Finite Group Databasebeta

← #128154

Subgroups — #128154

Order 512.

543 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21820C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1467C211yesmaximal, characteristic
4#7323C214yesmaximal, complemented
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7C2×#432—21no—
8C2×#432C411yescharacteristic
9C2×#431C411yescharacteristic
10#1985C412yes—
11#1985C2212yes—
12#1984—21no—
13#1984—21no—
14#1821—21no—
15#1817—21no—
16#1817—21no—
17#1801—21no—
18#1801C2212yes—
19#1784C412yes—
20C4×#101—21no—
21C4×#101C412yes—
22#1467—21no—
23#1467—21no—
24#1467C412yes—
25#1467C2212yes—
26C2×#324—21no—
27C2×#324C412yes—
28C23×Q8C2×C411yescharacteristic
29C2×C4×Q8—41no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8Q811yescharacteristic
40C2×C4×Q8C2311yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4Q811yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47#432—22no—
48#432—22no—
49#432—41no—
50#432—41no—
51#432—41no—
52#432—41no—
53#432—41no—
54#432—41no—
55#431—41no—
56#431—41no—
57#431—41no—
58#431—41no—
59#423—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8C2×C412yes—
63C2×#101—22no—
64C2×#101C2×C414yes—
65#407—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C4×M16—22no—
70C2×#97—41no—
71C2×#97C2×C412yes—
72C2×C4×C8—21no—
73#343—22no—
74#343—22no—
75#336—22no—
76#336—22no—
77#336—22no—
78#336—22no—
79#324—22no—
80#324—22no—
81#324—22no—
82#324C2×C414yes—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8C4211yescharacteristic
88C22×Q8C22⋊C411yescharacteristic
89C22×Q8C4⋊C411yescharacteristic
90C22×Q8C22×C411yescharacteristic
91C23×C4C4211yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C2×M16—24no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—42no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C4211yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C4⋊C411yescharacteristic
129C2×C4⋊C4C4⋊C411yescharacteristic
130C2×C4⋊C4C4⋊C411yescharacteristic
131C2×C4⋊C4C4⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×Q811yescharacteristic
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42C4211yescharacteristic
149C2×C42C22⋊C411yescharacteristic
150C2×C42C22⋊C411yescharacteristic
151C2×C42C22⋊C411yescharacteristic
152C2×C42C22⋊C411yescharacteristic
153C2×C42C22⋊C411yescharacteristic
154C2×C42C4⋊C411yescharacteristic
155C2×C42C4⋊C411yescharacteristic
156C2×C42C4⋊C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C4⋊C8—42no—
161C4⋊C8—42no—
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166#101—44no—
167#101—44no—
168C22⋊C8—24no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176#97—42no—
177#97—42no—
178#97—42no—
179#97—42no—
180C4×C8—42no—
181C4×C8—42no—
182C24#9511yescharacteristic
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—48no—
196C2×Q8—48no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4#9511yescharacteristic
220C22×C4#9511yescharacteristic
221C22×C4#9511yescharacteristic
222C22×C4#9511yescharacteristic
223C22×C4#9511yescharacteristic
224C22×C4#9511yescharacteristic
225C22×C4#9511yescharacteristic
226C22×C4#9911yescharacteristic
227C22×C4#9911yescharacteristic
228C22×C4#9911yescharacteristic
229C22×C4#9911yescharacteristic
230C22×C4C2×C4211yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237M16—44no—
238M16—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—84no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23#34211yescharacteristic
287C23#34211yescharacteristic
288C23C2×#9511yescharacteristic
289Q8—44no—
290Q8—44no—
291Q8—44no—
292Q8—44no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#34211yescharacteristic
340C2×C4#34211yescharacteristic
341C2×C4#34412yes—
342C2×C4C2×#9911yescharacteristic
343C2×C4C2×#9911yescharacteristic
344C2×C4#41312yes—
345C8—82no—
346C8—84no—
347C8—84no—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22#139512yes—
359C22C2×#34211yescharacteristic
360C22#174211yescharacteristic
361C22C2×#34411yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—22no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4#213312yes—
378C2—22no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2#810811yescharacteristic
383C2#1258611yescharacteristic
384C2C2×#139511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7323C2semidirect

Maximal subgroups

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

subgroupindex
#218202
#207272
C2×#14672
#73232