Finite Group Databasebeta

← #114112

Subgroups — #114112

Order 512.

1290 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 194 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
1C2×#2504C211yesmaximal, characteristic
2C2×#1824C212yesmaximal, complemented
3C2×#1519C212yesmaximal
4#10846C211yesmaximal, characteristic, complemented
5#7907C211yesmaximal, characteristic, complemented
6C22×#121C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9#2504—24no—
10#2504—24no—
11C2×#397—22no—
12C2×#390C412yescomplemented
13C2×#386—22no—
14C2×#382C412yescomplemented
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1824—24no—
18#1824—24no—
19C2×#342—22no—
20C2×#340—24no—
21#1519—24no—
22#1519—24no—
23C2×#327—24no—
24#1280—22no—
25C23×D4—21no—
26C23×D4C2×C411yescharacteristic
27C24×C4—21no—
28C24×C4C2311yescharacteristic
29C23×C8—22no—
30C2×#121—22no—
31C2×#121—22no—
32C2×#121—22no—
33C2×#121—22no—
34C2×#121—22no—
35C2×#121D414yes—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D414yes—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4D412yes—
50C22×C42C2×C412yes—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8D414yes—
57#397—216no—
58#390—24no—
59#390—24no—
60#386—216no—
61#382—24no—
62#382—24no—
63#380—22no—
64#380—22no—
65#380—22no—
66#380D414yes—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C2×#95—24no—
70C2×#95—22no—
71C2×#95D412yes—
72#342—44no—
73#340—216no—
74#327—216no—
75C25—21no—
76C25C2×D411yescharacteristic
77C22×D4—24no—
78C22×D4—21no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—44no—
82C22×D4—42no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4—44no—
86C22×D4—42no—
87C22×D4—42no—
88C22×D4C22⋊C412yes—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—24no—
92C23×C4—44no—
93C23×C4—42no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—42no—
97C23×C4—84no—
98C23×C4C22⋊C414yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4#4112yes—
106C23×C4#4112yes—
107C22×C8—24no—
108C22×C8—48no—
109C22×C8—42no—
110C22×C8—42no—
111#121—24no—
112#121—24no—
113#121—24no—
114#121—24no—
115#121—42no—
116#121—42no—
117#121—42no—
118#121—42no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—88no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—44no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163#95—216no—
164#95—24no—
165#95—24no—
166#95—42no—
167C24—22no—
168C24—42no—
169C24—42no—
170C24—44no—
171C24—44no—
172C24—41no—
173C24—41no—
174C24—42no—
175C24—42no—
176C24C2×C22⋊C411yescharacteristic
177C24#12012yes—
178C24#12112yes—
179C24#12411yescharacteristic
180C24#12711yescharacteristic
181C24#13514yes—
182C24AGL(1,ℤ/8)12yes—
183C24#13712yes—
184C2×D4—24no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—48no—
189C2×D4—48no—
190C2×D4—42no—
191C2×D4—44no—
192C2×D4—84no—
193C2×D4—84no—
194C2×D4—84no—
195C2×D4—84no—
196C2×D4—82no—
197C2×D4—82no—
198C2×D4—82no—
199C2×D4—81no—
200C2×D4—81no—
201C22×C4—24no—
202C22×C4—28no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—48no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—88no—
227C22×C4—88no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C4×D412yes—
236C22×C4C4×D412yes—
237C22×C4#12012yes—
238C22×C4#12012yes—
239C22×C4#12012yes—
240C22×C4#12112yes—
241C22×C4#12112yes—
242C22×C4#12112yes—
243C22×C4#12112yes—
244C22×C4#12112yes—
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4#12411yescharacteristic
248C22×C4#12711yescharacteristic
249C2×C8—48no—
250C2×C8—44no—
251C2×C8—84no—
252C2×C8—84no—
253C4⋊C4—28no—
254C4⋊C4—24no—
255C4⋊C4—48no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—84no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—48no—
266C22⋊C4—816no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C42—44no—
270C42—44no—
271C23—22no—
272C23—24no—
273C23—22no—
274C23—24no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—48no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—42no—
286C23—44no—
287C23—44no—
288C23—82no—
289C23—82no—
290C23—88no—
291C23—88no—
292C23—82no—
293C23—82no—
294C23—81no—
295C23—81no—
296C23—82no—
297C23#37911yescharacteristic
298C23#39011yescharacteristic
299C23#39011yescharacteristic
300C23#39212yes—
301C23#39412yes—
302C23#41612yes—
303C23#41712yes—
304C23#44914yes—
305C23#45214yes—
306C23#46314yes—
307C23#46414yes—
308C23#48812yes—
309C23#48912yes—
310C23#49512yes—
311C23#49611yescharacteristic
312C23#49711yescharacteristic
313D4—84no—
314D4—84no—
315D4—84no—
316D4—88no—
317D4—84no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—88no—
343C2×C4—88no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—168no—
353C2×C4#38612yes—
354C2×C4#38612yes—
355C2×C4#39212yes—
356C2×C4#39412yes—
357C2×C4#41014yes—
358C8—84no—
359C22—24no—
360C22—24no—
361C22—28no—
362C22—28no—
363C22—22no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—44no—
372C22—44no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—82no—
379C22—88no—
380C22—82no—
381C22—82no—
382C22—162no—
383C22—162no—
384C22—164no—
385C22#143811yescharacteristic
386C22#178212yes—
387C22#178812yes—
388C22#179512yes—
389C22#195811yescharacteristic
390C22#195911yescharacteristic
391C22#198312yes—
392C22#201212yes—
393C22#201512yes—
394C22#204214yes—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—82no—
399C4—84no—
400C4—84no—
401C4—164no—
402C4#189914yes—
403C2—22no—
404C2—21no—
405C2—21no—
406C2—22no—
407C2—22no—
408C2—24no—
409C2—162no—
410C2#837814yes—
411C2#1096611yescharacteristic
412C2#1100611yescharacteristic
413C2#1100911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1824C2semidirect
4. #10846C2semidirect
5. #7907C2semidirect
12. C2×#390C4semidirect
14. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#18242
C2×#15192
#108462
#79072