Finite Group Databasebeta

← #241402

Subgroups — #241402

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21964C211yesmaximal, characteristic
4#21365C211yesmaximal, characteristic
5#10580C212yesmaximal, complemented
6#10579C212yesmaximal, complemented
7#10578C212yesmaximal, complemented
8#10578C212yesmaximal, complemented
9#8042C211yesmaximal, characteristic
10#7901C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#2851—22no—
16#2646—22no—
17#2571—21no—
18#2571C2212yes—
19#2570—22no—
20#2570—22no—
21#2552—21no—
22#2552C2212yes—
23#2317—21no—
24#2317—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1949—22no—
30#1948—21no—
31#1948C2212yes—
32#1948C2212yes—
33#1947—21no—
34#1947C2212yes—
35#1947C2212yes—
36#1854—22no—
37#1853—21no—
38#1852—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1809—22no—
44#1776—22no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×M16—21no—
52C2×#125—22no—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432Q814yes—
62#431—22no—
63#431Q814yes—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70#423—22no—
71#423Q814yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×#97—21no—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#400—24no—
82#400—24no—
83#398—24no—
84#398—24no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#387—24no—
92#387—24no—
93#385—24no—
94#385—24no—
95#384—22no—
96#384—22no—
97#384D411yescharacteristic
98#384D411yescharacteristic
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#382—22no—
104#382—22no—
105#382—22no—
106#382—21no—
107#382—21no—
108#382—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380D412yes—
114#380D412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4C2312yes—
124C4×C4⋊C4C2312yes—
125C2×#95—21no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C2×M16—81no—
139C2×M16—81no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144#125—82no—
145#125—82no—
146#117—44no—
147#117—44no—
148#117—82no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—41no—
185C2×C42—81no—
186C2×C42—81no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q812yes—
191C2×C42C2×Q812yes—
192C2×C42C2×Q812yes—
193C2×C42#4112yes—
194C2×C42C2411yescharacteristic
195#107—44no—
196#107—82no—
197#107—82no—
198#106—44no—
199#106—82no—
200#106—82no—
201C4⋊C8—24no—
202C4⋊C8—24no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8C2×Q814yes—
206C4⋊C8C2×Q814yes—
207C22⋊C8—24no—
208C22⋊C8—24no—
209#97—24no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8C2×Q812yes—
213C4×C8C2×Q812yes—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—41no—
225#95—41no—
226C24#12011yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4C22⋊Q812yes—
253C22×C4C22⋊Q812yes—
254C22×C4C22⋊Q812yes—
255C22×C4C4⋊Q812yes—
256C22×C4C4⋊Q812yes—
257C22×C4C4⋊Q812yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261M16—84no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—82no—
311C42—44no—
312C42—44no—
313C42—42no—
314C42—42no—
315C42—82no—
316C42—82no—
317C42C22⋊Q812yes—
318C42C22⋊Q812yes—
319C42C22⋊Q812yes—
320C42C4⋊Q812yes—
321C42C4⋊Q812yes—
322C42C4⋊Q812yes—
323C42C22×Q811yescharacteristic
324C42C22×Q811yescharacteristic
325C42C22×Q811yescharacteristic
326C42C2×#4111yescharacteristic
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23#39512yes—
334C23C2×#12011yescharacteristic
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4#39512yes—
378C2×C4#39512yes—
379C2×C4#39512yes—
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×C4⋊Q811yescharacteristic
384C2×C4C2×C4⋊Q811yescharacteristic
385C2×C4C2×C4⋊Q811yescharacteristic
386C2×C4#57512yes—
387C2×C4#57512yes—
388C2×C4#57611yescharacteristic
389C2×C4#57711yescharacteristic
390C2×C4#57711yescharacteristic
391C2×C4#57811yescharacteristic
392C8—42no—
393C8—42no—
394C8—44no—
395C8—44no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22C2×#39511yescharacteristic
404C22#300511yescharacteristic
405C22#300511yescharacteristic
406C22#301211yescharacteristic
407C22#301311yescharacteristic
408C22#302111yescharacteristic
409C22#302211yescharacteristic
410C4—22no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—161no—
422C4—161no—
423C4#308012yes—
424C4#308111yescharacteristic
425C4#308211yescharacteristic
426C4#316311yescharacteristic
427C4#316411yescharacteristic
428C4#316612yes—
429C2—42no—
430C2#1884111yescharacteristic
431C2#1884811yescharacteristic
432C2#1885611yescharacteristic
433C2#1886511yescharacteristic
434C2#2355311yescharacteristic
435C2#2355711yescharacteristic
436C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
5. #10580C2semidirect
6. #10579C2semidirect
7. #10578C2semidirect
8. #10578C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#219842
#219642
#213652
#105802
#105792
#105782
#80422
#79012
#79002