Finite Group Databasebeta

← #475244

Subgroups — #475244

Order 512.

844 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#11358C212yesmaximal
3#11327C212yesmaximal
4#11231C211yesmaximal, characteristic
5#11231C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11208C212yesmaximal
8#10835C211yesmaximal, characteristic
9#10821C212yesmaximal
10C22×#106C2211yescharacteristic
11C22×#106C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C2×#389C2212yes—
15C2×#389C2212yes—
16C2×#382C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#336C2212yes—
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⋊C4D412yes—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C42C2312yes—
46C22×C42C2311yescharacteristic
47C2×#106—24no—
48C2×#106—41no—
49C2×#106—42no—
50C2×#106—41no—
51C2×C4⋊C8—41no—
52C2×C4⋊C8—41no—
53C2×C4⋊C8Q814yes—
54C2×#97—41no—
55C2×#97—41no—
56C2×#97Q814yes—
57#389—24no—
58#389—42no—
59#389—42no—
60#389—42no—
61#382—24no—
62#382—42no—
63C4×C4⋊C4—41no—
64C4×C4⋊C4—41no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95D411yescharacteristic
76C2×#95D411yescharacteristic
77#340—24no—
78#340—24no—
79#340—24no—
80#340—41no—
81#340—42no—
82#340—42no—
83#340—41no—
84#340—42no—
85#336—42no—
86#336—42no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8C2×Q814yes—
111C22×C8C2×Q812yes—
112C22×C8C2×Q812yes—
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—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4#4114yes—
145C2×C4⋊C4#4114yes—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42C2×Q812yes—
157C2×C42C2×Q812yes—
158#106—44no—
159#106—44no—
160C4⋊C8—44no—
161#97—44no—
162#95—41no—
163#95—41no—
164#95—44no—
165#95—44no—
166#95—44no—
167#95—44no—
168#95—44no—
169#95—44no—
170#95—44no—
171#95—82no—
172#95—82no—
173#95—82no—
174C24C22×D411yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
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—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C4⋊Q812yes—
226C22×C4C4⋊Q812yes—
227C22×C4C22×Q811yescharacteristic
228C22×C4C22×Q811yescharacteristic
229C22×C4C2×#4112yes—
230C22×C4C2×#4112yes—
231C22×C4H(5,𝔽2)11yescharacteristic
232C22×C4H(5,𝔽2)11yescharacteristic
233C22×C4H(5,𝔽2)12yes—
234C22×C4H(5,𝔽2)11yescharacteristic
235C22×C42−1+412yes—
236C22×C42−1+411yescharacteristic
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—48no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—88no—
272C4⋊C4—88no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C23C2×C4⋊Q811yescharacteristic
282C23#53411yescharacteristic
283C23#53711yescharacteristic
284C23#54612yes—
285C23#54812yes—
286C23#57512yes—
287C23#57512yes—
288C23#57712yes—
289C23#57712yes—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
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—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4#53312yes—
343C2×C4#53912yes—
344C2×C4#53912yes—
345C2×C4#54012yes—
346C2×C4#54012yes—
347C2×C4#54311yescharacteristic
348C2×C4#54311yescharacteristic
349C2×C4#54411yescharacteristic
350C2×C4#54412yes—
351C2×C4#54411yescharacteristic
352C2×C4#55111yescharacteristic
353C2×C4#55212yes—
354C2×C4#55412yes—
355C2×C4#55511yescharacteristic
356C2×C4#55712yes—
357C2×C4#56011yescharacteristic
358C2×C4#56312yes—
359C2×C4#56311yescharacteristic
360C8—44no—
361C8—42no—
362C8—42no—
363C22#255012yes—
364C22#265911yescharacteristic
365C22#269911yescharacteristic
366C22#270712yes—
367C22#272811yescharacteristic
368C22#316311yescharacteristic
369C22#316311yescharacteristic
370C22#316612yes—
371C22#319712yes—
372C22#320012yes—
373C22#325611yescharacteristic
374C22#325611yescharacteristic
375C22#326312yes—
376C22#330712yes—
377C22#331112yes—
378C22#331712yes—
379C22#332112yes—
380C22#337112yes—
381C22#337112yes—
382C22#337612yes—
383C22#337612yes—
384C4—42no—
385C4—42no—
386C4—84no—
387C4—84no—
388C4—82no—
389C4—82no—
390C4—162no—
391C4—162no—
392C4#261211yescharacteristic
393C4#266712yes—
394C4#274811yescharacteristic
395C4#279112yes—
396C4#280911yescharacteristic
397C4#284311yescharacteristic
398C2#1703611yescharacteristic
399C2#2880512yes—
400C2#2880812yes—
401C2#3029711yescharacteristic
402C2#3030211yescharacteristic
403C2#3173512yes—
404C2#3198112yes—
405C2#3198312yes—
406C2#3256012yes—
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
#113582
#113272
#112312
#112202
#112082
#108352
#108212