Finite Group Databasebeta

← #100393

Subgroups — #100393

Order 512.

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

normal subgroups only

This group has 618 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#33108C211yesmaximal, characteristic
2#13810C211yesmaximal, characteristic
3#9909C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#8706C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C2×#483—21no—
9#2857C2211yescharacteristic
10C2×C4×C4⋊C4—21no—
11C2×C4×C4⋊C4C2211yescharacteristic
12C2×#376—21no—
13#1981C412yescomplemented
14#1948C412yescomplemented
15#1927—22no—
16#1872—21no—
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1434—21no—
23#1434—22no—
24C4×#95C2211yescharacteristic
25#1294—22no—
26#1294—21no—
27#1294—21no—
28#1279—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C22⋊Q8—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C42—22no—
35C22×C42—21no—
36C22×C42—21no—
37C22×C42C2311yescharacteristic
38#483—42no—
39#483—42no—
40C2×#106—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C412yes—
43C2×#103—22no—
44C2×#103—22no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C4×C8—21no—
49#391—21no—
50#389—21no—
51#389—21no—
52#384C2×C411yescharacteristic
53#380—21no—
54#380—21no—
55C4×C4⋊C4—28no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—21no—
58C4×C4⋊C4C2×C412yes—
59C4×C4⋊C4D412yes—
60#376—42no—
61#376—42no—
62C2×#95—22no—
63C2×#95—21no—
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—
75C22×Q8—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—22no—
85C22×C8—41no—
86C22×C8—41no—
87C4⋊Q8—42no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C22⋊C4—41no—
112C2×C42—24no—
113C2×C42—28no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—42no—
117C2×C42—44no—
118C2×C42—42no—
119C2×C42—44no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130#106—44no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133#103—44no—
134#103—44no—
135C22⋊C8—22no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C4×C8—42no—
142#95—28no—
143#95—22no—
144#95—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—42no—
149#95—42no—
150#95—44no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—82no—
156#95SD1612yes—
157C24#12011yescharacteristic
158C2×Q8—42no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—42no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12311yescharacteristic
211C22×C4C2×SD1611yescharacteristic
212C22×C4#13511yescharacteristic
213C22×C4#13511yescharacteristic
214C22×C4#13511yescharacteristic
215C22×C4AGL(1,ℤ/8)11yescharacteristic
216C22×C4AGL(1,ℤ/8)11yescharacteristic
217C22×C4AGL(1,ℤ/8)11yescharacteristic
218C22×C4#13711yescharacteristic
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C2×C8—81no—
227C2×C8—81no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4#10414yes—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C42—22no—
248C42—22no—
249C42—41no—
250C42—41no—
251C42—44no—
252C42—44no—
253C42—416no—
254C42—42no—
255C42—42no—
256C42—42no—
257C42—42no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23#38611yescharacteristic
263C23#44911yescharacteristic
264C23#44911yescharacteristic
265C23#45011yescharacteristic
266C23#45211yescharacteristic
267C23#45511yescharacteristic
268C23#45611yescharacteristic
269Q8—82no—
270Q8—82no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4#41012yes—
333C2×C4#41611yescharacteristic
334C2×C4#41811yescharacteristic
335C2×C4C2×#10412yes—
336C2×C4C4×SD1611yescharacteristic
337C2×C4#44011yescharacteristic
338C2×C4#46111yescharacteristic
339C2×C4#46311yescharacteristic
340C2×C4#48211yescharacteristic
341C2×C4#48211yescharacteristic
342C8—82no—
343C8—82no—
344C8—161no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22#164211yescharacteristic
358C22#164611yescharacteristic
359C22#165011yescharacteristic
360C22#165411yescharacteristic
361C22#187511yescharacteristic
362C22#187711yescharacteristic
363C22#188211yescharacteristic
364C22#189611yescharacteristic
365C22#189911yescharacteristic
366C22#190511yescharacteristic
367C22#190611yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—44no—
375C4—44no—
376C4—41no—
377C4—41no—
378C4—84no—
379C4—82no—
380C4—82no—
381C4—161no—
382C4—161no—
383C4#150012yes—
384C4#150512yes—
385C2—21no—
386C2—21no—
387C2—21no—
388C2—21no—
389C2#898811yescharacteristic
390C2#914911yescharacteristic
391C2#918211yescharacteristic
392C2#923111yescharacteristic
393C2#925611yescharacteristic
394C2#932411yescharacteristic
395C2#936311yescharacteristic
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
13. #1981C4semidirect
14. #1948C4semidirect

Maximal subgroups

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

subgroupindex
#331082
#138102
#99092
#98852
#87062
#70392
#70282