Finite Group Databasebeta

← #391352

Subgroups — #391352

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic
2#14264C212yesmaximal
3#11327C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11292C212yesmaximal, complemented
6#11219C212yesmaximal, complemented
7#11200C212yesmaximal
8#11095C211yesmaximal, characteristic
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#380C2212yes—
18C2×#380C2212yes—
19#2075C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1444—22no—
31#1442—22no—
32#1439—22no—
33#1439—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C24×C4C2311yescharacteristic
37C23×C8C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C2×#117—42no—
41C2×#117—42no—
42C2×#117—41no—
43C2×#117—41no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42C2312yes—
56C2×#107—41no—
57C2×#107—41no—
58C2×#107C2×C414yes—
59C2×#106—42no—
60C2×#106—41no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—41no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8C2×C414yes—
67#389—42no—
68#389—42no—
69#389—42no—
70#389—42no—
71#380—24no—
72#380—42no—
73#380—42no—
74#380C2×C418yes—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83#340—42no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—41no—
88#340—42no—
89#340C2×C414yes—
90#340C2×C418yes—
91#336—42no—
92#336—42no—
93C25C2×D411yescharacteristic
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C22×C412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4#4112yes—
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C22×C412yes—
128C22×C8C22×C412yes—
129#117—48no—
130#117—44no—
131#117—44no—
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—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C414yes—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4—82no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181#107—44no—
182#106—44no—
183C22⋊C8—44no—
184C22⋊C8—44no—
185#95—28no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193C24—22no—
194C24—21no—
195C24—22no—
196C24—21no—
197C24—22no—
198C24—22no—
199C24—22no—
200C24—22no—
201C24C4×D412yes—
202C24C22×D411yescharacteristic
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C4×D412yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C4H(5,𝔽2)12yes—
274C22×C4H(5,𝔽2)12yes—
275C22×C42−1+412yes—
276C22×C42−1+412yes—
277C22×C42−1+411yescharacteristic
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—48no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—88no—
313C4⋊C4—88no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—88no—
317C22⋊C4—88no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C23—24no—
323C23—22no—
324C23—24no—
325C23—22no—
326C23—22no—
327C23—41no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23C2×C4×D411yescharacteristic
344C23#53411yescharacteristic
345C23#53412yes—
346C23#53612yes—
347C23#53711yescharacteristic
348C23#57612yes—
349C23#57712yes—
350C23#57712yes—
351C23#57812yes—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#51811yescharacteristic
406C2×C4#52012yes—
407C2×C4#52012yes—
408C2×C4#52011yescharacteristic
409C2×C4#53811yescharacteristic
410C2×C4#53912yes—
411C2×C4#53912yes—
412C2×C4#53911yescharacteristic
413C2×C4#56012yes—
414C2×C4#56011yescharacteristic
415C2×C4H(3,𝔽4)11yescharacteristic
416C2×C4#56212yes—
417C2×C4#56212yes—
418C2×C4#56312yes—
419C2×C4#56311yescharacteristic
420C2×C4#56312yes—
421C2×C4#56311yescharacteristic
422C2×C4#56312yes—
423C8—42no—
424C8—42no—
425C8—82no—
426C22—42no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—42no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22#232212yes—
436C22#232511yescharacteristic
437C22#281211yescharacteristic
438C22#281412yes—
439C22#281711yescharacteristic
440C22#294712yes—
441C22#294811yescharacteristic
442C22#295211yescharacteristic
443C22#319812yes—
444C22#319812yes—
445C22#320012yes—
446C22#320112yes—
447C22#320112yes—
448C22#320312yes—
449C22#324112yes—
450C22#324212yes—
451C22#324312yes—
452C22#324412yes—
453C22#325712yes—
454C22#326111yescharacteristic
455C22#326311yescharacteristic
456C4—42no—
457C4—84no—
458C4—84no—
459C4—82no—
460C4—82no—
461C4—162no—
462C4—162no—
463C4#237611yescharacteristic
464C4#237911yescharacteristic
465C4#238312yes—
466C4#282712yes—
467C4#283411yescharacteristic
468C4#283511yescharacteristic
469C2—82no—
470C2#1462711yescharacteristic
471C2#2209212yes—
472C2#2209312yes—
473C2#2219311yescharacteristic
474C2#2219811yescharacteristic
475C2#3271312yes—
476C2#3275112yes—
477C2#3275712yes—
478C2#3279312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11327C2semidirect
4. #11316C2semidirect
5. #11292C2semidirect
6. #11219C2semidirect
19. #2075C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
#142642
#113272
#113162
#112922
#112192
#112002
#110952
#107132