Finite Group Databasebeta

← #142008

Subgroups — #142008

Order 512.

702 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 118 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#15135C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7250C214yesmaximal
5C2×#399—21no—
6C2×#389—21no—
7C2×#384C411yescharacteristic, complemented
8C2×#380—21no—
9C2×#380—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1445—21no—
16#1444C412yescomplemented
17#1439C2212yes—
18C4⋊#95C411yescharacteristic, complemented
19#1280—21no—
20C24×C4D411yescharacteristic
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4C2×C412yes—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4D411yescharacteristic
30C22×C22⋊C4C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C4×C8—41no—
35#399—42no—
36#399—42no—
37#389—44no—
38#384—41no—
39#384—41no—
40#380—24no—
41#380—44no—
42#380—42no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—41no—
49C2×#95C2×C412yes—
50C2×#95C2×C411yescharacteristic
51C2×#95D411yescharacteristic
52#342—24no—
53#342—42no—
54#336—41no—
55#323—22no—
56#323Q814yes—
57C25C2×D411yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4SD1612yes—
71C23×C4Q1612yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—82no—
90C2×C4⋊C4—82no—
91C2×C4⋊C4—82no—
92C2×C4⋊C4—82no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—82no—
107C2×C22⋊C4D812yes—
108C2×C22⋊C4SD1612yes—
109C2×C22⋊C4C2×Q812yes—
110C2×C22⋊C4#4112yes—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—42no—
116C2×C42—42no—
117C22⋊C8—24no—
118C22⋊C8—42no—
119C4×C8—82no—
120#95—41no—
121#95—41no—
122#95—42no—
123#95—44no—
124#95—44no—
125#95—44no—
126#95—42no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24—41no—
135C24C2×C22⋊C411yescharacteristic
136C24#12411yescharacteristic
137C24C4⋊Q811yescharacteristic
138C24C2×D811yescharacteristic
139C24C2×SD1611yescharacteristic
140C24C2×SD1611yescharacteristic
141C24C2×Q1611yescharacteristic
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—84no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4#9911yescharacteristic
197C22×C4#9911yescharacteristic
198C22×C4#10212yes—
199C22×C4#10212yes—
200C22×C4#10312yes—
201C22×C4#10312yes—
202C22×C4#10412yes—
203C22×C4C4×Q812yes—
204C22×C4C22⋊Q812yes—
205C2×C8—44no—
206C2×C8—82no—
207C2×C8—82no—
208C4⋊C4—48no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4—88no—
212C4⋊C4—84no—
213C4⋊C4—88no—
214C4⋊C4—82no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—84no—
222C22⋊C4—84no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—81no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23#39111yescharacteristic
268C23C2×#9911yescharacteristic
269C23#41011yescharacteristic
270C23C2×#10211yescharacteristic
271C23C2×#10311yescharacteristic
272C23C2×#10411yescharacteristic
273C23#42111yescharacteristic
274C23#48611yescharacteristic
275C23#48711yescharacteristic
276C23#49811yescharacteristic
277C23#50011yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—162no—
329C2×C4—162no—
330C2×C4—162no—
331C2×C4—162no—
332C2×C4#32812yes—
333C2×C4#32812yes—
334C2×C4#35212yes—
335C2×C4#35312yes—
336C8—84no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
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—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22C2×#32811yescharacteristic
376C22#150211yescharacteristic
377C22#151811yescharacteristic
378C22#151911yescharacteristic
379C22#198611yescharacteristic
380C22#199011yescharacteristic
381C22#199611yescharacteristic
382C22C2×#35211yescharacteristic
383C22#212211yescharacteristic
384C22C2×#35311yescharacteristic
385C22#212911yescharacteristic
386C4—22no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—84no—
393C4—164no—
394C4—162no—
395C4#135214yes—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—41no—
401C2—41no—
402C2—41no—
403C2—41no—
404C2#988411yescharacteristic
405C2C2×#135211yescharacteristic
406C2#1171711yescharacteristic
407C2#1187411yescharacteristic
408C2#1187711yescharacteristic
409C2#1329211yescharacteristic
410C2#1329811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#384C4semidirect
16. #1444C4semidirect
18. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151352
#108462
C4×#3232
#72502