Finite Group Databasebeta

← #140939

Subgroups — #140939

Order 512.

626 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 92 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#15050C211yesmaximal, characteristic
2C2×#1351C212yesmaximal
3C2×#1344C212yesmaximal
4#10801C211yesmaximal, characteristic
5C4×#323C211yesmaximal, characteristic
6C2×#397C2211yescharacteristic
7C2×#393C2211yescharacteristic
8C2×#391C411yescharacteristic, complemented
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1446C411yescharacteristic, complemented
15#1445C411yescharacteristic, complemented
16#1443—21no—
17#1440—21no—
18#1434—21no—
19#1434C411yescharacteristic, complemented
20C22⋊#95—21no—
21C22⋊#95—21no—
22#1351—24no—
23#1344—24no—
24#1280—21no—
25C23×Q8C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—22no—
36C2×C4×C8—41no—
37#397—22no—
38#397—22no—
39#393—22no—
40#393—22no—
41#391—41no—
42#391—41no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—41no—
56C2×#95—41no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95D411yescharacteristic
61#342—24no—
62#342—22no—
63#342—22no—
64#336—41no—
65#323—22no—
66#323D414yes—
67C25C2×D411yescharacteristic
68C22×Q8—22no—
69C22×Q8—42no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—81no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—41no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—81no—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C22×C8—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—81no—
113C2×C22⋊C4C2×D412yes—
114C2×C22⋊C4#4112yes—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—42no—
120C2×C42—81no—
121C22⋊C8—24no—
122C22⋊C8—42no—
123C4×C8—82no—
124#95—22no—
125#95—22no—
126#95—41no—
127#95—41no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—81no—
138#95—81no—
139#95—82no—
140#95—82no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24—41no—
149C24C2×C22⋊C411yescharacteristic
150C24#12411yescharacteristic
151C24#12711yescharacteristic
152C24#13512yes—
153C24#13512yes—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—82no—
157C2×Q8—84no—
158C2×Q8—84no—
159C2×Q8—82no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4#9911yescharacteristic
230C22×C4#9911yescharacteristic
231C22×C4#10412yes—
232C22×C4C4×D412yes—
233C22×C4#12112yes—
234C2×C8—44no—
235C2×C8—82no—
236C2×C8—82no—
237C4⋊C4—82no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C42—42no—
250C42—81no—
251C42—81no—
252C42—81no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23#39011yescharacteristic
284C23C2×#9911yescharacteristic
285C23#41011yescharacteristic
286C23#41611yescharacteristic
287C23#41611yescharacteristic
288C23C2×#10411yescharacteristic
289C23#42111yescharacteristic
290C23#48812yes—
291C23#49512yes—
292Q8—84no—
293Q8—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—84no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—84no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—84no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—162no—
360C2×C4—162no—
361C2×C4—162no—
362C2×C4—162no—
363C2×C4#35412yes—
364C8—84no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—81no—
390C22—81no—
391C22—82no—
392C22—82no—
393C22#150011yescharacteristic
394C22#150111yescharacteristic
395C22#151611yescharacteristic
396C22#152111yescharacteristic
397C22#195811yescharacteristic
398C22#196411yescharacteristic
399C22#198011yescharacteristic
400C22#212011yescharacteristic
401C22#212111yescharacteristic
402C22C2×#35411yescharacteristic
403C22#212911yescharacteristic
404C4—22no—
405C4—41no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4—161no—
413C4—161no—
414C4—164no—
415C4—161no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—42no—
421C2—42no—
422C2#968111yescharacteristic
423C2#1172811yescharacteristic
424C2#1173111yescharacteristic
425C2#1184211yescharacteristic
426C2#1184511yescharacteristic
427C2#1321911yescharacteristic
428C2#1322411yescharacteristic
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
8. C2×#391C4semidirect
14. #1446C4semidirect
15. #1445C4semidirect
19. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#150502
C2×#13512
C2×#13442
#108012
C4×#3232