Finite Group Databasebeta

← #100365

Subgroups — #100365

Order 512.

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

normal subgroups only

This group has 586 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#33149C211yesmaximal, characteristic
2#13812C211yesmaximal, characteristic
3#9908C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#8707C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C2×C4×C4⋊C4—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1987C412yescomplemented
12#1947C412yescomplemented
13#1921—22no—
14#1852—21no—
15#1824—21no—
16#1797C2211yescharacteristic
17#1797C2211yescharacteristic
18C4×C22⋊C8C2211yescharacteristic
19C2×#328—22no—
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1434—21no—
23#1434—22no—
24C4×#95—21no—
25C4×#95C2211yescharacteristic
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—21no—
35C22×C42—22no—
36C22×C42—21no—
37C22×C42C2311yescharacteristic
38C2×#107—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8C2×C412yes—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C4×C8—21no—
45#391—21no—
46#389—21no—
47#389—21no—
48#384C2×C411yescharacteristic
49#380—21no—
50#380—21no—
51C4×C4⋊C4—28no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4C2×C412yes—
55C4×C4⋊C4D412yes—
56C2×#95—22no—
57C2×#95—21no—
58C2×#95—22no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95D411yescharacteristic
63C2×#95D411yescharacteristic
64C2×#95D412yes—
65#340—22no—
66#340—22no—
67#340—22no—
68#336—21no—
69#336—21no—
70#336—21no—
71#328—44no—
72C22×Q8—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C4⋊Q8—42no—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
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—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C22⋊C412yes—
105C2×C4⋊C4C22⋊C412yes—
106C2×C22⋊C4—41no—
107C2×C42—24no—
108C2×C42—28no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—82no—
116C2×C42—82no—
117C2×C42—82no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122#107—44no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C22⋊C8—22no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C4×C8—42no—
132#95—28no—
133#95—22no—
134#95—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—42no—
139#95—42no—
140#95—44no—
141#95—41no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—82no—
146#95Q1612yes—
147C24#12011yescharacteristic
148C2×Q8—42no—
149C2×Q8—81no—
150C2×Q8—81no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4C2×Q1611yescharacteristic
199C22×C4#13511yescharacteristic
200C22×C4#13511yescharacteristic
201C22×C4#13511yescharacteristic
202C22×C4#13711yescharacteristic
203C22×C4#13711yescharacteristic
204C22×C4#13711yescharacteristic
205C22×C4#13711yescharacteristic
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—82no—
229C4⋊C4#10414yes—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—22no—
233C42—22no—
234C42—41no—
235C42—41no—
236C42—44no—
237C42—44no—
238C42—82no—
239C42—82no—
240C42—88no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23#38611yescharacteristic
246C23#45111yescharacteristic
247C23#45211yescharacteristic
248C23#45211yescharacteristic
249C23#45211yescharacteristic
250C23#45311yescharacteristic
251C23#45411yescharacteristic
252Q8—162no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4#41012yes—
313C2×C4#41611yescharacteristic
314C2×C4#41911yescharacteristic
315C2×C4C2×#10412yes—
316C2×C4C4×Q1611yescharacteristic
317C2×C4#44111yescharacteristic
318C2×C4#46211yescharacteristic
319C2×C4#46411yescharacteristic
320C2×C4#48511yescharacteristic
321C2×C4#48511yescharacteristic
322C8—82no—
323C8—82no—
324C8—161no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22#163011yescharacteristic
338C22#163211yescharacteristic
339C22#163611yescharacteristic
340C22#163811yescharacteristic
341C22#187311yescharacteristic
342C22#187811yescharacteristic
343C22#188311yescharacteristic
344C22#189611yescharacteristic
345C22#189911yescharacteristic
346C22#190411yescharacteristic
347C22#190611yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—44no—
355C4—44no—
356C4—41no—
357C4—41no—
358C4—84no—
359C4—82no—
360C4—82no—
361C4—161no—
362C4—161no—
363C4#150012yes—
364C4#150612yes—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2#897411yescharacteristic
370C2#914511yescharacteristic
371C2#917711yescharacteristic
372C2#923211yescharacteristic
373C2#925711yescharacteristic
374C2#932511yescharacteristic
375C2#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
11. #1987C4semidirect
12. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331492
#138122
#99082
#96792
#87072
#70392
#70282