Finite Group Databasebeta

← #101201

Subgroups — #101201

Order 512.

584 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 96 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#15165C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7641C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C2×#396—21no—
7C2×#389C2211yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9#1924—24no—
10#1797C2212yes—
11#1797C2212yes—
12#1779—21no—
13#1749—21no—
14C2×#323—21no—
15#1444C411yescharacteristic
16#1442C411yescharacteristic
17#1441—21no—
18#1439C411yescharacteristic
19#1439C411yescharacteristic
20#1436—21no—
21C4⋊#95C2211yescharacteristic
22C22⋊#95—21no—
23#1280—21no—
24#1271—24no—
25C24×C4D411yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42C2311yescharacteristic
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—22no—
44#396—44no—
45#389—22no—
46#389—22no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60#336—22no—
61#336—22no—
62#336D414yes—
63#323—42no—
64C25C22⋊C411yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—81no—
74C23×C4—81no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—24no—
82C22×C8—42no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—81no—
109C2×C22⋊C4—81no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—81no—
113C2×C42—81no—
114C2×C42C2×D412yes—
115C2×C42#4112yes—
116C4⋊C8—82no—
117C4⋊C8—82no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#97—82no—
125#95—22no—
126#95—22no—
127#95—41no—
128#95—41no—
129#95—42no—
130#95—42no—
131#95—41no—
132#95—41no—
133#95—42no—
134#95—81no—
135#95—81no—
136#95—81no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24#10112yes—
148C24C2×C22⋊C411yescharacteristic
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
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—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—42no—
189C22×C4—42no—
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—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4C4×D412yes—
209C22×C4#12011yescharacteristic
210C22×C4#12112yes—
211C22×C4#12311yescharacteristic
212C22×C4#13512yes—
213C22×C4AGL(1,ℤ/8)12yes—
214C22×C4AGL(1,ℤ/8)11yescharacteristic
215C22×C4#13712yes—
216C22×C4#13711yescharacteristic
217C2×C8—44no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C22⋊C4—44no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—84no—
234C22⋊C4—84no—
235C42—42no—
236C42—42no—
237C42—42no—
238C42—81no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23#38611yescharacteristic
267C23C2×#10111yescharacteristic
268C23#41311yescharacteristic
269C23#41611yescharacteristic
270C23#41711yescharacteristic
271C23#41811yescharacteristic
272C23#41911yescharacteristic
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—84no—
308C2×C4—84no—
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—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4#45212yes—
335C2×C4#46812yes—
336C2×C4#46912yes—
337C2×C4#48512yes—
338C8—84no—
339C8—162no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#155211yescharacteristic
364C22#155311yescharacteristic
365C22#155611yescharacteristic
366C22#155911yescharacteristic
367C22#187311yescharacteristic
368C22#188411yescharacteristic
369C22#189111yescharacteristic
370C22#189511yescharacteristic
371C22#190111yescharacteristic
372C22#191111yescharacteristic
373C22#191311yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—42no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—161no—
385C4—161no—
386C2—21no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—82no—
391C2#903411yescharacteristic
392C2#913811yescharacteristic
393C2#920111yescharacteristic
394C2#938611yescharacteristic
395C2#944911yescharacteristic
396C2#948911yescharacteristic
397C2#951711yescharacteristic
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. #7641C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219062
#151652
#85572
#76412
#70392