Finite Group Databasebeta

← #128786

Subgroups — #128786

Order 512.

686 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 124 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#14317C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3C4×#342C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5#7250C212yesmaximal
6C2×#399C2211yescharacteristic
7C2×#389C411yescharacteristic, complemented
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445C411yescharacteristic, complemented
15#1442C411yescharacteristic, complemented
16#1441—21no—
17#1439—21no—
18#1439C2211yescharacteristic
19#1436—21no—
20#1434C411yescharacteristic, complemented
21C22⋊#95—21no—
22#1352C418yescomplemented
23#1280—21no—
24C24×C4D411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—22no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4D411yescharacteristic
33C22×C22⋊C4C2311yescharacteristic
34C22×C42—41no—
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C22⋊C8—22no—
38C2×C4×C8—41no—
39#399—22no—
40#399C2×C414yes—
41#389—41no—
42#389—41no—
43C4×C22⋊C4—24no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—42no—
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×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60#342—24no—
61#342—22no—
62#342C2×C414yes—
63#336—41no—
64#323—22no—
65#323C2×C414yes—
66C25C2×D411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4SD1612yes—
82C23×C4Q1612yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—42no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4C4212yescomplemented
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—81no—
112C2×C22⋊C4—81no—
113C2×C22⋊C4—81no—
114C2×C22⋊C4—81no—
115C2×C22⋊C4C22×C412yes—
116C2×C22⋊C4#4112yes—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—44no—
122C2×C42—81no—
123C2×C42—82no—
124C2×C42—82no—
125C2×C42—81no—
126C22⋊C8—24no—
127C22⋊C8—42no—
128C4×C8—82no—
129#95—21no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—44no—
134#95—41no—
135#95—41no—
136#95—42no—
137#95—42no—
138#95—81no—
139#95—81no—
140#95—81no—
141#95—81no—
142#95—81no—
143#95C4212yescomplemented
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24C2×C22⋊C411yescharacteristic
153C24C4×D412yes—
154C24C2×SD1611yescharacteristic
155C24C2×Q1611yescharacteristic
156C24#13512yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—44no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
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—81no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—84no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4#9911yescharacteristic
227C22×C4#9911yescharacteristic
228C22×C4#10312yes—
229C22×C4#10312yes—
230C22×C4#10412yes—
231C22×C4#10412yes—
232C22×C4C2×C4211yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4C4×D412yes—
235C2×C8—44no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—84no—
241C4⋊C4—82no—
242C4⋊C4—84no—
243C22⋊C4—28no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C42—42no—
253C42—81no—
254C42—81no—
255C42—81no—
256C42—88no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—21no—
269C23—42no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23C4×C22⋊C411yescharacteristic
288C23C2×#9911yescharacteristic
289C23#41011yescharacteristic
290C23C2×#10311yescharacteristic
291C23#41611yescharacteristic
292C23C2×#10411yescharacteristic
293C23C2×#10411yescharacteristic
294C23C4×SD1612yes—
295C23C4×Q1612yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—48no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—81no—
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—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
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—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#32812yes—
369C2×C4#32812yes—
370C8—84no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22C2×#32811yescharacteristic
400C22#150011yescharacteristic
401C22#150111yescharacteristic
402C22#150311yescharacteristic
403C22C4×#9911yescharacteristic
404C22C4×#10411yescharacteristic
405C22C4×#10311yescharacteristic
406C22#212111yescharacteristic
407C22#212311yescharacteristic
408C22#212711yescharacteristic
409C22#212811yescharacteristic
410C4—22no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—82no—
418C4—88no—
419C4—82no—
420C4—81no—
421C4—161no—
422C4—161no—
423C2—21no—
424C2—21no—
425C2—21no—
426C2—21no—
427C2—42no—
428C2—42no—
429C2C4×#32811yescharacteristic
430C2#1171611yescharacteristic
431C2#1171811yescharacteristic
432C2#1172911yescharacteristic
433C2#1173211yescharacteristic
434C2#1262111yescharacteristic
435C2#1262811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C22×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×#389C4semidirect
14. #1445C4semidirect
15. #1442C4semidirect
20. #1434C4semidirect
22. #1352C4semidirect
97. C2×C4⋊C4C42semidirect
143. #95C42semidirect

Maximal subgroups

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

subgroupindex
#143172
C2×#13522
C4×#3422
C4×#3232
#72502