Finite Group Databasebeta

← #138304

Subgroups — #138304

Order 512.

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

normal subgroups only

This group has 686 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
1C2×C4×#101C211yesmaximal, characteristic
2#14880C211yesmaximal, characteristic
3C2×#1346C212yesmaximal
4#8303C211yesmaximal, characteristic
5#7251C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C22×#101—21no—
8C2×C4×M16—21no—
9C2×#398C411yescharacteristic
10C2×#398C2211yescharacteristic
11C2×#395—21no—
12C2×#389C411yescharacteristic
13#1817—21no—
14C4×#101—24no—
15C2×#343C2212yes—
16C2×#324—21no—
17C2×#324C2212yes—
18#1441—21no—
19#1441C411yescharacteristic
20#1441C2211yescharacteristic
21#1436—21no—
22#1436—21no—
23#1434C411yescharacteristic
24#1346—24no—
25C23×Q8D411yescharacteristic
26C22×M16—22no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—41no—
30C2×C4×Q8—41no—
31C2×C4×Q8—42no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4C2311yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42D411yescharacteristic
43C2×#101—24no—
44C2×#101—24no—
45C2×#101—24no—
46C2×C22⋊C8—22no—
47C4×M16—24no—
48C4×M16—42no—
49C2×#97—41no—
50C2×C4×C8—41no—
51#398—22no—
52#398—22no—
53#398—42no—
54#395—44no—
55#389—41no—
56#389—41no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95C2×C411yescharacteristic
67C2×#95C2×C411yescharacteristic
68#343—22no—
69#343D414yes—
70#336—41no—
71#336—41no—
72#324—22no—
73#324—22no—
74#324—42no—
75C22×Q8—22no—
76C22×Q8—22no—
77C22×Q8—21no—
78C22×Q8—21no—
79C22×Q8—22no—
80C22×Q8—22no—
81C22×Q8—41no—
82C22×Q8—41no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C2×M16—24no—
95C2×M16—24no—
96C2×M16—24no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C22×C8—42no—
101C22×C8—42no—
102C4×Q8—44no—
103C4×Q8—44no—
104C4×Q8—44no—
105C4×Q8—44no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4#4112yes—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—22no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—41no—
143C2×C42—41no—
144#101—216no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147#97—44no—
148C4×C8—44no—
149#95—24no—
150#95—22no—
151#95—41no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—41no—
157#95—41no—
158#95—42no—
159#95—42no—
160#95—42no—
161C24C2×C22⋊C411yescharacteristic
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
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—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4#9911yescharacteristic
233C22×C4#9911yescharacteristic
234C22×C4#9912yes—
235C22×C4#10412yes—
236C22×C4#10412yes—
237C22×C4C4×D412yes—
238C22×C4#12011yescharacteristic
239C22×C4#12112yes—
240C22×C4#12311yescharacteristic
241C22×C4#13512yes—
242C22×C4AGL(1,ℤ/8)12yes—
243M16—28no—
244M16—44no—
245M16—44no—
246M16—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—82no—
260C4⋊C4—84no—
261C4⋊C4—84no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23#38611yescharacteristic
277C23C2×#9911yescharacteristic
278C23C2×#9911yescharacteristic
279C23#41611yescharacteristic
280C23#41811yescharacteristic
281C23C2×#10411yescharacteristic
282C23C2×#10411yescharacteristic
283Q8—48no—
284Q8—48no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#35612yes—
358C2×C4#44912yes—
359C2×C4#48212yes—
360C8—48no—
361C8—162no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22#150111yescharacteristic
375C22#150111yescharacteristic
376C22#150711yescharacteristic
377C22#150711yescharacteristic
378C22#187511yescharacteristic
379C22#189611yescharacteristic
380C22#189811yescharacteristic
381C22#212011yescharacteristic
382C22#212311yescharacteristic
383C22C2×#35611yescharacteristic
384C22#213311yescharacteristic
385C4—22no—
386C4—22no—
387C4—22no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—82no—
395C4—82no—
396C4—82no—
397C4—161no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#915011yescharacteristic
403C2#1173611yescharacteristic
404C2#1174011yescharacteristic
405C2#1177811yescharacteristic
406C2#1178211yescharacteristic
407C2#1308111yescharacteristic
408C2#1309411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×C4×#1012
#148802
C2×#13462
#83032
#72512