Finite Group Databasebeta

← #100422

Subgroups — #100422

Order 512.

658 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 108 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#32906C211yesmaximal, characteristic
2#13931C211yesmaximal, characteristic
3#9909C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#8311C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#462C412yescomplemented
10C2×#451—21no—
11#2857C2211yescharacteristic
12C2×#391—21no—
13#1878—22no—
14#1872—21no—
15#1803C412yescomplemented
16#1797C2211yescharacteristic
17#1797C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C2×Q8⋊C8—22no—
20#1445—22no—
21C4⋊#95—21no—
22C4⋊#95C2211yescharacteristic
23#1434—21no—
24#1434—21no—
25C4×#95C2211yescharacteristic
26#1294—22no—
27#1294—21no—
28#1294—21no—
29C23×Q8D411yescharacteristic
30C22×Q16—22no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×C22⋊Q8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—44no—
36C2×C4×Q8C2×C412yes—
37C2×C4×Q8D412yes—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42—21no—
43C22×C42C2311yescharacteristic
44#462—44no—
45#451—42no—
46#451—42no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8C2×C412yes—
49C2×#103—22no—
50C2×#103—22no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C4×C8—21no—
55#391—21no—
56#391—44no—
57#389—21no—
58#389—21no—
59#384C2×C411yescharacteristic
60#380—21no—
61#380—21no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—22no—
68C2×#95D411yescharacteristic
69C2×#95D411yescharacteristic
70C2×#95D412yes—
71#340—22no—
72#336—21no—
73#336—21no—
74#336—21no—
75Q8⋊C8—44no—
76C22×Q8—24no—
77C22×Q8—21no—
78C22×Q8—21no—
79C22×Q8—21no—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—42no—
84C22×Q8C22⋊C412yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×Q16—44no—
94C2×Q16—44no—
95C2×Q16—44no—
96C22×C8—22no—
97C22×C8—41no—
98C22×C8—41no—
99C4⋊Q8—42no—
100C22⋊Q8—41no—
101C22⋊Q8—41no—
102C22⋊Q8—41no—
103C22⋊Q8—41no—
104C4×Q8—24no—
105C4×Q8—24no—
106C4×Q8—42no—
107C4×Q8—42no—
108C4×Q8—44no—
109C4×Q8—44no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4C22⋊C412yes—
126C2×C22⋊C4—41no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—42no—
130C2×C42—44no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42#4111yescharacteristic
139C4⋊C8—42no—
140C4⋊C8—42no—
141#103—44no—
142#103—44no—
143C22⋊C8—22no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C4×C8—42no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—42no—
154#95—42no—
155#95—44no—
156#95—44no—
157#95—44no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—82no—
163#95Q1612yescomplemented
164C24#12011yescharacteristic
165C2×Q8—24no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8#9914yes—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4C2×Q1611yescharacteristic
231C22×C4#13511yescharacteristic
232C22×C4#13511yescharacteristic
233C22×C4#13511yescharacteristic
234C22×C4AGL(1,ℤ/8)11yescharacteristic
235C22×C4AGL(1,ℤ/8)11yescharacteristic
236C22×C4#13711yescharacteristic
237C22×C4#13711yescharacteristic
238Q16—88no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—81no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C42—22no—
266C42—22no—
267C42—41no—
268C42—41no—
269C42—44no—
270C42—44no—
271C42—82no—
272C42—82no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23#38611yescharacteristic
278C23#44911yescharacteristic
279C23#44911yescharacteristic
280C23#45111yescharacteristic
281C23#45211yescharacteristic
282C23#45511yescharacteristic
283C23#45611yescharacteristic
284Q8—24no—
285Q8—24no—
286Q8—44no—
287Q8—44no—
288Q8—42no—
289Q8—42no—
290Q8—82no—
291Q8—82no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C2×#9912yes—
352C2×C4#41611yescharacteristic
353C2×C4#41811yescharacteristic
354C2×C4#42112yes—
355C2×C4C4×Q1611yescharacteristic
356C2×C4#44111yescharacteristic
357C2×C4#46211yescharacteristic
358C2×C4#46411yescharacteristic
359C2×C4#48211yescharacteristic
360C2×C4#48211yescharacteristic
361C8—82no—
362C8—82no—
363C8—161no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22#164011yescharacteristic
377C22#164411yescharacteristic
378C22#164811yescharacteristic
379C22#165211yescharacteristic
380C22#187511yescharacteristic
381C22#187811yescharacteristic
382C22#188211yescharacteristic
383C22#189711yescharacteristic
384C22#189811yescharacteristic
385C22#190511yescharacteristic
386C22#190611yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—44no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4—162no—
400C4—161no—
401C4—161no—
402C4#151012yes—
403C4#151612yes—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2#898611yescharacteristic
409C2#915811yescharacteristic
410C2#918211yescharacteristic
411C2#923611yescharacteristic
412C2#925811yescharacteristic
413C2#932811yescharacteristic
414C2#936411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
9. C2×#462C4semidirect
15. #1803C4semidirect
163. #95Q16semidirect

Maximal subgroups

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

subgroupindex
#329062
#139312
#99092
#98852
#83112
#70392
#70282