Finite Group Databasebeta

← #479716

Subgroups — #479716

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15235C212yesmaximal
2#11378C212yesmaximal
3#11346C211yesmaximal, characteristic
4#11325C212yesmaximal
5#11325C212yesmaximal
6#11316C212yesmaximal
7#11315C212yesmaximal
8#11256C211yesmaximal, characteristic
9#11256C211yesmaximal, characteristic
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2212yes—
13C2×#382C2212yes—
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×#340C2211yescharacteristic
16C2×#340C2212yes—
17C2×#340C2211yescharacteristic
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#336C2212yes—
23C2×#336C2212yes—
24C2×#336C2211yescharacteristic
25C2×#336C2211yescharacteristic
26#1443—22no—
27#1443—22no—
28#1442—22no—
29#1442—22no—
30C4⋊#95C2212yes—
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C23×C8C2312yes—
36C23×C8C2311yescharacteristic
37C23×C8C2311yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4C2312yes—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C42C2311yescharacteristic
45C22×C42C2312yes—
46C2×#107—42no—
47C2×#107—42no—
48C2×#107—41no—
49C2×#107—41no—
50C2×#106—24no—
51C2×#106—42no—
52C2×#106—42no—
53#382—42no—
54#382—42no—
55C4×C4⋊C4—41no—
56C4×C4⋊C4—41no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95D412yes—
70#340—22no—
71#340—22no—
72#340—24no—
73#340—24no—
74#340—42no—
75#340—42no—
76#340—42no—
77#336—24no—
78#336—24no—
79#336—41no—
80#336—41no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4C2×D412yes—
86C23×C4C2×D412yes—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2411yescharacteristic
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8#4112yes—
105C22×C8#4112yes—
106C22×C8#4114yes—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146#107—44no—
147#107—44no—
148#106—48no—
149#95—42no—
150#95—44no—
151#95—44no—
152#95—44no—
153#95—44no—
154#95—44no—
155#95—44no—
156#95—82no—
157#95—82no—
158#95—82no—
159#95—82no—
160#95—82no—
161#95—82no—
162C24C22×D411yescharacteristic
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
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—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4#13512yes—
215C22×C4C2×#4112yes—
216C22×C4C2×#4111yescharacteristic
217C22×C4C2×#4111yescharacteristic
218C22×C4H(5,𝔽2)11yescharacteristic
219C22×C4H(5,𝔽2)12yes—
220C22×C4H(5,𝔽2)11yescharacteristic
221C22×C42−1+411yescharacteristic
222C22×C42−1+411yescharacteristic
223C22×C42−1+412yes—
224C22×C42−1+412yes—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C42—42no—
261C42—42no—
262C42—42no—
263C42—42no—
264C42—41no—
265C42—41no—
266C42—42no—
267C23#53411yescharacteristic
268C23#53612yes—
269C23#54711yescharacteristic
270C23#54711yescharacteristic
271C23#55012yes—
272C23C2×#13511yescharacteristic
273C23#57512yes—
274C23#57511yescharacteristic
275C23#57611yescharacteristic
276C23#57712yes—
277C23#57811yescharacteristic
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#53812yes—
331C2×C4#53911yescharacteristic
332C2×C4#53911yescharacteristic
333C2×C4#53912yes—
334C2×C4#53912yes—
335C2×C4#54112yes—
336C2×C4#54112yes—
337C2×C4#55212yes—
338C2×C4#55212yes—
339C2×C4#55511yescharacteristic
340C2×C4#55911yescharacteristic
341C2×C4H(3,𝔽4)11yescharacteristic
342C2×C4#56311yescharacteristic
343C2×C4#56312yes—
344C2×C4#56312yes—
345C2×C4#56312yes—
346C2×C4#56311yescharacteristic
347C2×C4#56311yescharacteristic
348C8—42no—
349C8—42no—
350C8—44no—
351C22#272112yes—
352C22#272112yes—
353C22#272411yescharacteristic
354C22#273311yescharacteristic
355C22#281711yescharacteristic
356C22#319111yescharacteristic
357C22#319711yescharacteristic
358C22#320311yescharacteristic
359C22#320411yescharacteristic
360C22#323212yes—
361C22#324012yes—
362C22#324112yes—
363C22#324212yes—
364C22#334111yescharacteristic
365C22#334211yescharacteristic
366C22#334812yes—
367C22#335012yes—
368C22#335211yescharacteristic
369C22#335511yescharacteristic
370C22#339412yes—
371C22#340112yes—
372C22#340212yes—
373C22#340312yes—
374C4—42no—
375C4—81no—
376C4—82no—
377C4—82no—
378C4—162no—
379C4—162no—
380C4—162no—
381C4—162no—
382C4#276911yescharacteristic
383C4#276911yescharacteristic
384C4#277112yes—
385C4#283312yes—
386C4#283511yescharacteristic
387C4#283511yescharacteristic
388C2#1722811yescharacteristic
389C2#3238512yes—
390C2#3238712yes—
391C2#3239312yes—
392C2#3239512yes—
393C2#3246211yescharacteristic
394C2#3247111yescharacteristic
395C2#3267111yescharacteristic
396C2#3267711yescharacteristic
397C2#3279111yescharacteristic
398C2#3279511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#152352
#113782
#113462
#113252
#113162
#113152
#112562