Finite Group Databasebeta

← #474669

Subgroups — #474669

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1944C211yesmaximal, characteristic
2C2×#1940C211yesmaximal, characteristic
3#15194C212yesmaximal
4#11327C212yesmaximal
5#11256C212yesmaximal
6#11228C212yesmaximal
7#11227C212yesmaximal
8#11227C212yesmaximal
9#10821C211yesmaximal, characteristic
10C22×#107C2211yescharacteristic
11C22×#106C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1944—24no—
20#1940—24no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30#1445—22no—
31#1444—22no—
32#1441—22no—
33#1441—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C23×C8C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C42C2312yes—
49C22×C42C2311yescharacteristic
50C2×#107—22no—
51C2×#107—41no—
52C2×#107—41no—
53C2×#106—22no—
54C2×#106—24no—
55C2×C4⋊C8—24no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—41no—
58C2×C4⋊C8—41no—
59C2×C4⋊C8—42no—
60C2×C4⋊C8—42no—
61C2×#97—41no—
62C2×#97—41no—
63C2×#97D414yes—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—42no—
69#389—42no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95D411yescharacteristic
82C2×#95D411yescharacteristic
83#340—24no—
84#340—41no—
85#340—41no—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#340Q818yes—
91#336—24no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2411yescharacteristic
101C22×C8—22no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C22×C8C2×Q814yes—
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—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4C2×Q814yes—
152C2×C4⋊C4C2×Q814yes—
153C2×C4⋊C4#4114yes—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42C2×D412yes—
165C2×C42#4112yes—
166#107—44no—
167#106—28no—
168C4⋊C8—44no—
169C4⋊C8—48no—
170#97—44no—
171#95—22no—
172#95—22no—
173#95—44no—
174#95—44no—
175#95—44no—
176#95—44no—
177#95—82no—
178#95—82no—
179#95—82no—
180#95—82no—
181#95—82no—
182C24C22×D411yescharacteristic
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
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—24no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
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×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4#12112yes—
233C22×C4#12112yes—
234C22×C4#13714yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×Q812yes—
237C22×C4C2×#4112yes—
238C22×C4C2×#4111yescharacteristic
239C22×C4H(5,𝔽2)11yescharacteristic
240C22×C4H(5,𝔽2)11yescharacteristic
241C22×C4H(5,𝔽2)12yes—
242C22×C42−1+411yescharacteristic
243C22×C42−1+411yescharacteristic
244C22×C42−1+412yes—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8C4⋊Q814yes—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—88no—
280C4⋊C4—88no—
281C4⋊C4—88no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C23C2×#12111yescharacteristic
291C23#53711yescharacteristic
292C23#53711yescharacteristic
293C23#54612yes—
294C23D4×Q812yes—
295C23C2×#13712yes—
296C23#57512yes—
297C23#57612yes—
298C23#57712yes—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
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—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#52511yescharacteristic
352C2×C4C2×C4⋊Q811yescharacteristic
353C2×C4#53411yescharacteristic
354C2×C4#53511yescharacteristic
355C2×C4#53611yescharacteristic
356C2×C4#53611yescharacteristic
357C2×C4#53911yescharacteristic
358C2×C4#53911yescharacteristic
359C2×C4#54112yes—
360C2×C4#54112yes—
361C2×C4#54412yes—
362C2×C4#54412yes—
363C2×C4#55312yes—
364C2×C4#55412yes—
365C2×C4#55412yes—
366C2×C4#55712yes—
367C2×C4#56212yes—
368C2×C4#56212yes—
369C8—24no—
370C8—42no—
371C8—82no—
372C22#253511yescharacteristic
373C22#254611yescharacteristic
374C22#261811yescharacteristic
375C22#270412yes—
376C22#270412yes—
377C22#305612yes—
378C22#306212yes—
379C22#325212yes—
380C22#325511yescharacteristic
381C22#325611yescharacteristic
382C22#325712yes—
383C22#325911yescharacteristic
384C22#326311yescharacteristic
385C22#330912yes—
386C22#331112yes—
387C22#331412yes—
388C22#332012yes—
389C22#338512yes—
390C22#338712yes—
391C22#338812yes—
392C22#338912yes—
393C4—42no—
394C4—42no—
395C4—44no—
396C4—84no—
397C4—84no—
398C4—82no—
399C4—84no—
400C4—162no—
401C4#258311yescharacteristic
402C4#263411yescharacteristic
403C4#266011yescharacteristic
404C4#280212yes—
405C4#280212yes—
406C4#281611yescharacteristic
407C2#1699211yescharacteristic
408C2#2815512yes—
409C2#2868612yes—
410C2#2978311yescharacteristic
411C2#2978511yescharacteristic
412C2#3188312yes—
413C2#3188412yes—
414C2#3189012yes—
415C2#3189312yes—
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
C2×#19442
C2×#19402
#151942
#113272
#112562
#112282
#112272
#108212