Finite Group Databasebeta

← #413185

Subgroups — #413185

Order 512.

2816 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2816 conjugacy classes of subgroups — too many to draw. Only its 314 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#35001C211yesmaximal, characteristic
2C2×#2501C212yesmaximal, complemented
3C2×#2022C218yesmaximal, complemented
4C2×#1964C212yesmaximal, complemented
5#13904C211yesmaximal, characteristic, complemented
6C4×#342C211yesmaximal, characteristic, complemented
7C2×#534C2214yes—
8C22×#123C2218yes—
9C22×#121C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C22×#99C2218yes—
12#2856—22no—
13#2656—22no—
14#2539—24no—
15#2501—28no—
16#2501—22no—
17#2501—28no—
18#2433—22no—
19C2×#399—24no—
20C2×#390—22no—
21C2×#384—22no—
22C2×#381C2214yes—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#2022—232no—
27#1964—28no—
28C2×#342C2214yes—
29#1445—22no—
30C4⋊#95—22no—
31C23×D4D412yes—
32C23×D4C2314yes—
33C24×C4C2312yes—
34C24×C4C2311yescharacteristic
35#534—216no—
36#534—48no—
37C2×#127—22no—
38C2×#123—216no—
39C2×#123—28no—
40C2×#123—216no—
41C2×#123—232no—
42C2×#123—416no—
43C2×#121—22no—
44C2×#121—28no—
45C2×#121—22no—
46C2×#121—22no—
47C2×#121—24no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—22no—
51C2×#121—24no—
52C2×#120—28no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D414yescomplemented
63C22×C22⋊C4—28no—
64C22×C22⋊C4D414yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C42—22no—
69C22×C42D412yes—
70C2×#99—216no—
71C2×#99—232no—
72#399—232no—
73#399—28no—
74#392—28no—
75#390—24no—
76#390—28no—
77#390—216no—
78#384—22no—
79#384—216no—
80#381—216no—
81#380—24no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4D418yescomplemented
86C2×#95—22no—
87C2×#95—28no—
88C2×#95—21no—
89C2×#95D412yes—
90#342D4116yescomplemented
91C25C2×D414yes—
92C25C2411yescharacteristic
93C22×D4—24no—
94C22×D4—24no—
95C22×D4—24no—
96C22×D4—24no—
97C22×D4—24no—
98C22×D4—216no—
99C22×D4—24no—
100C22×D4—216no—
101C22×D4—28no—
102C22×D4—22no—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—24no—
106C22×D4—48no—
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—28no—
112C23×C4—22no—
113C23×C4—28no—
114C23×C4—22no—
115C23×C4—44no—
116C23×C4C2×D414yes—
117C23×C4C2×D414yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D414yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D414yes—
125C23×C4#4112yes—
126#127—84no—
127#123—464no—
128#123—432no—
129#123—432no—
130#123—432no—
131#121—416no—
132#121—44no—
133#121—44no—
134#121—44no—
135#121—44no—
136#121—44no—
137#121—44no—
138#121—44no—
139#121—44no—
140#121—416no—
141#120—432no—
142C2×C4⋊C4—232no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—432no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—232no—
167C2×C22⋊C4—216no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—432no—
175C2×C22⋊C4C2×D418yes—
176C2×C22⋊C4C2×D4116yes—
177C2×C22⋊C4C2×D418yes—
178C2×C42—22no—
179C2×C42—28no—
180C2×C42—22no—
181C2×C42—48no—
182C2×C42—44no—
183#99—432no—
184#95—28no—
185#95—24no—
186#95—42no—
187#95—416no—
188C24—24no—
189C24—24no—
190C24—24no—
191C24—216no—
192C24—42no—
193C24—42no—
194C24—48no—
195C24—48no—
196C24C22×D412yes—
197C24C22×D411yescharacteristic
198C24C22×D411yescharacteristic
199C24C22×D412yes—
200C24C22×D414yes—
201C24C2×#4111yescharacteristic
202C24H(5,𝔽2)14yes—
203C2×D4—416no—
204C2×D4—416no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—48no—
209C2×D4—44no—
