Finite Group Databasebeta

← #135209

Subgroups — #135209

Order 512.

770 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 106 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#13785C211yesmaximal, characteristic
2#8301C212yesmaximal
3#7189C212yesmaximal, complemented
4#7189C212yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#390C411yescharacteristic
7C2×#390C411yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×#376C411yescharacteristic
10#1779—22no—
11C2×#324C2214yes—
12C2×#323—22no—
13#1437—22no—
14C4⋊#95C2211yescharacteristic
15C4⋊#95C2211yescharacteristic
16C22⋊#95—22no—
17C23×D4C2×C411yescharacteristic
18C24×C4D412yes—
19C2×C4×D4—22no—
20C2×C4×D4—22no—
21C2×C4×D4—42no—
22C2×C4×D4—42no—
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4C2311yescharacteristic
26C22×C22⋊C4—22no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4D412yes—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C2×C22⋊C8—24no—
34C2×C22⋊C8—42no—
35C2×C22⋊C8—42no—
36#390—22no—
37#390—22no—
38#390—22no—
39#390C814yescomplemented
40C4×C4⋊C4—42no—
41#376—42no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47#324—24no—
48#324—24no—
49#323—44no—
50C25C22⋊C412yes—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4—41no—
56C22×D4—41no—
57C22×D4—41no—
58C22×D4—41no—
59C22×D4C2×C812yes—
60C22×D4M1612yes—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—42no—
69C23×C4—82no—
70C23×C4—82no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D412yes—
75C22×C8—44no—
76C4×D4—48no—
77C4×D4—48no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4C2×C812yes—
87C2×C4⋊C4M1612yes—
88C2×C4⋊C4#4112yes—
89C2×C4⋊C4#4112yes—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—28no—
93C2×C22⋊C4—28no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—82no—
96C2×C22⋊C4—82no—
97C2×C22⋊C4—82no—
98C2×C22⋊C4—82no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42—42no—
110C2×C42—82no—
111C2×C42C2×C812yes—
112C2×C42M1612yes—
113C22⋊C8—44no—
114C22⋊C8—44no—
115C22⋊C8—84no—
116C22⋊C8—84no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—82no—
121#95—82no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—42no—
129C24—42no—
130C24—42no—
131C24—42no—
132C24C22⋊C814yes—
133C24C2×C22⋊C411yescharacteristic
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—42no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
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—82no—
195C22×C4—82no—
196C22×C4—88no—
197C22×C4—88no—
198C22×C4#9912yes—
199C22×C4#9912yes—
200C22×C4#11711yescharacteristic
201C22×C4#11711yescharacteristic
202C22×C4#12314yes—
203C22×C4C22×C811yescharacteristic
204C22×C4C2×M1611yescharacteristic
205C22×C4C2×M1611yescharacteristic
206C22×C4C2×M1611yescharacteristic
207C2×C8—84no—
208C2×C8—84no—
209C2×C8—84no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—48no—
217C22⋊C4—88no—
218C22⋊C4—88no—
219C42—24no—
220C42—24no—
221C42—42no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42—84no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—44no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—44no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23#38111yescharacteristic
254C23C2×C22⋊C811yescharacteristic
255C23#40711yescharacteristic
256C23C2×#9911yescharacteristic
257C23C2×#9911yescharacteristic
258C23#41311yescharacteristic
259C23#41311yescharacteristic
260D4—44no—
261D4—44no—
262D4—44no—
263D4—44no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—88no—
302C2×C4—88no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#43111yescharacteristic
318C2×C4#43111yescharacteristic
319C2×C4#43211yescharacteristic
320C2×C4#43211yescharacteristic
321C8—164no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—48no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—84no—
339C22—84no—
340C22—84no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22#145911yescharacteristic
346C22#145911yescharacteristic
347C22#146511yescharacteristic
348C22#146511yescharacteristic
349C22#181711yescharacteristic
350C22#181911yescharacteristic
351C22#182211yescharacteristic
352C22#212711yescharacteristic
353C22#212711yescharacteristic
354C22#213411yescharacteristic
355C22#213511yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—44no—
361C4—42no—
362C4—42no—
363C4—82no—
364C4—82no—
365C4—82no—
366C4—82no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—84no—
372C2#870011yescharacteristic
373C2#1147911yescharacteristic
374C2#1148111yescharacteristic
375C2#1153311yescharacteristic
376C2#1153511yescharacteristic
377C2#1285811yescharacteristic
378C2#1286311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
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
3. #7189C2semidirect
4. #7189C2semidirect
39. #390C8semidirect

Maximal subgroups

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

subgroupindex
#137852
#83012
#71892