Finite Group Databasebeta

← #100397

Subgroups — #100397

Order 512.

506 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 104 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#13936C211yesmaximal, characteristic
3#9908C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#8707C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1987C412yescomplemented
11#1947C412yescomplemented
12#1921—22no—
13#1852—21no—
14#1824—21no—
15#1797C2211yescharacteristic
16#1797C2211yescharacteristic
17C4×C22⋊C8C2211yescharacteristic
18C2×#328—22no—
19#1443—22no—
20C4⋊#95—21no—
21C4⋊#95—21no—
22C4⋊#95C2211yescharacteristic
23#1434—21no—
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—21no—
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4D411yescharacteristic
34C22×C42—21no—
35C22×C42—21no—
36C22×C42C2311yescharacteristic
37C2×#107—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C412yes—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C4×C8—21no—
44#391—21no—
45#389—21no—
46#389—21no—
47#384C2×C411yescharacteristic
48#380—21no—
49#380—21no—
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4C2×C412yes—
53C4×C4⋊C4D412yes—
54C2×#95—21no—
55C2×#95—22no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95D411yescharacteristic
62C2×#95D411yescharacteristic
63C2×#95D412yes—
64#340—22no—
65#340—22no—
66#340—22no—
67#336—21no—
68#336—21no—
69#336—21no—
70#328—44no—
71C22×Q8—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—22no—
81C22×C8—41no—
82C22×C8—41no—
83C4⋊Q8—42no—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4C22⋊C412yes—
104C2×C4⋊C4C22⋊C412yes—
105C2×C22⋊C4—41no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42—82no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118#107—44no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C22⋊C8—22no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C4×C8—42no—
128#95—22no—
129#95—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—42no—
134#95—42no—
135#95—44no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—82no—
141#95—82no—
142#95—82no—
143#95Q1612yes—
144C24#12011yescharacteristic
145C2×Q8—42no—
146C2×Q8—81no—
147C2×Q8—81no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4C2×Q1611yescharacteristic
196C22×C4#13511yescharacteristic
197C22×C4#13511yescharacteristic
198C22×C4#13511yescharacteristic
199C22×C4#13711yescharacteristic
200C22×C4#13711yescharacteristic
201C22×C4#13711yescharacteristic
202C22×C4#13711yescharacteristic
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C42—22no—
230C42—22no—
231C42—41no—
232C42—41no—
233C42—82no—
234C42—82no—
235C42—82no—
236C42—82no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#38611yescharacteristic
242C23#45111yescharacteristic
243C23#45211yescharacteristic
244C23#45211yescharacteristic
245C23#45211yescharacteristic
246C23#45311yescharacteristic
247C23#45411yescharacteristic
248Q8—162no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#41012yes—
309C2×C4#41611yescharacteristic
310C2×C4#41911yescharacteristic
311C2×C4#42112yes—
312C2×C4C4×Q1611yescharacteristic
313C2×C4#44111yescharacteristic
314C2×C4#46211yescharacteristic
315C2×C4#46411yescharacteristic
316C2×C4#48511yescharacteristic
317C2×C4#48511yescharacteristic
318C8—82no—
319C8—82no—
320C8—161no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22#163011yescharacteristic
334C22#163211yescharacteristic
335C22#163611yescharacteristic
336C22#163811yescharacteristic
337C22#187311yescharacteristic
338C22#187811yescharacteristic
339C22#188311yescharacteristic
340C22#189711yescharacteristic
341C22#189911yescharacteristic
342C22#190411yescharacteristic
343C22#190611yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—82no—
353C4—82no—
354C4—82no—
355C4—82no—
356C4—162no—
357C4—161no—
358C4—161no—
359C4#151512yes—
360C4#152112yes—
361C2—21no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2#897411yescharacteristic
366C2#915311yescharacteristic
367C2#917711yescharacteristic
368C2#923611yescharacteristic
369C2#925711yescharacteristic
370C2#932911yescharacteristic
371C2#936711yescharacteristic
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
10. #1987C4semidirect
11. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#331492
#139362
#99082
#96792
#87072
#70392
#70282