Finite Group Databasebeta

← #101019

Subgroups — #101019

Order 512.

744 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 112 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#21934C211yesmaximal, characteristic
2#15125C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#8342C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C2×#407—21no—
8C2×#393C412yes—
9C2×#380—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1851—24no—
12#1797—21no—
13#1797C2212yes—
14#1797C2212yes—
15C2×#333—24no—
16C2×#323—21no—
17#1446C412yes—
18#1442—21no—
19#1439—21no—
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8C2×C412yes—
24C24×C4D411yescharacteristic
25C22×M16—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—44no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C22⋊C4—22no—
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42C2311yescharacteristic
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8D414yes—
40#407—42no—
41#407—42no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45#393—44no—
46#380—44no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—21no—
52C2×#95—22no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C412yes—
55C2×#95D411yescharacteristic
56#336—22no—
57#336—22no—
58#336—22no—
59#333—48no—
60#323—42no—
61C25C22⋊C411yescharacteristic
62C22×Q8—24no—
63C22×Q8—44no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—82no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4SD1612yes—
79C23×C4Q1612yes—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—44no—
84C2×M16—82no—
85C2×M16—82no—
86C22×C8—24no—
87C22×C8—42no—
88C4⋊Q8—28no—
89C4⋊Q8—44no—
90C4⋊Q8—48no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4—82no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42C2×D412yes—
116C2×C42#4112yes—
117C4⋊C8—44no—
118C4⋊C8—44no—
119C4⋊C8—82no—
120C22⋊C8—44no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#95—22no—
126#95—22no—
127#95—42no—
128#95—44no—
129#95—44no—
130#95—41no—
131#95—41no—
132#95—82no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24#10312yes—
144C24C2×C22⋊C411yescharacteristic
145C2×Q8—28no—
146C2×Q8—44no—
147C2×Q8—84no—
148C2×Q8—84no—
149C2×Q8—88no—
150C2×Q8—88no—
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—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4#10012yes—
199C22×C4#10312yes—
200C22×C4C4×D412yes—
201C22×C4#12011yescharacteristic
202C22×C4#12112yes—
203C22×C4#12311yescharacteristic
204C22×C4C2×SD1611yescharacteristic
205C22×C4C2×Q1611yescharacteristic
206C22×C4AGL(1,ℤ/8)11yescharacteristic
207C22×C4AGL(1,ℤ/8)11yescharacteristic
208C22×C4#13711yescharacteristic
209C22×C4#13712yes—
210C22×C4#13711yescharacteristic
211M16—88no—
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C4⋊C4—44no—
219C4⋊C4—48no—
220C4⋊C4—44no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—44no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—88no—
227C42—24no—
228C42—42no—
229C42—42no—
230C42—81no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23#38611yescharacteristic
259C23C2×#10011yescharacteristic
260C23#41311yescharacteristic
261C23C2×#10311yescharacteristic
262C23#41711yescharacteristic
263C23#41711yescharacteristic
264C23#41911yescharacteristic
265Q8—88no—
266Q8—88no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4#45011yescharacteristic
317C2×C4#45111yescharacteristic
318C2×C4#45512yes—
319C2×C4B(4,𝔽2)12yes—
320C2×C4#46812yes—
321C2×C4#47012yes—
322C2×C4#48311yescharacteristic
323C2×C4#48411yescharacteristic
324C8—84no—
325C8—162no—
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#154811yescharacteristic
350C22#155011yescharacteristic
351C22#155911yescharacteristic
352C22#156011yescharacteristic
353C22#187211yescharacteristic
354C22#188511yescharacteristic
355C22#188911yescharacteristic
356C22#189411yescharacteristic
357C22#190011yescharacteristic
358C22#190911yescharacteristic
359C22#191411yescharacteristic
360C4—21no—
361C4—21no—
362C4—22no—
363C4—22no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—84no—
368C4—81no—
369C4—162no—
370C4#160112yes—
371C4#160212yes—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2#901311yescharacteristic
378C2#913511yescharacteristic
379C2#919111yescharacteristic
380C2#939911yescharacteristic
381C2#945111yescharacteristic
382C2#947111yescharacteristic
383C2#950411yescharacteristic
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. #8342C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219342
#151252
#85572
#83422
#70392