Finite Group Databasebeta

← #100246

Subgroups — #100246

Order 512.

462 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 100 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#33154C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic
3#10808C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic
5#7687C211yesmaximal, characteristic
6#7045C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C4×#117C2211yescharacteristic
10#1988C411yescharacteristic, complemented
11#1988C411yescharacteristic, complemented
12#1948C411yescharacteristic, complemented
13#1947C411yescharacteristic, complemented
14#1939—21no—
15#1923—22no—
16#1921—21no—
17#1920—21no—
18#1854—21no—
19#1831—21no—
20#1825—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23C8×C4⋊C4—21no—
24C4×C22⋊C8C2211yescharacteristic
25#1744—21no—
26C2×#339—21no—
27C2×#339—21no—
28#1728C2211yescharacteristic
29C4×#95C2211yescharacteristic
30#1294—22no—
31#1272—21no—
32#1272—21no—
33C22×M16—21no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C22⋊Q8—21no—
36C2×#117D411yescharacteristic
37C2×#117D411yescharacteristic
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40C2×#107—21no—
41C2×#106—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8D412yes—
49C2×C4×C8—22no—
50C2×C4×C8—21no—
51#391—21no—
52#389—21no—
53#389—21no—
54#384C2×C411yescharacteristic
55#380—21no—
56#380—21no—
57C4×C4⋊C4C2×C411yescharacteristic
58C4×C4⋊C4C2×C411yescharacteristic
59C4×C4⋊C4D411yescharacteristic
60C4×C4⋊C4D411yescharacteristic
61C4×C22⋊C4—21no—
62C4×C22⋊C4—21no—
63#376—21no—
64C2×#95—21no—
65C2×#95—21no—
66C43—21no—
67#340—21no—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#339—24no—
74#339—42no—
75#336—22no—
76#336—22no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336D411yescharacteristic
82#336D411yescharacteristic
83C22×Q8—21no—
84C23×C4—41no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—21no—
99C22×C8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C22⋊Q8—82no—
103#117—21no—
104#117—21no—
105#117—22no—
106#117—41no—
107#117SD1612yes—
108#117Q1612yes—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4C22⋊C411yescharacteristic
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—41no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42#4111yescharacteristic
140#107—42no—
141#106—42no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—82no—
147C22⋊C8—42no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—42no—
151C4×C8—41no—
152C4×C8—41no—
153C4×C8—44no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—41no—
164C24#12111yescharacteristic
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—81no—
168C2×Q8—81no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C4×D411yescharacteristic
196C22×C4C4×D411yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4C2×SD1611yescharacteristic
201C22×C4C2×Q1611yescharacteristic
202C22×C4#13511yescharacteristic
203C22×C4#13511yescharacteristic
204M16—42no—
205M16—42no—
206M16—81no—
207M16—81no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—42no—
223C2×C8—41no—
224C2×C8—81no—
225C2×C8—81no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C4⋊C4#10312yes—
240C4⋊C4#10312yes—
241C4⋊C4#10412yes—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—42no—
245C22⋊C4—82no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C42—42no—
255C42—42no—
256C42—42no—
257C42—42no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—81no—
263C42—81no—
264C42—81no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#38611yescharacteristic
272C23#46511yescharacteristic
273C23#46711yescharacteristic
274Q8—161no—
275Q8—161no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—44no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4#41011yescharacteristic
326C2×C4#41011yescharacteristic
327C2×C4C2×#10311yescharacteristic
328C2×C4#41611yescharacteristic
329C2×C4C2×#10411yescharacteristic
330C2×C4#42111yescharacteristic
331C2×C4#44311yescharacteristic
332C2×C4#44411yescharacteristic
333C2×C4#45511yescharacteristic
334C2×C4#45611yescharacteristic
335C2×C4#46511yescharacteristic
336C2×C4#46711yescharacteristic
337C8—42no—
338C8—81no—
339C8—82no—
340C8—82no—
341C8—81no—
342C8—81no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22#187111yescharacteristic
354C22#187611yescharacteristic
355C22#188211yescharacteristic
356C22#189211yescharacteristic
357C22#189911yescharacteristic
358C22#190311yescharacteristic
359C22#190611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4—42no—
371C4—84no—
372C4—81no—
373C4—161no—
374C4—161no—
375C4#150011yescharacteristic
376C4#150311yescharacteristic
377C4#151811yescharacteristic
378C4#152111yescharacteristic
379C2—41no—
380C2—41no—
381C2#897211yescharacteristic
382C2#911211yescharacteristic
383C2#916111yescharacteristic
384C2#920711yescharacteristic
385C2#924211yescharacteristic
386C2#931511yescharacteristic
387C2#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
10. #1988C4semidirect
11. #1988C4semidirect
12. #1948C4semidirect
13. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331542
#226322
#108082
#99082
#76872
#70452
#70282