Finite Group Databasebeta

← #127828

Subgroups — #127828

Order 512.

623 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21820C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×C4×#101C211yesmaximal, characteristic
4#7284C214yesmaximal, complemented
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7C2×#432C411yescharacteristic
8C2×#431C411yescharacteristic
9C22×#101—21no—
10C2×C4×M16—21no—
11#1985C412yes—
12#1985C2212yes—
13#1984—21no—
14#1984—21no—
15#1821C412yes—
16#1817—21no—
17#1817—21no—
18#1801—21no—
19#1801C412yes—
20#1784—21no—
21C4×#101—21no—
22C4×#101—21no—
23C4×#101—21no—
24C4×#101C412yes—
25C4×#101C2212yes—
26C4×#101C2212yes—
27C2×#343C412yes—
28C23×Q8D411yescharacteristic
29C22×M16—22no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C2×C4×Q8Q811yescharacteristic
41C2×C4×Q8C2311yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4Q811yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—41no—
49#432—41no—
50#432—41no—
51#432—41no—
52#431—41no—
53#431—41no—
54#431—41no—
55#431—41no—
56#423—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8C2×C412yes—
59C2×#101—28no—
60C2×#101—22no—
61C2×#101—22no—
62C2×#101—22no—
63C2×#101—22no—
64#407—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—22no—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16—41no—
71C4×M16—41no—
72C2×#97—41no—
73C2×#97C2×C412yes—
74C2×C4×C8—21no—
75C2×C4×C8—41no—
76#343—22no—
77#343—22no—
78#343C2×C414yes—
79#343C2×C414yes—
80#336—22no—
81#336—22no—
82#336—22no—
83#336—22no—
84C22×Q8—24no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8C22⋊C411yescharacteristic
89C22×Q8C22⋊C411yescharacteristic
90C22×Q8C4⋊C411yescharacteristic
91C22×Q8C2×D411yescharacteristic
92C23×C4C4211yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C2×M16—28no—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—44no—
103C2×M16—42no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—42no—
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—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×Q8—41no—
122C4×Q8—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4C4211yescharacteristic
130C2×C4⋊C4C4211yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C4⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C4⋊C411yescharacteristic
137C2×C4⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C22×C411yescharacteristic
139C2×C4⋊C4C22×C411yescharacteristic
140C2×C4⋊C4C2×Q811yescharacteristic
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42C4211yescharacteristic
154C2×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C4⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—42no—
168C4⋊C8—42no—
169#101—28no—
170#101—28no—
171#101—44no—
172#101—44no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179#97—42no—
180#97—42no—
181#97—42no—
182#97—42no—
183C4×C8—42no—
184C4×C8—42no—
185C4×C8—42no—
186C4×C8—42no—
187C24#9511yescharacteristic
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—48no—
200C2×Q8—48no—
201C2×Q8#10414yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4#9511yescharacteristic
225C22×C4#9511yescharacteristic
226C22×C4#9511yescharacteristic
227C22×C4#9511yescharacteristic
228C22×C4#9511yescharacteristic
229C22×C4#9511yescharacteristic
230C22×C4#9511yescharacteristic
231C22×C4#9911yescharacteristic
232C22×C4#9911yescharacteristic
233C22×C4#9911yescharacteristic
234C22×C4#9911yescharacteristic
235C22×C4C2×C4211yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241C22×C4C2×C4⋊C411yescharacteristic
242M16—28no—
243M16—48no—
244M16—44no—
245M16—44no—
246M16—44no—
247C2×C8—44no—
248C2×C8—48no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—42no—
253C2×C8—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
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—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C23—22no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23#34211yescharacteristic
295C23#34211yescharacteristic
296C23C2×#9511yescharacteristic
297Q8—44no—
298Q8—44no—
299Q8—44no—
300Q8—44no—
301C2×C4—24no—
302C2×C4—24no—
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—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4#33912yes—
346C2×C4#33912yes—
347C2×C4#34211yescharacteristic
348C2×C4#34211yescharacteristic
349C2×C4C2×#9911yescharacteristic
350C2×C4C2×#9911yescharacteristic
351C2×C4C2×#10412yes—
352C2×C4#42112yes—
353C8—48no—
354C8—82no—
355C8—84no—
356C8—82no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22#139212yes—
367C22C2×#33911yescharacteristic
368C22#173311yescharacteristic
369C22C2×#34211yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4#213312yes—
386C2—22no—
387C2—21no—
388C2—21no—
389C2—41no—
390C2#805111yescharacteristic
391C2C2×#139211yescharacteristic
392C2#1258611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C2×#342
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #7284C2semidirect

Maximal subgroups

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

subgroupindex
#218202
#207272
C2×C4×#1012
#72842