Finite Group Databasebeta

← #100364

Subgroups — #100364

Order 512.

644 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 116 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#33148C211yesmaximal, characteristic
2#13821C211yesmaximal, characteristic
3#9908C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#8706C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C2×#389—21no—
10C2×C4×C4⋊C4—21no—
11C2×C4×C4⋊C4—21no—
12C2×C4×C4⋊C4C2211yescharacteristic
13#1987C412yescomplemented
14#1948C412yescomplemented
15#1920—22no—
16#1853—21no—
17#1823—21no—
18#1797C2211yescharacteristic
19#1797C2211yescharacteristic
20C4×C22⋊C8C2211yescharacteristic
21C2×#328—22no—
22#1436—22no—
23C4⋊#95C2211yescharacteristic
24#1434—21no—
25C4×#95C2211yescharacteristic
26#1294—21no—
27#1294—21no—
28#1279—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C22⋊Q8—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C42—21no—
36C22×C42—21no—
37C22×C42—22no—
38C22×C42C2311yescharacteristic
39C2×#106—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8C2×C412yes—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C4×C8—21no—
46#391—21no—
47#389—28no—
48#389—21no—
49#389—21no—
50#384C2×C411yescharacteristic
51#380—21no—
52#380—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—42no—
56C4×C4⋊C4—42no—
57C4×C4⋊C4—42no—
58C4×C4⋊C4—42no—
59C4×C4⋊C4C2×C412yes—
60C4×C4⋊C4D412yes—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95D411yescharacteristic
67C2×#95D411yescharacteristic
68C2×#95D412yes—
69#340—22no—
70#340—22no—
71#340—22no—
72#336—21no—
73#336—21no—
74#336—21no—
75#328—44no—
76C22×Q8—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C4⋊Q8—42no—
89C22⋊Q8—42no—
90C22⋊Q8—42no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4C22⋊C412yes—
111C2×C4⋊C4SD1612yes—
112C2×C4⋊C4Q1612yes—
113C2×C22⋊C4—41no—
114C2×C42—24no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—44no—
126C2×C42—44no—
127C2×C42—82no—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132#106—44no—
133C4⋊C8—42no—
134C4⋊C8—42no—
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#95SD1612yes—
156C24#12011yescharacteristic
157C2×Q8—42no—
158C2×Q8—81no—
159C2×Q8—81no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—24no—
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—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4C4×D411yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12311yescharacteristic
209C22×C4C2×SD1611yescharacteristic
210C22×C4C2×SD1611yescharacteristic
211C22×C4C2×Q1611yescharacteristic
212C22×C4#13511yescharacteristic
213C22×C4AGL(1,ℤ/8)11yescharacteristic
214C22×C4AGL(1,ℤ/8)11yescharacteristic
215C22×C4#13711yescharacteristic
216C22×C4#13711yescharacteristic
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—416no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—82no—
240C4⋊C4#10314yes—
241C4⋊C4#10414yes—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—41no—
248C42—44no—
249C42—44no—
250C42—42no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—88no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23#38611yescharacteristic
260C23#45011yescharacteristic
261C23#45011yescharacteristic
262C23#45111yescharacteristic
263C23#45211yescharacteristic
264C23#45311yescharacteristic
265C23#45411yescharacteristic
266Q8—162no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—81no—
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#41012yes—
329C2×C4C2×#10311yescharacteristic
330C2×C4#41711yescharacteristic
331C2×C4C2×#10412yes—
332C2×C4C4×SD1611yescharacteristic
333C2×C4#44011yescharacteristic
334C2×C4#46111yescharacteristic
335C2×C4#46311yescharacteristic
336C2×C4#48311yescharacteristic
337C2×C4#48411yescharacteristic
338C8—82no—
339C8—82no—
340C8—161no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#162911yescharacteristic
354C22#163211yescharacteristic
355C22#163511yescharacteristic
356C22#163711yescharacteristic
357C22#187211yescharacteristic
358C22#187711yescharacteristic
359C22#188311yescharacteristic
360C22#189611yescharacteristic
361C22#189911yescharacteristic
362C22#190511yescharacteristic
363C22#190611yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—44no—
371C4—44no—
372C4—44no—
373C4—41no—
374C4—41no—
375C4—84no—
376C4—82no—
377C4—161no—
378C4—161no—
379C4#150312yes—
380C4#150912yes—
381C2—21no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2#897311yescharacteristic
386C2#914411yescharacteristic
387C2#917611yescharacteristic
388C2#923111yescharacteristic
389C2#925611yescharacteristic
390C2#932611yescharacteristic
391C2#936611yescharacteristic
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. #1987C4semidirect
14. #1948C4semidirect

Maximal subgroups

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

subgroupindex
#331482
#138212
#99082
#96792
#87062
#70392
#70282