Finite Group Databasebeta

← #101167

Subgroups — #101167

Order 512.

556 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 84 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#21906C211yesmaximal, characteristic
2#15045C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7627C212yesmaximal, complemented
5#7036C212yesmaximal, complemented
6C2×#389C2211yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8#1924—24no—
9#1797—21no—
10#1779—21no—
11#1749C2212yes—
12#1749C2212yes—
13C2×#323—21no—
14#1442—21no—
15#1442C411yescharacteristic
16#1442C411yescharacteristic
17#1439—21no—
18#1439C411yescharacteristic
19#1439C411yescharacteristic
20#1434—21no—
21#1434C2211yescharacteristic
22C22⋊#95—21no—
23#1280—21no—
24#1271—24no—
25C24×C4D411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42C2311yescharacteristic
35C2×C4⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×#97—22no—
41C2×#97—22no—
42#389—22no—
43#389—22no—
44C4×C22⋊C4—22no—
45C4×C22⋊C4—22no—
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×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#336—22no—
60#336—22no—
61#336D414yes—
62#323—24no—
63C25C22⋊C411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4—42no—
73C23×C4—81no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—24no—
81C22×C8—42no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—81no—
99C2×C22⋊C4—81no—
100C2×C22⋊C4—81no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42—81no—
105C2×C42C2×D412yes—
106C2×C42#4112yes—
107C4⋊C8—82no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—82no—
113C22⋊C8—82no—
114#97—82no—
115#97—82no—
116#95—22no—
117#95—22no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—41no—
125#95—41no—
126#95—81no—
127#95—81no—
128#95—81no—
129#95—81no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—41no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24#10412yes—
141C24C2×C22⋊C411yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
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—81no—
201C22×C4—81no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4#10112yes—
205C22×C4C4×D412yes—
206C22×C4#12011yescharacteristic
207C22×C4#12112yes—
208C22×C4#12311yescharacteristic
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—82no—
214C2×C8—82no—
215C4⋊C4—24no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C4⋊C4—84no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—84no—
222C22⋊C4—84no—
223C42—42no—
224C42—42no—
225C42—42no—
226C42—81no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—81no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23#38611yescharacteristic
255C23C2×#10111yescharacteristic
256C23#41311yescharacteristic
257C23C2×#10411yescharacteristic
258C23#42111yescharacteristic
259C23#42111yescharacteristic
260C23#42111yescharacteristic
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—84no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—84no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
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—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4#45512yes—
324C8—84no—
325C8—84no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—44no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#152811yescharacteristic
350C22#153211yescharacteristic
351C22#153511yescharacteristic
352C22#153711yescharacteristic
353C22#187111yescharacteristic
354C22#188511yescharacteristic
355C22#189111yescharacteristic
356C22#189711yescharacteristic
357C22#190111yescharacteristic
358C22#191011yescharacteristic
359C22#191311yescharacteristic
360C4—22no—
361C4—42no—
362C4—41no—
363C4—42no—
364C4—42no—
365C4—82no—
366C4—81no—
367C4—81no—
368C4—161no—
369C4—161no—
370C4—161no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—82no—
376C2#902911yescharacteristic
377C2#912511yescharacteristic
378C2#918511yescharacteristic
379C2#941111yescharacteristic
380C2#945311yescharacteristic
381C2#949411yescharacteristic
382C2#951811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7627C2semidirect
5. #7036C2semidirect

Maximal subgroups

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

subgroupindex
#219062
#150452
#81522
#76272
#70362