Finite Group Databasebeta

← #100982

Subgroups — #100982

Order 512.

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

normal subgroups only

This group has 568 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#15016C211yesmaximal, characteristic
3#8718C212yesmaximal, complemented
4#8152C211yesmaximal, characteristic
5#7036C212yesmaximal, complemented
6C2×#399C411yescharacteristic
7C2×#397C411yescharacteristic
8C2×#382C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1924—24no—
11#1797—21no—
12#1779—21no—
13#1749C2212yes—
14#1749C2212yes—
15C2×#323—21no—
16#1445C411yescharacteristic
17#1445C411yescharacteristic
18#1442—21no—
19#1440—21no—
20#1436—21no—
21#1434C2211yescharacteristic
22C22⋊#95—21no—
23#1295—24no—
24#1280—21no—
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×#97D414yes—
42#399—42no—
43#397—42no—
44#382—22no—
45#382—22no—
46C4×C22⋊C4—22no—
47C4×C22⋊C4—22no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60C2×#95D411yescharacteristic
61#336—22no—
62#336—22no—
63#336—22no—
64#323—24no—
65C25C22⋊C411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—24no—
83C22×C8—42no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—81no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—42no—
108C2×C42—81no—
109C2×C42C2×D412yes—
110C2×C42#4112yes—
111C4⋊C8—82no—
112C22⋊C8—24no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—82no—
117C22⋊C8—82no—
118#97—44no—
119#97—44no—
120#95—22no—
121#95—22no—
122#95—41no—
123#95—41no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—42no—
131#95—41no—
132#95—81no—
133#95—81no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24—41no—
144C24#10012yes—
145C24C2×C22⋊C411yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
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—42no—
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—42no—
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—81no—
205C22×C4—81no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4#10412yes—
209C22×C4C4×D412yes—
210C22×C4#12011yescharacteristic
211C22×C4#12112yes—
212C22×C4#12311yescharacteristic
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—82no—
218C2×C8—82no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—84no—
225C22⋊C4—84no—
226C42—24no—
227C42—42no—
228C42—42no—
229C42—81no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23#38611yescharacteristic
258C23C2×#10011yescharacteristic
259C23#41311yescharacteristic
260C23C2×#10411yescharacteristic
261C23#42111yescharacteristic
262C23#42111yescharacteristic
263C23#42111yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—84no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—84no—
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—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4#45812yes—
327C8—84no—
328C8—84no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#152911yescharacteristic
353C22#153011yescharacteristic
354C22#153311yescharacteristic
355C22#153511yescharacteristic
356C22#187111yescharacteristic
357C22#188411yescharacteristic
358C22#189011yescharacteristic
359C22#189711yescharacteristic
360C22#190011yescharacteristic
361C22#191111yescharacteristic
362C22#191511yescharacteristic
363C4—22no—
364C4—41no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—161no—
372C4—161no—
373C4—161no—
374C2—21no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—82no—
379C2#901211yescharacteristic
380C2#912411yescharacteristic
381C2#918911yescharacteristic
382C2#939011yescharacteristic
383C2#944711yescharacteristic
384C2#948111yescharacteristic
385C2#950711yescharacteristic
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
3. #8718C2semidirect
5. #7036C2semidirect

Maximal subgroups

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

subgroupindex
#219062
#150162
#87182
#81522
#70362