Finite Group Databasebeta

← #474157

Subgroups — #474157

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3#14282C211yesmaximal, characteristic, complemented
4#14260C211yesmaximal, characteristic
5#11315C212yesmaximal, complemented
6#11231C212yesmaximal, complemented
7#11208C211yesmaximal, characteristic, complemented
8#11121C212yesmaximal
9#11076C212yesmaximal
10#11051C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#2058—24no—
24#2023—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1446—22no—
34#1445—22no—
35#1440—22no—
36#1439—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C23×D4C2311yescharacteristic
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×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—42no—
65C2×#106—42no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—42no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74C2×#102—42no—
75#390—22no—
76#390—22no—
77#390—22no—
78#390—42no—
79#390Q814yescomplemented
80#389—22no—
81#389—22no—
82#382—42no—
83#376—42no—
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—
103C25—21no—
104C22×Q8—24no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—82no—
108C22×Q8—82no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4C2×Q812yes—
119C23×C4—21no—
120C23×C4—22no—
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×C22⋊C4—28no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42C2×D412yes—
195C2×C42C2×Q812yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198#106—48no—
199C4⋊C8—48no—
200#102—48no—
201#102—48no—
202#95—42no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—82no—
208#95—82no—
209#95—82no—
210#95—82no—
211#95—82no—
212#95—82no—
213C24—22no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24C22×D411yescharacteristic
225C2×Q8—48no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×Q8—88no—
229C2×Q8—88no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—42no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4C22⋊Q814yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4C2×#4111yescharacteristic
304C22×C4H(5,𝔽2)12yes—
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)11yescharacteristic
307C22×C42−1+411yescharacteristic
308C22×C42−1+412yes—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—84no—
341C22⋊C4—88no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C22⋊C4—88no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—42no—
350C42—42no—
351C23—42no—
352C23—44no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—82no—
358C23—81no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C2×#12111yescharacteristic
375C23C2×C22⋊Q811yescharacteristic
376C23#53211yescharacteristic
377C23#53411yescharacteristic
378C23#53511yescharacteristic
379C23#53611yescharacteristic
380C23#53711yescharacteristic
381C23#57612yes—
382C23#57712yes—
383C23#57712yes—
384C23#57712yes—
385Q8—816no—
386D4—44no—
387D4—44no—
388D4—82no—
389D4—82no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4#53412yes—
444C2×C4#53812yes—
445C2×C4#53812yes—
446C2×C4#54212yes—
447C2×C4#54212yes—
448C2×C4#54412yes—
449C2×C4#54812yes—
450C2×C4D4×Q811yescharacteristic
451C2×C4#55011yescharacteristic
452C2×C4#55112yes—
453C2×C4#55412yes—
454C2×C4#55411yescharacteristic
455C2×C4#55611yescharacteristic
456C2×C4#55612yes—
457C2×C4#56212yes—
458C2×C4#56312yes—
459C8—44no—
460C8—82no—
461C22—82no—
462C22—84no—
463C22—82no—
464C22—82no—
465C22—84no—
466C22—161no—
467C22—161no—
468C22—162no—
469C22—162no—
470C22—161no—
471C22—161no—
472C22#253811yescharacteristic
473C22#254511yescharacteristic
474C22#256411yescharacteristic
475C22#261711yescharacteristic
476C22#262811yescharacteristic
477C22#266211yescharacteristic
478C22#281411yescharacteristic
479C22#304111yescharacteristic
480C22#304211yescharacteristic
481C22#304311yescharacteristic
482C22#304311yescharacteristic
483C22#308112yes—
484C22#308412yes—
485C22#317312yes—
486C22#317912yes—
487C22#319812yes—
488C22#320012yes—
489C22#322112yes—
490C22#322412yes—
491C22#324112yes—
492C22#324412yes—
493C22#325712yes—
494C22#326312yes—
495C4—42no—
496C4—42no—
497C4—44no—
498C4—84no—
499C4—82no—
500C4—82no—
501C4—161no—
502C4—162no—
503C4—161no—
504C4#272912yes—
505C4#273012yes—
506C4#275912yes—
507C4#278812yes—
508C2—162no—
509C2#1690111yescharacteristic
510C2#2828911yescharacteristic
511C2#2829011yescharacteristic
512C2#2866211yescharacteristic
513C2#2866711yescharacteristic
514C2#2917412yes—
515C2#2972211yescharacteristic
516C2#2972711yescharacteristic
517C2#2999012yes—
518C2#3035412yes—
519C2#3275112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
1. C2×#2058C2semidirect
3. #14282C2semidirect
5. #11315C2semidirect
6. #11231C2semidirect
7. #11208C2semidirect
11. #10833C2semidirect
79. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
C2×#20232
#142822
#142602
#113152
#112312
#112082
#111212
#110762
#110512
#108332