Finite Group Databasebeta

← #101155

Subgroups — #101155

Order 512.

676 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 92 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#20526C211yesmaximal, characteristic
2#14969C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7627C212yesmaximal, complemented
5#7036C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#389C2211yescharacteristic
8C2×#387—21no—
9C2×#385—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1752—24no—
12#1749—21no—
13#1749C2212yes—
14#1749C2212yes—
15C2×#323—21no—
16#1445—21no—
17#1444C411yescharacteristic
18#1444C411yescharacteristic
19#1439—21no—
20#1439C411yescharacteristic
21#1439C411yescharacteristic
22#1434C2211yescharacteristic
23#1280—21no—
24#1271—24no—
25C24×C4D411yescharacteristic
26C23×C8—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42C2311yescharacteristic
38C2×C22⋊C8—24no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—22no—
44C2×#97—22no—
45C2×#97—22no—
46#389—22no—
47#389—22no—
48#387—44no—
49#385—44no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95D411yescharacteristic
63#336—22no—
64#336—22no—
65#336D414yes—
66#323—24no—
67C25C22⋊C411yescharacteristic
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—42no—
77C23×C4—81no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—24no—
85C22×C8—28no—
86C22×C8—42no—
87C22×C8—82no—
88C22×C8—82no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4—81no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—81no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—42no—
118C2×C42—81no—
119C2×C42C2×D412yes—
120C2×C42#4112yes—
121C22⋊C8—28no—
122C22⋊C8—24no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126#97—82no—
127#97—82no—
128#97—82no—
129#95—24no—
130#95—22no—
131#95—22no—
132#95—41no—
133#95—41no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95—42no—
139#95—81no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24#10112yes—
149C24#10412yes—
150C24#10412yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
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—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—41no—
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#10412yes—
216C22×C4C4×D412yes—
217C22×C4#12011yescharacteristic
218C22×C4#12112yes—
219C22×C4#12311yescharacteristic
220C2×C8—28no—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—88no—
225C2×C8—82no—
226C2×C8—82no—
227C4⋊C4—24no—
228C4⋊C4—42no—
229C4⋊C4—48no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C4⋊C4—82no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—48no—
237C22⋊C4—84no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—81no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23#33014yes—
269C23#38611yescharacteristic
270C23C2×#10111yescharacteristic
271C23#41311yescharacteristic
272C23C2×#10411yescharacteristic
273C23C2×#10411yescharacteristic
274C23C2×#10411yescharacteristic
275C23#42111yescharacteristic
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—84no—
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—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#45312yes—
340C8—48no—
341C8—84no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—24no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22C2×#33011yescharacteristic
366C22#152411yescharacteristic
367C22#153111yescharacteristic
368C22#153211yescharacteristic
369C22#187111yescharacteristic
370C22#188711yescharacteristic
371C22#189111yescharacteristic
372C22#189611yescharacteristic
373C22#190111yescharacteristic
374C22#190911yescharacteristic
375C22#191311yescharacteristic
376C4—22no—
377C4—42no—
378C4—41no—
379C4—42no—
380C4—42no—
381C4—82no—
382C4—82no—
383C4—81no—
384C4—82no—
385C4—81no—
386C4—161no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—44no—
392C2#903111yescharacteristic
393C2#912111yescharacteristic
394C2#918311yescharacteristic
395C2#940911yescharacteristic
396C2#945511yescharacteristic
397C2#949111yescharacteristic
398C2#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. #7627C2semidirect
5. #7036C2semidirect

Maximal subgroups

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

subgroupindex
#205262
#149692
#81522
#76272
#70362