Finite Group Databasebeta

← #316145

Subgroups — #316145

Order 512.

641 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29877C211yesmaximal, characteristic
2#27908C211yesmaximal, characteristic
3#27892C211yesmaximal, characteristic
4#27878C211yesmaximal, characteristic
5#27875C211yesmaximal, characteristic
6#21984C211yesmaximal, characteristic
7#10684C211yesmaximal, characteristic, complemented
8#10681C211yesmaximal, characteristic, complemented
9#10680C211yesmaximal, characteristic, complemented
10#10675C211yesmaximal, characteristic, complemented
11#9841C211yesmaximal, characteristic, complemented
12#9828C211yesmaximal, characteristic, complemented
13#8821C211yesmaximal, characteristic, complemented
14#8820C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#485—22no—
18C2×#431C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#2097—22no—
24#2088—22no—
25#2077—22no—
26#2077—22no—
27#2076—22no—
28#2036—22no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1956C2211yescharacteristic
39#1949—22no—
40#1928—22no—
41#1921—22no—
42#1920—22no—
43#1854—22no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×#117—21no—
68C2×#117—21no—
69C22×C42C2311yescharacteristic
70#498—22no—
71#498—22no—
72#485—48no—
73#431—22no—
74#431—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×C4⋊C8D412yes—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×C22⋊C8D412yes—
89C2×C22⋊C8D412yes—
90C2×C22⋊C8D412yes—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—21no—
98#391—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—21no—
110#382—21no—
111#380—21no—
112#380—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—
123#340—22no—
124#340—22no—
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#336D412yes—
136#336D412yes—
137#322—22no—
138#322—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C22⋊Q8—44no—
159C22⋊Q8—82no—
160#117—44no—
161#117—82no—
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—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2411yescharacteristic
200#107—44no—
201#107—44no—
202#107—82no—
203#107—82no—
204#106—44no—
205#106—44no—
206#106—44no—
207#106—82no—
208C4⋊C8—24no—
209#103—44no—
210#103—44no—
211#103—44no—
212#103—82no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—42no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8#4112yes—
227C4×C8#4112yes—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236C24#12711yescharacteristic
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4#12711yescharacteristic
266C22×C4#12711yescharacteristic
267C22×C4#12711yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
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×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8#12314yes—
288C2×C8AGL(1,ℤ/8)14yes—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
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—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—82no—
328C42—82no—
329C42C2×#4111yescharacteristic
330C42C2×#4111yescharacteristic
331C422−1+411yescharacteristic
332C422−1+411yescharacteristic
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23C2×#12711yescharacteristic
341Q8—82no—
342Q8—82no—
343Q8—82no—
344Q8—82no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
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—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4C2×#12311yescharacteristic
385C2×C4#53611yescharacteristic
386C2×C4#54711yescharacteristic
387C2×C4#54711yescharacteristic
388C2×C4#54711yescharacteristic
389C2×C4#54711yescharacteristic
390C2×C4C2×AGL(1,ℤ/8)12yes—
391C2×C4#57512yes—
392C2×C4#57712yes—
393C2×C4#57712yes—
394C2×C4#57812yes—
395C2×C4#57812yes—
396C8—24no—
397C8—24no—
398C8—42no—
399C8—42no—
400C8—42no—
401C8—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22#250111yescharacteristic
410C22#315011yescharacteristic
411C22#315111yescharacteristic
412C22#315411yescharacteristic
413C22#315411yescharacteristic
414C22#315611yescharacteristic
415C22#315611yescharacteristic
416C4—41no—
417C4—41no—
418C4—41no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4#309712yes—
428C4#324212yes—
429C4#334312yes—
430C4#334812yes—
431C4#335012yes—
432C4#335312yes—
433C2—81no—
434C2#1969411yescharacteristic
435C2#1971111yescharacteristic
436C2#1972511yescharacteristic
437C2#1973911yescharacteristic
438C2#2792211yescharacteristic
439C2#2793711yescharacteristic
440C2#2794411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. #10684C2semidirect
8. #10681C2semidirect
9. #10680C2semidirect
10. #10675C2semidirect
11. #9841C2semidirect
12. #9828C2semidirect
13. #8821C2semidirect
14. #8820C2semidirect

Maximal subgroups

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

subgroupindex
#298772
#279082
#278922
#278782
#278752
#219842
#106842
#106812
#106802
#106752
#98412
#98282
#88212
#88202
#81492