Finite Group Databasebeta

← #100264

Subgroups — #100264

Order 512.

468 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33114C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic
3#10808C211yesmaximal, characteristic
4#9909C211yesmaximal, characteristic
5#7685C211yesmaximal, characteristic
6#7045C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C2×#485—21no—
9#2857C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1981C411yescharacteristic, complemented
12#1981C411yescharacteristic, complemented
13#1948C411yescharacteristic, complemented
14#1947C411yescharacteristic, complemented
15#1939—21no—
16#1928—21no—
17#1927—21no—
18#1924—22no—
19#1873—21no—
20#1831—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23#1770—21no—
24C4×C22⋊C8C2211yescharacteristic
25#1745—21no—
26C2×#339—21no—
27C2×#339—21no—
28#1728C2211yescharacteristic
29C4×#95C2211yescharacteristic
30#1294—22no—
31#1271—21no—
32#1271—21no—
33C22×M16—21no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C22⋊Q8—21no—
36C2×#117D411yescharacteristic
37C2×#117D411yescharacteristic
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40#485—42no—
41#485—42no—
42C2×#107—21no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2×C411yescharacteristic
47C2×C4⋊C8C2×C411yescharacteristic
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8D412yes—
55C2×#97—22no—
56C2×C4×C8—21no—
57#391—21no—
58#389—21no—
59#389—21no—
60#384C2×C411yescharacteristic
61#380—21no—
62#380—21no—
63C4×C4⋊C4C2×C411yescharacteristic
64C4×C4⋊C4C2×C411yescharacteristic
65C4×C4⋊C4D411yescharacteristic
66C4×C4⋊C4D411yescharacteristic
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69#376—21no—
70C2×#95—21no—
71C2×#95—21no—
72C43—21no—
73#340—21no—
74#340—21no—
75#339—24no—
76#339—42no—
77#336—22no—
78#336—22no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336D411yescharacteristic
84#336D411yescharacteristic
85C22×Q8—21no—
86C23×C4—41no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—21no—
101C22×C8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C22⋊Q8—41no—
105C22⋊Q8—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108#117—21no—
109#117—21no—
110#117—21no—
111#117—21no—
112#117—22no—
113#117—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—41no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—21no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C22×C411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145#107—42no—
146#106—42no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—82no—
152#103—42no—
153#103—42no—
154#103—42no—
155#103—42no—
156C22⋊C8—42no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—42no—
160#97—44no—
161C4×C8—41no—
162C4×C8—41no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—22no—
168#95—22no—
169#95—41no—
170#95—41no—
171#95—41no—
172#95—41no—
173C24#12111yescharacteristic
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4#12011yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12311yescharacteristic
211C22×C4AGL(1,ℤ/8)11yescharacteristic
212C22×C4AGL(1,ℤ/8)11yescharacteristic
213C22×C4AGL(1,ℤ/8)11yescharacteristic
214C22×C4#13711yescharacteristic
215M16—42no—
216M16—42no—
217M16—81no—
218M16—81no—
219C2×C8—22no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—81no—
236C2×C8—81no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4#10412yes—
254C22⋊C4—42no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—41no—
264C42—41no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—41no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—81no—
276C42—81no—
277C42—81no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23#38611yescharacteristic
285C23#46811yescharacteristic
286C23#46911yescharacteristic
287Q8—81no—
288Q8—81no—
289Q8—81no—
290Q8—81no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4#41011yescharacteristic
341C2×C4#41011yescharacteristic
342C2×C4#41711yescharacteristic
343C2×C4#41811yescharacteristic
344C2×C4C2×#10411yescharacteristic
345C2×C4#42111yescharacteristic
346C2×C4#44311yescharacteristic
347C2×C4#44411yescharacteristic
348C2×C4#45511yescharacteristic
349C2×C4#45611yescharacteristic
350C2×C4#46811yescharacteristic
351C2×C4#46911yescharacteristic
352C8—42no—
353C8—81no—
354C8—82no—
355C8—82no—
356C8—81no—
357C8—81no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22#187111yescharacteristic
369C22#187611yescharacteristic
370C22#188211yescharacteristic
371C22#189511yescharacteristic
372C22#189911yescharacteristic
373C22#190211yescharacteristic
374C22#190611yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4—84no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—161no—
389C4—161no—
390C4#150511yescharacteristic
391C4#150911yescharacteristic
392C4#151211yescharacteristic
393C4#151411yescharacteristic
394C2—41no—
395C2—41no—
396C2#897111yescharacteristic
397C2#911811yescharacteristic
398C2#916011yescharacteristic
399C2#921011yescharacteristic
400C2#924011yescharacteristic
401C2#932011yescharacteristic
402C2#936311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
11. #1981C4semidirect
12. #1981C4semidirect
13. #1948C4semidirect
14. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331142
#226362
#108082
#99092
#76852
#70452
#70282