Finite Group Databasebeta

← #398655

Subgroups — #398655

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1865C211yesmaximal, characteristic
2C2×#1805C212yesmaximal
3C2×#1803C211yesmaximal, characteristic
4#15242C211yesmaximal, characteristic
5Q8×#95C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic
8#11268C212yesmaximal
9#11231C212yesmaximal
10#11227C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23#1865—24no—
24#1805—28no—
25#1803—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1445—22no—
35#1441—21no—
36#1441—21no—
37C4⋊#95—21no—
38C4⋊#95—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4×#95—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—24no—
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—24no—
53C2×C4×Q8—42no—
54C2×C4×Q8—42no—
55C2×C4×Q8—42no—
56C2×C4×Q8D414yes—
57C2×C4×Q8Q818yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×#106—22no—
73C2×#106—42no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8—42no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—22no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103D414yes—
87#389—22no—
88#389—22no—
89#384—22no—
90#384—22no—
91#382—24no—
92#382—42no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104C2×#95D412yes—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—24no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8C2×D412yes—
119C22×Q8C2×D412yes—
120C22×Q8C2×Q814yes—
121C22×Q8#4112yes—
122C22×Q8#4114yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8C2×D412yes—
151C4×Q8—216no—
152C4×Q8—28no—
153C4×Q8—44no—
154C4×Q8—48no—
155C4×Q8—48no—
156C4×Q8—48no—
157C4×Q8—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×Q814yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—28no—
204C2×C42—44no—
205C2×C42—44no—
206C2×C42—44no—
207C2×C42C2×D412yes—
208C2×C42C2×Q814yes—
209C2×C42#4112yes—
210C2×C42#4114yes—
211#106—44no—
212C4⋊C8—48no—
213C4⋊C8—44no—
214#103—28no—
215#103—48no—
216#103—44no—
217#95—22no—
218#95—22no—
219#95—42no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—42no—
225#95—82no—
226#95—81no—
227#95—82no—
228#95—82no—
229C24C22×D411yescharacteristic
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8#12114yes—
244C2×Q8C22⋊Q818yes—
245C2×Q8#12314yes—
246C2×Q8#12518yes—
247C2×Q8C4⋊Q814yes—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4C22⋊Q814yes—
305C22×C4C22⋊Q814yes—
306C22×C4#13512yes—
307C22×C4#13712yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×Q812yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)12yes—
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C42−1+411yescharacteristic
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—48no—
342C4⋊C4—416no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C42—24no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—48no—
366C42—416no—
367C23C2×#12111yescharacteristic
368C23C2×C22⋊Q812yes—
369C23D4211yescharacteristic
370C23#54612yes—
371C23#54811yescharacteristic
372C23C2×#13511yescharacteristic
373C23C2×#13711yescharacteristic
374C23#57512yes—
375C23#57712yes—
376C23#57711yescharacteristic
377C23#57811yescharacteristic
378Q8—28no—
379Q8—24no—
380Q8—28no—
381Q8—24no—
382Q8—24no—
383Q8#39818yes—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4#46414yes—
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×C22⋊Q812yes—
446C2×C4C2×#12311yescharacteristic
447C2×C4C2×#12512yes—
448C2×C4C2×C4⋊Q811yescharacteristic
449C2×C4#53211yescharacteristic
450C2×C4#53312yes—
451C2×C4#53411yescharacteristic
452C2×C4#53711yescharacteristic
453C2×C4#54312yes—
454C2×C4#54412yes—
455C2×C4#54612yes—
456C2×C4#54711yescharacteristic
457C2×C4#55112yes—
458C2×C4#55112yes—
459C2×C4#55212yes—
460C2×C4#55311yescharacteristic
461C2×C4#55312yes—
462C8—42no—
463C8—81no—
464C8—82no—
465C22#241211yescharacteristic
466C22#241611yescharacteristic
467C22#243712yes—
468C22#244212yes—
469C22#285711yescharacteristic
470C22C2×#46411yescharacteristic
471C22#304011yescharacteristic
472C22#304311yescharacteristic
473C22#304611yescharacteristic
474C22#307612yes—
475C22#307812yes—
476C22#308212yes—
477C22#308412yes—
478C22#327112yes—
479C22#327611yescharacteristic
480C22#328411yescharacteristic
481C22#330112yes—
482C22#331712yes—
483C22#332012yes—
484C22#332712yes—
485C22#336411yescharacteristic
486C22#336911yescharacteristic
487C22#337112yes—
488C4—28no—
489C4—24no—
490C4—44no—
491C4—44no—
492C4—42no—
493C4—84no—
494C4—82no—
495C4—82no—
496C4—162no—
497C4—161no—
498C4C2×#39811yescharacteristic
499C4#243812yes—
500C4#249811yescharacteristic
501C4#254311yescharacteristic
502C4#256912yes—
503C4#265911yescharacteristic
504C2#1476611yescharacteristic
505C2#2453311yescharacteristic
506C2#2454311yescharacteristic
507C2#2472911yescharacteristic
508C2#2474711yescharacteristic
509C2#2595312yes—
510C2#2598112yes—
511C2#2611612yes—
512C2#2612112yes—
513C2#3320211yescharacteristic
514C2#3320411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#18652
C2×#18052
C2×#18032
#152422
Q8×#952
#113712
#113222
#112682
#112312
#112272
#107892