Finite Group Databasebeta

← #476219

Subgroups — #476219

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2059C212yesmaximal
2C2×#2059C212yesmaximal
3#15218C212yesmaximal
4#11378C212yesmaximal
5#11346C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#11256C211yesmaximal, characteristic
9#11256C211yesmaximal, characteristic
10C22×#125C2212yes—
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C2×#389C2212yes—
15C2×#389C2212yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#2059—28no—
18#2059—28no—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30#1444—22no—
31#1444—22no—
32#1441—22no—
33#1441—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C23×C8C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C2×#125—24no—
41C2×#125—24no—
42C2×#125—42no—
43C2×#125—42no—
44C2×#125—42no—
45C2×#125—42no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×#107—24no—
59C2×#107—24no—
60C2×#107—41no—
61C2×#107—41no—
62C2×#106—42no—
63C2×#106—42no—
64C2×#106D418yes—
65#389—24no—
66#389—24no—
67#389—42no—
68#389—42no—
69C4×C4⋊C4—41no—
70C4×C4⋊C4—41no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95D412yes—
80#340—24no—
81#340—24no—
82#340—24no—
83#340—24no—
84#340—42no—
85#340Q814yes—
86#340Q814yes—
87#336—24no—
88#336—24no—
89#336—41no—
90#336—41no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2411yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D414yes—
115C22×C8C2×Q812yes—
116C22×C8C2×Q812yes—
117#125—48no—
118#125—48no—
119#125—48no—
120#125—48no—
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—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×Q814yes—
161C2×C4⋊C4C2×Q814yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172#107—44no—
173#107—44no—
174#106—48no—
175#95—42no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—44no—
180#95—82no—
181#95—82no—
182#95—82no—
183#95—82no—
184C24C22×D411yescharacteristic
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
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—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4#13512yes—
237C22×C4C22×D412yes—
238C22×C4C22×Q811yescharacteristic
239C22×C4C22×Q811yescharacteristic
240C22×C4H(5,𝔽2)11yescharacteristic
241C22×C4H(5,𝔽2)12yes—
242C22×C4H(5,𝔽2)11yescharacteristic
243C22×C42−1+411yescharacteristic
244C22×C42−1+411yescharacteristic
245C22×C42−1+412yes—
246C22×C42−1+412yes—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—41no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—416no—
275C4⋊C4—416no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—88no—
285C4⋊C4—88no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—41no—
292C42—41no—
293C23#53411yescharacteristic
294C23#53612yes—
295C23D4212yes—
296C23D4×Q811yescharacteristic
297C23D4×Q811yescharacteristic
298C23C2×#13511yescharacteristic
299C23#57512yes—
300C23#57511yescharacteristic
301C23#57611yescharacteristic
302C23#57712yes—
303C23#57811yescharacteristic
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
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—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#53412yes—
357C2×C4#53612yes—
358C2×C4#53612yes—
359C2×C4#54411yescharacteristic
360C2×C4#54411yescharacteristic
361C2×C4#54412yes—
362C2×C4#54412yes—
363C2×C4D4211yescharacteristic
364C2×C4D4×Q812yes—
365C2×C4D4×Q812yes—
366C2×C4Q8211yescharacteristic
367C2×C4H(3,𝔽4)11yescharacteristic
368C2×C4#56311yescharacteristic
369C2×C4#56312yes—
370C2×C4#56312yes—
371C2×C4#56312yes—
372C2×C4#56311yescharacteristic
373C2×C4#56311yescharacteristic
374C8—44no—
375C8—42no—
376C8—42no—
377C22#267712yes—
378C22#267712yes—
379C22#267911yescharacteristic
380C22#273411yescharacteristic
381C22#281711yescharacteristic
382C22#319111yescharacteristic
383C22#319711yescharacteristic
384C22#320311yescharacteristic
385C22#320411yescharacteristic
386C22#323212yes—
387C22#324012yes—
388C22#324112yes—
389C22#324212yes—
390C22#326812yes—
391C22#326912yes—
392C22#327512yes—
393C22#328412yes—
394C22#338011yescharacteristic
395C22#338311yescharacteristic
396C22#338812yes—
397C22#338812yes—
398C22#338911yescharacteristic
399C22#339011yescharacteristic
400C4—42no—
401C4—81no—
402C4—82no—
403C4—84no—
404C4—82no—
405C4—84no—
406C4—84no—
407C4—84no—
408C4#267711yescharacteristic
409C4#267711yescharacteristic
410C4#273412yes—
411C4#281712yes—
412C4#284411yescharacteristic
413C4#284411yescharacteristic
414C2#1708511yescharacteristic
415C2#3095112yes—
416C2#3096012yes—
417C2#3096312yes—
418C2#3097212yes—
419C2#3105911yescharacteristic
420C2#3106611yescharacteristic
421C2#3268611yescharacteristic
422C2#3269211yescharacteristic
423C2#3279111yescharacteristic
424C2#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
C2×#20592
#152182
#113782
#113462
#112932
#112922
#112562