210C2×D4—48no—
211C2×D4—48no—
212C2×D4—48no—
213C2×D4—48no—
214C2×D4—48no—
215C2×D4—416no—
216C2×D4—416no—
217C2×D4—416no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—48no—
221C2×D4—84no—
222C2×D4—88no—
223C2×D4—88no—
224C2×D4—88no—
225C22×C4—24no—
226C22×C4—28no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—216no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—216no—
237C22×C4—216no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—416no—
255C22×C4—48no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4#12014yes—
261C22×C4#12012yes—
262C22×C4#12011yescharacteristic
263C22×C4#12014yes—
264C22×C4#12011yescharacteristic
265C22×C4#12112yes—
266C22×C4#12114yes—
267C22×C4#12114yes—
268C22×C4#12112yes—
269C22×C4#12712yes—
270C22×C4#12718yes—
271C22×C4#12712yes—
272C4⋊C4—464no—
273C4⋊C4—416no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—84no—
280C22⋊C4—28no—
281C22⋊C4—216no—
282C22⋊C4—216no—
283C22⋊C4—464no—
284C22⋊C4—432no—
285C22⋊C4—48no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C22⋊C4#127116yescomplemented
289C42—416no—
290C42—44no—
291C42—82no—
292C23—22no—
293C23—24no—
294C23—22no—
295C23—24no—
296C23—41no—
297C23—42no—
298C23—41no—
299C23—41no—
300C23—42no—
301C23—48no—
302C23—41no—
303C23—48no—
304C23—48no—
305C23—48no—
306C23—44no—
307C23—48no—
308C23—48no—
309C23—48no—
310C23—48no—
311C23—48no—
312C23—832no—
313C23—84no—
314C23—84no—
315C23C2×#12012yes—
316C23C2×#12011yescharacteristic
317C23C2×#12111yescharacteristic
318C23C2×#12112yes—
319C23C2×#12711yescharacteristic
320C23C2×#12712yes—
321C23C2×#12714yes—
322C23#53512yes—
323C23#53514yes—
324C23D4218yes—
325C23D4214yes—
326C23#55014yes—
327D4—88no—
328D4—84no—
329D4—84no—
330D4—88no—
331D4—88no—
332D4—88no—
333D4—88no—
334D4—1616no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—216no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—432no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—416no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—84no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—88no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—88no—
368C2×C4—82no—
369C2×C4#39212yes—
370C2×C4#39214yes—
371C2×C4#39212yes—
372C2×C4#45814yes—
373C22—22no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—24no—
382C22—216no—
383C22—24no—
384C22—44no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—816no—
389C22—84no—
390C22—84no—
391C22—84no—
392C22—816no—
393C22—168no—
394C22C2×#39211yescharacteristic
395C22#243314yes—
396C22#243312yes—
397C22#254112yes—
398C22#254114yes—
399C22#265711yescharacteristic
400C22#286711yescharacteristic
401C22C2×#45812yes—
402C22#302812yes—
403C4—22no—
404C4—24no—
405C4—416no—
406C4—48no—
407C4—84no—
408C4—84no—
409C4—162no—
410C4#202214yescomplemented
411C2—24no—
412C2—21no—
413C2—24no—
414C2—21no—
415C2—22no—
416C2—164no—
417C2#1514211yescharacteristic
418C2C2×#202211yescharacteristic
419C2#2344611yescharacteristic
420C2#2577214yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC25C24
prime radical#35001C2
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2501C2semidirect
3. C2×#2022C2semidirect
4. C2×#1964C2semidirect
5. #13904C2semidirect
6. C4×#342C2semidirect
62. C22×C4⋊C4D4semidirect
85. C4×C22⋊C4D4semidirect
90. #342D4semidirect
288. C22⋊C4#127semidirect
410. C4#2022semidirect

Maximal subgroups

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

subgroupindex
#350012
C2×#25012
C2×#20222
C2×#19642
#139042
C4×#3422