Finite Group Databasebeta

← #127021

Subgroups — #127021

Order 512.

878 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 154 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×#1465C211yesmaximal, characteristic
2C2×#1459C211yesmaximal, characteristic
3#13941C211yesmaximal, characteristic
4#7189C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#407—21no—
7C22×C22⋊C8—21no—
8C2×#390C412yes—
9C2×#382C411yescharacteristic
10C2×C4×C4⋊C4C411yescharacteristic
11#1465—22no—
12#1465C414yescomplemented
13#1459—22no—
14#1459C414yescomplemented
15C2×#324C2212yes—
16C2×#324C2212yes—
17C2×#323—21no—
18C2×#323—21no—
19#1439—21no—
20#1439—21no—
21C4⋊#95C2212yes—
22C22⋊#95—21no—
23C22⋊#95—21no—
24C23×D4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—41no—
28C23×C8—41no—
29C2×C4×D4—42no—
30C2×C4×D4—41no—
31C2×C4×D4—41no—
32C2×C4×D4C2×C412yes—
33C2×C4×D4C2×C412yes—
34C2×C4×D4D412yes—
35C2×C4×D4Q812yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4—22no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4D411yescharacteristic
42C22×C22⋊C4D411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C412yes—
45#407—24no—
46#407—42no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—42no—
51C2×C22⋊C8—41no—
52C2×C22⋊C8—41no—
53#390—44no—
54#382—41no—
55#382—41no—
56C4×C4⋊C4—42no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62#324—22no—
63#324—22no—
64#324C2×C414yes—
65#324C2×C414yes—
66#323—24no—
67#323—24no—
68C25C22⋊C411yescharacteristic
69C25C22⋊C411yescharacteristic
70C22×D4—22no—
71C22×D4—22no—
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4C4212yes—
77C22×D4C4⋊C412yes—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—82no—
87C23×C4—82no—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C4⋊C412yes—
93C23×C4C4⋊C412yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C2×M16—44no—
98C2×M16—84no—
99C22×C8—42no—
100C22×C8—44no—
101C22×C8—42no—
102C22×C8—84no—
103C4×D4—42no—
104C4×D4—42no—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108C4×D4—42no—
109C4×D4—42no—
110C4×D4—42no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C22×C411yescharacteristic
119C2×C4⋊C4C22×C411yescharacteristic
120C2×C4⋊C4C2×D411yescharacteristic
121C2×C4⋊C4C2×Q811yescharacteristic
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—44no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—82no—
131C2×C22⋊C4—82no—
132C2×C22⋊C4C22⋊C412yes—
133C2×C22⋊C4C22⋊C412yes—
134C2×C22⋊C4C4⋊C412yes—
135C2×C22⋊C4C4⋊C412yes—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—24no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42C4212yes—
143C2×C42C4⋊C412yes—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—42no—
147C22⋊C8—44no—
148C22⋊C8—42no—
149C22⋊C8—44no—
150#95—42no—
151#95—42no—
152#95—41no—
153#95—41no—
154#95—82no—
155#95—82no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—42no—
167C24—42no—
168C24—42no—
169C24—42no—
170C24#9512yes—
171C24#9512yes—
172C24#10012yes—
173C24#10112yes—
174C24C2×C22⋊C411yescharacteristic
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4#9512yes—
232C22×C4#9512yes—
233C22×C4#9911yescharacteristic
234C22×C4#9911yescharacteristic
235C22×C4#9912yes—
236C22×C4C2×C4211yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241C22×C4C2×C4⋊C411yescharacteristic
242C22×C4#13112yes—
243C22×C4#13112yes—
244M16—88no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—88no—
248C2×C8—84no—
249C2×C8—84no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—48no—
255C4⋊C4—84no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—88no—
261C22⋊C4—88no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—84no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—44no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—82no—
299C23—82no—
300C23—82no—
301C23—82no—
302C23#34314yes—
303C23C2×#9511yescharacteristic
304C23#40811yescharacteristic
305C23#40811yescharacteristic
306C23C2×#9911yescharacteristic
307C23C2×#9911yescharacteristic
308C23C2×#10011yescharacteristic
309C23C2×#10111yescharacteristic
310D4—48no—
311D4—48no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#34212yes—
363C2×C4#34212yes—
364C2×C4#40512yes—
365C2×C4#42412yes—
366C8—84no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—24no—
377C22—44no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—44no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22#147111yescharacteristic
395C22#147111yescharacteristic
396C22#147211yescharacteristic
397C22#147211yescharacteristic
398C22#172911yescharacteristic
399C22C2×#34211yescharacteristic
400C22C2×#34311yescharacteristic
401C22#212712yes—
402C22#213212yes—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—42no—
412C4—84no—
413C4—162no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—44no—
419C2—82no—
420C2#791511yescharacteristic
421C2#1156612yes—
422C2#1157212yes—
423C2#1258211yescharacteristic
424C2#1258511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7189C2semidirect
12. #1465C4semidirect
14. #1459C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14652
C2×#14592
#139412
#71892