Finite Group Databasebeta

← #479294

Subgroups — #479294

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15221C212yesmaximal
2#11377C211yesmaximal, characteristic
3#11372C212yesmaximal
4#11346C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11315C212yesmaximal
7#11256C211yesmaximal, characteristic
8#11219C212yesmaximal
9#11208C212yesmaximal
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C2×#389C2212yes—
13C2×#389C2212yes—
14C2×#382C2212yes—
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×#340C2212yes—
17C2×#340C2212yes—
18C2×#340C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#336C2211yescharacteristic
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28#1445—22no—
29#1444—22no—
30#1442—22no—
31#1442—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C23×C8C2312yes—
36C23×C8C2311yescharacteristic
37C23×C8C2311yescharacteristic
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C42C2311yescharacteristic
47C22×C42C2312yes—
48C2×#107—41no—
49C2×#107—41no—
50C2×#107—42no—
51C2×#107—41no—
52C2×#107Q814yes—
53#389—24no—
54#389—42no—
55#389—42no—
56#389—42no—
57#382—24no—
58#382—42no—
59C4×C4⋊C4—41no—
60C4×C4⋊C4—41no—
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×#95—22no—
70C2×#95—22no—
71C2×#95D412yes—
72#340—24no—
73#340—22no—
74#340—24no—
75#340—41no—
76#340—42no—
77#340—42no—
78#340—42no—
79#340—42no—
80#336—41no—
81#336—41no—
82#336—42no—
83#336—42no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4C2×D412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2411yescharacteristic
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C2×Q812yes—
109C22×C8#4112yes—
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—22no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—82no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4C2×Q814yes—
144C2×C4⋊C4#4114yes—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155#107—44no—
156#107—44no—
157#95—42no—
158#95—44no—
159#95—44no—
160#95—44no—
161#95—44no—
162#95—44no—
163#95—44no—
164#95—44no—
165#95—82no—
166#95—82no—
167#95—82no—
168C24C22×D411yescharacteristic
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
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—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4#13512yes—
221C22×C4C22×Q811yescharacteristic
222C22×C4C2×#4111yescharacteristic
223C22×C4C2×#4112yes—
224C22×C4H(5,𝔽2)11yescharacteristic
225C22×C4H(5,𝔽2)12yes—
226C22×C4H(5,𝔽2)11yescharacteristic
227C22×C4H(5,𝔽2)11yescharacteristic
228C22×C42−1+412yes—
229C22×C42−1+411yescharacteristic
230C22×C42−1+412yes—
231C2×C8—22no—
232C2×C8—24no—
233C2×C8—22no—
234C2×C8—24no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—88no—
267C4⋊C4—88no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—41no—
273C42—42no—
274C42—41no—
275C23#53411yescharacteristic
276C23#53612yes—
277C23#54612yes—
278C23D4×Q811yescharacteristic
279C23#55011yescharacteristic
280C23C2×#13511yescharacteristic
281C23#57512yes—
282C23#57511yescharacteristic
283C23#57612yes—
284C23#57811yescharacteristic
285C23#57811yescharacteristic
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
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—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4#53911yescharacteristic
339C2×C4#53912yes—
340C2×C4#53912yes—
341C2×C4#54012yes—
342C2×C4#54112yes—
343C2×C4#54311yescharacteristic
344C2×C4#54412yes—
345C2×C4#55212yes—
346C2×C4#55411yescharacteristic
347C2×C4#55511yescharacteristic
348C2×C4#55712yes—
349C2×C4#56011yescharacteristic
350C2×C4#56211yescharacteristic
351C2×C4#56212yes—
352C2×C4#56312yes—
353C2×C4#56311yescharacteristic
354C2×C4#56312yes—
355C2×C4#56311yescharacteristic
356C8—42no—
357C8—42no—
358C8—82no—
359C22#269412yes—
360C22#269911yescharacteristic
361C22#271312yes—
362C22#273211yescharacteristic
363C22#281711yescharacteristic
364C22#319111yescharacteristic
365C22#319711yescharacteristic
366C22#319911yescharacteristic
367C22#320211yescharacteristic
368C22#323212yes—
369C22#324012yes—
370C22#324312yes—
371C22#324312yes—
372C22#330512yes—
373C22#331312yes—
374C22#331512yes—
375C22#332212yes—
376C22#338311yescharacteristic
377C22#338912yes—
378C22#339011yescharacteristic
379C22#339311yescharacteristic
380C22#340211yescharacteristic
381C22#340312yes—
382C4—42no—
383C4—81no—
384C4—84no—
385C4—84no—
386C4—82no—
387C4—82no—
388C4—162no—
389C4—162no—
390C4#276911yescharacteristic
391C4#279112yes—
392C4#280911yescharacteristic
393C4#283411yescharacteristic
394C4#283612yes—
395C4#284111yescharacteristic
396C2#1719711yescharacteristic
397C2#3157912yes—
398C2#3159312yes—
399C2#3171911yescharacteristic
400C2#3173311yescharacteristic
401C2#3214512yes—
402C2#3215312yes—
403C2#3265311yescharacteristic
404C2#3266211yescharacteristic
405C2#3279111yescharacteristic
406C2#3279211yescharacteristic
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
#152212
#113772
#113722
#113462
#113282
#113152
#112562
#112192
#112082