Finite Group Databasebeta

← #241337

Subgroups — #241337

Order 512.

772 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#37642C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21984C211yesmaximal, characteristic
4#21959C211yesmaximal, characteristic
5#10578C212yesmaximal, complemented
6#10576C212yesmaximal, complemented
7#10575C212yesmaximal, complemented
8#10574C212yesmaximal, complemented
9#8044C211yesmaximal, characteristic
10#7900C211yesmaximal, characteristic
11#7893C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#2844—22no—
16#2628—22no—
17#2628—22no—
18#2628—22no—
19#2571—21no—
20#2571—21no—
21#2571C2212yes—
22#2551—21no—
23#2551—21no—
24#2551C2212yes—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1949—22no—
30#1948—21no—
31#1948—21no—
32#1948C2212yes—
33#1948C2212yes—
34#1948C2212yes—
35#1947C2212yes—
36#1854—22no—
37#1854—22no—
38#1847—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1836—21no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1807—22no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C23×C8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117—21no—
58C2×#117D412yes—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432Q814yescomplemented
62#431—22no—
63#431Q814yescomplemented
64C2×#107—22no—
65C2×#107—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8Q814yescomplemented
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#400—24no—
84#399—24no—
85#395—24no—
86#395—24no—
87#395—24no—
88#393—24no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#384D411yescharacteristic
96#384D411yescharacteristic
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#382D412yes—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380D412yes—
117#380D412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132C22×Q8—24no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—24no—
141C22×C8—81no—
142C22×C8—81no—
143C4⋊Q8—82no—
144C4⋊Q8—84no—
145C4⋊Q8—82no—
146C4⋊Q8—84no—
147#117—81no—
148#117—82no—
149#117—81no—
150#117—82no—
151#117—82no—
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—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—41no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—41no—
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—82no—
196#107—82no—
197#106—82no—
198#106—82no—
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
227C2×Q8—84no—
228C2×Q8—84no—
229C2×Q8—84no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4C22⋊Q812yes—
250C22×C4C22⋊Q812yes—
251C22×C4C22⋊Q812yes—
252C22×C4C4⋊Q812yes—
253C22×C4C4⋊Q812yes—
254C22×C4C4⋊Q812yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—84no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42—44no—
310C42—42no—
311C42—44no—
312C42—42no—
313C42—82no—
314C42C22⋊Q812yes—
315C42C22⋊Q812yes—
316C42C22⋊Q812yes—
317C42C4⋊Q812yes—
318C42C4⋊Q812yes—
319C42C4⋊Q812yes—
320C42C22×Q811yescharacteristic
321C42C22×Q811yescharacteristic
322C42C22×Q811yescharacteristic
323C42C2×#4111yescharacteristic
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#39512yes—
331C23C2×#12011yescharacteristic
332Q8—168no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
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—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4#39512yes—
374C2×C4#39512yes—
375C2×C4#39512yes—
376C2×C4C2×C22⋊Q811yescharacteristic
377C2×C4C2×C22⋊Q811yescharacteristic
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4C2×C4⋊Q811yescharacteristic
380C2×C4C2×C4⋊Q811yescharacteristic
381C2×C4C2×C4⋊Q811yescharacteristic
382C2×C4#57611yescharacteristic
383C2×C4#57711yescharacteristic
384C2×C4#57711yescharacteristic
385C2×C4#57712yes—
386C2×C4#57712yes—
387C2×C4#57811yescharacteristic
388C8—42no—
389C8—42no—
390C8—44no—
391C8—44no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22C2×#39511yescharacteristic
400C22#300911yescharacteristic
401C22#301011yescharacteristic
402C22#301211yescharacteristic
403C22#301311yescharacteristic
404C22#301911yescharacteristic
405C22#302011yescharacteristic
406C4—22no—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—22no—
411C4—82no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—161no—
418C4—161no—
419C4#308111yescharacteristic
420C4#308211yescharacteristic
421C4#308412yes—
422C4#316312yes—
423C4#316311yescharacteristic
424C4#316411yescharacteristic
425C2—42no—
426C2#1893511yescharacteristic
427C2#1893811yescharacteristic
428C2#1894911yescharacteristic
429C2#1895211yescharacteristic
430C2#2355611yescharacteristic
431C2#2355711yescharacteristic
432C2#2356111yescharacteristic
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. #10578C2semidirect
6. #10576C2semidirect
7. #10575C2semidirect
8. #10574C2semidirect
61. #432Q8semidirect
63. #431Q8semidirect
73. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#219882
#219842
#219592
#105782
#105762
#105752
#105742
#80442
#79002
#78932