Finite Group Databasebeta

← #101039

Subgroups — #101039

Order 512.

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

normal subgroups only

This group has 672 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#15135C211yesmaximal, characteristic
3#8724C212yesmaximal, complemented
4#8557C211yesmaximal, characteristic
5#7039C212yesmaximal, complemented
6C2×#407—21no—
7C2×#399—21no—
8C2×#389—21no—
9C2×#384C2211yescharacteristic
10C2×#380—21no—
11C2×C4×C22⋊C4C2211yescharacteristic
12#1851—24no—
13#1797—21no—
14#1797C2212yes—
15#1797C2212yes—
16C2×#323—21no—
17#1445—21no—
18#1444C412yes—
19#1439C412yes—
20C4⋊#95C2211yescharacteristic
21#1296—24no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C22×M16—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4C2×C412yes—
30C22×C22⋊C4—22no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42C2311yescharacteristic
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8D414yes—
38#407—42no—
39#407—42no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43#399—44no—
44#389—44no—
45#384—22no—
46#384—22no—
47#380—44no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C412yes—
55C2×#95D411yescharacteristic
56#336—22no—
57#336—22no—
58#336—22no—
59#323—42no—
60C25C22⋊C411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4SD1612yes—
73C23×C4Q1612yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—44no—
78C2×M16—82no—
79C2×M16—82no—
80C22×C8—24no—
81C22×C8—42no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—82no—
98C2×C4⋊C4—82no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—82no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42C2×D412yes—
114C2×C42#4112yes—
115C4⋊C8—44no—
116C4⋊C8—44no—
117C4⋊C8—82no—
118C22⋊C8—44no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123#95—22no—
124#95—22no—
125#95—42no—
126#95—44no—
127#95—44no—
128#95—42no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24—41no—
139C24#10312yes—
140C24C2×C22⋊C411yescharacteristic
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4#10112yes—
189C22×C4#10312yes—
190C22×C4C4×D412yes—
191C22×C4#12011yescharacteristic
192C22×C4#12112yes—
193C22×C4#12311yescharacteristic
194C22×C4C2×SD1611yescharacteristic
195C22×C4C2×Q1611yescharacteristic
196C22×C4AGL(1,ℤ/8)11yescharacteristic
197C22×C4AGL(1,ℤ/8)12yes—
198C22×C4AGL(1,ℤ/8)11yescharacteristic
199C22×C4#13711yescharacteristic
200C22×C4#13711yescharacteristic
201M16—88no—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—84no—
211C4⋊C4—82no—
212C4⋊C4—88no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C22⋊C4—44no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—88no—
220C42—24no—
221C42—42no—
222C42—42no—
223C42—81no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23—81no—
251C23#38611yescharacteristic
252C23C2×#10111yescharacteristic
253C23#41311yescharacteristic
254C23C2×#10311yescharacteristic
255C23#41711yescharacteristic
256C23#41711yescharacteristic
257C23#41811yescharacteristic
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#45011yescharacteristic
308C2×C4#45111yescharacteristic
309C2×C4#45312yes—
310C2×C4#45812yes—
311C2×C4#46812yes—
312C2×C4#46912yes—
313C2×C4#48311yescharacteristic
314C2×C4#48411yescharacteristic
315C8—84no—
316C8—162no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—44no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#154911yescharacteristic
341C22#155011yescharacteristic
342C22#155711yescharacteristic
343C22#156111yescharacteristic
344C22#187211yescharacteristic
345C22#188711yescharacteristic
346C22#189011yescharacteristic
347C22#189511yescharacteristic
348C22#190111yescharacteristic
349C22#190911yescharacteristic
350C22#191411yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—84no—
359C4—81no—
360C4—162no—
361C4#161312yes—
362C4#161412yes—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—82no—
368C2#902111yescharacteristic
369C2#913711yescharacteristic
370C2#919111yescharacteristic
371C2#940511yescharacteristic
372C2#945511yescharacteristic
373C2#947411yescharacteristic
374C2#951011yescharacteristic
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. #8724C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219342
#151352
#87242
#85572
#70392