Finite Group Databasebeta

← #474154

Subgroups — #474154

Order 512.

1000 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×#2058C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3#14282C211yesmaximal, characteristic
4#14261C211yesmaximal, characteristic
5#11316C212yesmaximal
6#11231C212yesmaximal
7#11208C211yesmaximal, characteristic
8#11122C212yesmaximal
9#11077C212yesmaximal
10#11052C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#2058—24no—
24#2024—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1446—22no—
34#1445—22no—
35#1443—22no—
36#1442—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C23×Q8C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8D414yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#106—24no—
60C2×#106—42no—
61C2×#106—42no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—42no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71#391—22no—
72#391—22no—
73#391—22no—
74#391—42no—
75#391Q814yescomplemented
76#389—22no—
77#389—22no—
78#382—42no—
79#376—42no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D412yes—
94C2×#95D412yes—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336—22no—
102#336—42no—
103C22×Q8—24no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—82no—
116C22×Q8—82no—
117C22×Q8C2×Q812yes—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8#4114yes—
143C4⋊Q8—48no—
144C4⋊Q8—48no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×Q812yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42C2×D412yes—
185C2×C42C2×Q812yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188#106—48no—
189C4⋊C8—48no—
190#103—48no—
191#103—48no—
192#95—28no—
193#95—42no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—44no—
198#95—42no—
199#95—42no—
200#95—42no—
201#95—82no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207C24C22×D411yescharacteristic
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—48no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—88no—
228C2×Q8—88no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4C22⋊Q814yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4H(5,𝔽2)12yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C42−1+411yescharacteristic
298C22×C42−1+412yes—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—84no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—42no—
336C42—42no—
337C23C2×#12111yescharacteristic
338C23C2×C22⋊Q811yescharacteristic
339C23#53211yescharacteristic
340C23#53411yescharacteristic
341C23#53511yescharacteristic
342C23#53611yescharacteristic
343C23#53711yescharacteristic
344C23#57712yes—
345C23#57712yes—
346C23#57712yes—
347C23#57812yes—
348Q8—44no—
349Q8—44no—
350Q8—816no—
351Q8—82no—
352Q8—82no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
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—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4#53412yes—
414C2×C4#53812yes—
415C2×C4#53812yes—
416C2×C4#54212yes—
417C2×C4#54212yes—
418C2×C4#54412yes—
419C2×C4#54812yes—
420C2×C4D4×Q811yescharacteristic
421C2×C4#55011yescharacteristic
422C2×C4#55112yes—
423C2×C4#55412yes—
424C2×C4#55411yescharacteristic
425C2×C4#55611yescharacteristic
426C2×C4#55612yes—
427C2×C4#56212yes—
428C2×C4#56312yes—
429C8—44no—
430C8—82no—
431C22#253811yescharacteristic
432C22#254511yescharacteristic
433C22#256411yescharacteristic
434C22#261711yescharacteristic
435C22#262811yescharacteristic
436C22#266211yescharacteristic
437C22#281411yescharacteristic
438C22#304211yescharacteristic
439C22#304211yescharacteristic
440C22#304311yescharacteristic
441C22#304411yescharacteristic
442C22#308212yes—
443C22#308412yes—
444C22#317312yes—
445C22#317812yes—
446C22#320012yes—
447C22#320112yes—
448C22#322212yes—
449C22#322512yes—
450C22#324212yes—
451C22#324412yes—
452C22#325812yes—
453C22#326312yes—
454C4—42no—
455C4—42no—
456C4—44no—
457C4—82no—
458C4—84no—
459C4—82no—
460C4—82no—
461C4—161no—
462C4—162no—
463C4—161no—
464C4#272912yes—
465C4#273012yes—
466C4#275912yes—
467C4#278812yes—
468C2#1690111yescharacteristic
469C2#2828911yescharacteristic
470C2#2829111yescharacteristic
471C2#2866211yescharacteristic
472C2#2866611yescharacteristic
473C2#2917012yes—
474C2#2972311yescharacteristic
475C2#2972811yescharacteristic
476C2#2999312yes—
477C2#3035212yes—
478C2#3275212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
75. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
C2×#20242
#142822
#142612
#113162
#112312
#112082
#111222
#110772
#110522
#108332