×#102—22no—
39C2×#334—24no—
40C2×#334C2212yes—
41C2×#334C2212yes—
42#1551—24no—
43C2×#333C2211yescharacteristic
44C2×#333C2211yescharacteristic
45C2×#331C2211yescharacteristic
46C2×#331C2211yescharacteristic
47#1296C2212yes—
48#1296C2212yes—
49C22×#41—21no—
50C23×C8—21no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54#532—22no—
55#532—22no—
56#532—22no—
57#532Q814yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#127C2311yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×#117—21no—
66C22×C42C2311yescharacteristic
67#432—22no—
68#432—22no—
69C2×#107—22no—
70C2×#106—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D414yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×C22⋊C8—21no—
89C2×#97—21no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8D412yes—
93C2×C4×C8C2312yes—
94#384C2312yes—
95#383—21no—
96#382—21no—
97#382D412yes—
98#381—21no—
99#380—21no—
100#380—21no—
101#380D411yescharacteristic
102#380D411yescharacteristic
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—21no—
107#376—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#336—22no—
115#336—22no—
116#334—24no—
117#334—24no—
118#333—22no—
119#333—22no—
120#331—22no—
121#331—22no—
122C2×#41—24no—
123C2×#41—42no—
124C2×#41—82no—
125C2×#41—82no—
126C22×Q8—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—41no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C22×C8—22no—
136C22×C8—24no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—24no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—82no—
147C22×C8—82no—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C2×Q812yes—
154#127—21no—
155#127—21no—
156#127—21no—
157#127C2×Q812yes—
158#124—24no—
159#124—42no—
160#121—24no—
161#121—24no—
162#121—42no—
163#121—42no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—42no—
167C4×D4—42no—
168#117—82no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—22no—
182C2×C42—41no—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×Q812yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189C2×C42#4111yescharacteristic
190C2×C42C2411yescharacteristic
191#107—82no—
192#106—82no—
193C4⋊C8—24no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—44no—
203C4⋊C8—44no—
204C4⋊C8—42no—
205C4⋊C8C2×D412yes—
206C4⋊C8C2×D412yes—
207#102—44no—
208#102—44no—
209#102—82no—
210#102—82no—
211C22⋊C8—42no—
212#97—42no—
213C4×C8—24no—
214C4×C8—22no—
215C4×C8—22no—
216#95—21no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—41no—
222#95—41no—
223C24—41no—
224C24#12111yescharacteristic
225#41—88no—
226#41—88no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—81no—
230C2×Q8—81no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—44no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—81no—
243C2×D4—81no—
244C2×D4—82no—
245C2×D4—82no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4#12112yes—
260C22×C4#12112yes—
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4C22⋊Q812yes—
265C22×C4C22⋊Q812yes—
266C22×C4C22⋊Q812yes—
267C22×C4#12612yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C2×C8—28no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—84no—
290C2×C8—84no—
291C2×C8#13514yes—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—82no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—82no—
310C42—42no—
311C42—42no—
312C42—82no—
313C42#12112yes—
314C42#12112yes—
315C42C22⋊Q812yes—
316C42C22⋊Q812yes—
317C42C22⋊Q812yes—
318C42#12612yes—
319C42C22×D411yescharacteristic
320C42C22×Q811yescharacteristic
321C42C2×#4111yescharacteristic
322C42C2×#4111yescharacteristic
323C23—22no—
324C23—22no—
325C23—42no—
326C23—82no—
327C23—82no—
328C23—81no—
329C23—81no—
330C23#39612yes—
331C23#47112yes—
332C23#47212yes—
333C23C2×#12111yescharacteristic
334Q8—82no—
335Q8—82no—
336D4—44no—
337D4—44no—
338D4—82no—
339D4—82no—
340D4—88no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—88no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4#39612yes—
373C2×C4#39612yes—
374C2×C4#39612yes—
375C2×C4C2×#12111yescharacteristic
376C2×C4C2×#12111yescharacteristic
377C2×C4C2×C22⋊Q811yescharacteristic
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4C2×C22⋊Q811yescharacteristic
380C2×C4C2×#12611yescharacteristic
381C2×C4C2×#13512yes—
382C2×C4#57512yes—
383C2×C4#57712yes—
384C2×C4#57712yes—
385C8—24no—
386C8—42no—
387C8—42no—
388C8—42no—
389C8—82no—
390C8—81no—
391C8—81no—
392C22—24no—
393C22—24no—
394C22—22no—
395C22—22no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—162no—
400C22—162no—
401C22#206314yes—
402C22C2×#39611yescharacteristic
403C22#303512yes—
404C22#306012yes—
405C22C2×#47111yescharacteristic
406C22C2×#47211yescharacteristic
407C4—22no—
408C4—21no—
409C4—21no—
410C4—22no—
411C4—22no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—162no—
416C4#304211yescharacteristic
417C4#304311yescharacteristic
418C4#305011yescharacteristic
419C4#305411yescharacteristic
420C4#307612yes—
421C4#308412yes—
422C2—24no—
423C2—162no—
424C2#1922112yes—
425C2#1922712yes—
426C2C2×#206311yescharacteristic
427C2#2360911yescharacteristic
428C2#2361011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×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
1. #21962C2semidirect
4. #21351C2semidirect
5. C22×#334C2semidirect
7. #10597C2semidirect
8. #10595C2semidirect
9. #10589C2semidirect
10. #10587C2semidirect
11. #8044C2semidirect

Maximal subgroups

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

subgroupindex
#219622
#218882
#214012
#213512
C22×#3342
C2×#15512
#105972
#105952
#105892
#105872
#80442