Finite Group Databasebeta

← #142209

Subgroups — #142209

Order 512.

622 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 90 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#14969C211yesmaximal, characteristic
2#10801C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7250C212yesmaximal
5#7250C212yesmaximal
6C2×#389C411yescharacteristic, complemented
7C2×#387—21no—
8C2×#385—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1445—21no—
15#1444C411yescharacteristic, complemented
16#1444C411yescharacteristic, complemented
17#1439—21no—
18#1439C2211yescharacteristic
19#1439C2211yescharacteristic
20#1434C411yescharacteristic, complemented
21C22⋊#95—21no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
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×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38#389—41no—
39#389—41no—
40#387—44no—
41#385—44no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—41no—
52C2×#95—41no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57#342—22no—
58#342—42no—
59#342Q814yes—
60#336—41no—
61#323—24no—
62C25C2×D411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—41no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4—81no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—81no—
114C2×C22⋊C4C2×Q812yes—
115C2×C22⋊C4#4112yes—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—42no—
121C2×C42—81no—
122C4⋊C8—82no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#95—24no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—81no—
136#95—82no—
137#95—82no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24C2×C22⋊C411yescharacteristic
147C24C22⋊Q812yes—
148C24#13512yes—
149C24AGL(1,ℤ/8)11yescharacteristic
150C24#13711yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
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—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
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—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—44no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4#9911yescharacteristic
222C22×C4#9911yescharacteristic
223C22×C4C4×Q812yes—
224C22×C4#12411yescharacteristic
225C22×C4C4⋊Q811yescharacteristic
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—48no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—84no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C42—42no—
248C42—81no—
249C42—81no—
250C42—81no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23#39111yescharacteristic
282C23C2×#9911yescharacteristic
283C23#41011yescharacteristic
284C23#41611yescharacteristic
285C23#41711yescharacteristic
286C23#42111yescharacteristic
287C23#42111yescharacteristic
288C23#47812yes—
289C23#47912yes—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—84no—
329C2×C4—81no—
330C2×C4—84no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
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—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—84no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—162no—
358C2×C4—162no—
359C2×C4—162no—
360C2×C4—162no—
361C2×C4#35312yes—
362C8—162no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—22no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—81no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22#151211yescharacteristic
392C22#151311yescharacteristic
393C22#151611yescharacteristic
394C22#152111yescharacteristic
395C22#198911yescharacteristic
396C22#199511yescharacteristic
397C22#199611yescharacteristic
398C22#212011yescharacteristic
399C22#212111yescharacteristic
400C22C2×#35311yescharacteristic
401C22#212911yescharacteristic
402C4—22no—
403C4—41no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—82no—
408C4—82no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4—164no—
414C4—161no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—42no—
420C2—42no—
421C2#991311yescharacteristic
422C2#1182111yescharacteristic
423C2#1182611yescharacteristic
424C2#1184211yescharacteristic
425C2#1184311yescharacteristic
426C2#1329711yescharacteristic
427C2#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
6. C2×#389C4semidirect
15. #1444C4semidirect
16. #1444C4semidirect
20. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#149692
#108012
#85572
#72502