Finite Group Databasebeta

← #415349

Subgroups — #415349

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5#11392C214yesmaximal
6#11276C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10839C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C2×#544C2212yes—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2868—21no—
15#2859—21no—
16#2660—22no—
17#2552—21no—
18#2552—21no—
19#2551—21no—
20#2551—21no—
21#2440—22no—
22#2440—22no—
23#2317C2212yes—
24C2×#391—22no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1993—22no—
30#1990—21no—
31#1990—21no—
32#1988—21no—
33#1988—21no—
34#1987—21no—
35#1987—21no—
36C2×#343C2214yes—
37#1736C2212yes—
38#1736C2214yes—
39#1736C2212yes—
40C2×#339C2212yes—
41C2×#339C2214yes—
42C2×#339C2212yes—
43C4⋊#95—22no—
44C4⋊#95C2214yes—
45C23×Q8—21no—
46C22×M16C2312yes—
47C22×M16C2312yes—
48#544—28no—
49#544—44no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8Q814yescomplemented
62C2×#125—22no—
63C2×#125—22no—
64C2×C22⋊Q8—24no—
65C2×#117C2312yes—
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C4×M16—22no—
82C4×M16—22no—
83C2×#97—21no—
84C2×C4×C8—21no—
85#398—24no—
86#398—24no—
87#395—24no—
88#395—24no—
89#391—48no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#385—24no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#382—22no—
107#382D412yes—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118#376—21no—
119C2×#95—28no—
120C43—21no—
121C43—22no—
122C43—22no—
123#343—44no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—24no—
130#339—24no—
131#339—44no—
132#339Q818yescomplemented
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—42no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2411yescharacteristic
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16C2×Q814yes—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—48no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—84no—
170C4⋊Q8—84no—
171#125—48no—
172#125—44no—
173#125—44no—
174C22⋊Q8—416no—
175#117—22no—
176#117—22no—
177#117—24no—
178#117—24no—
179#117C2×Q814yes—
180#117#4114yes—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×Q814yes—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×Q814yes—
242C2×C42C2×Q812yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C4⋊C8—42no—
247C4⋊C8—42no—
248#97—42no—
249C4×C8—42no—
250#95—24no—
251#95—24no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—416no—
259C24C22×D411yescharacteristic
260C2×Q8—416no—
261C2×Q8—48no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×Q8—88no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—82no—
270C2×Q8—82no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4C22⋊Q812yes—
302C22×C4C22⋊Q812yes—
303C22×C4#12412yes—
304C22×C4#12412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C22×Q812yes—
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4H(5,𝔽2)12yes—
314C22×C42−1+412yes—
315M16—24no—
316M16—24no—
317M16—42no—
318M16—42no—
319M16—44no—
320M16—44no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4#12412yes—
356C4⋊C4#12412yes—
357C4⋊C4C4⋊Q812yes—
358C4⋊C4C4⋊Q812yes—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—44no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—48no—
367C42—48no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42C22⋊Q812yes—
377C42C22⋊Q812yes—
378C42C22⋊Q812yes—
379C42C22⋊Q812yes—
380C42#12412yes—
381C42#12412yes—
382C42#12412yes—
383C42#12412yes—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23C2×#12011yescharacteristic
389C23C2×C22⋊Q811yescharacteristic
390C23C2×#12411yescharacteristic
391C23#54612yes—
392C23D4×Q812yes—
393Q8—88no—
394Q8—88no—
395Q8—168no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—28no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—48no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—88no—
438C2×C4#39312yes—
439C2×C4#39312yes—
440C2×C4#39312yes—
441C2×C4#39312yes—
442C2×C4#45312yes—
443C2×C4#45412yes—
444C2×C4#45512yes—
445C2×C4#45612yes—
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4C2×#12411yescharacteristic
449C2×C4C2×#12411yescharacteristic
450C2×C4C2×#12411yescharacteristic
451C2×C4C2×C4⋊Q811yescharacteristic
452C2×C4#53312yes—
453C2×C4#53312yes—
454C2×C4#53511yescharacteristic
455C2×C4#53611yescharacteristic
456C2×C4#54212yes—
457C2×C4#54212yes—
458C2×C4#54612yes—
459C2×C4D4×Q812yes—
460C2×C4#55412yes—
461C2×C4#55612yes—
462C2×C4#55712yes—
463C2×C4Q8212yes—
464C8—44no—
465C8—82no—
466C22—22no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—22no—
474C22C2×#39311yescharacteristic
475C22#243411yescharacteristic
476C22#243511yescharacteristic
477C22#245112yes—
478C22#249212yes—
479C22C2×#45311yescharacteristic
480C22C2×#45511yescharacteristic
481C22C2×#45411yescharacteristic
482C22C2×#45611yescharacteristic
483C4—22no—
484C4—22no—
485C4—48no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—84no—
491C4—82no—
492C4—82no—
493C4—84no—
494C4#202714yes—
495C4#202814yes—
496C4#245112yes—
497C4#249212yes—
498C4#256711yescharacteristic
499C4#256811yescharacteristic
500C4#264011yescharacteristic
501C4#286211yescharacteristic
502C2—22no—
503C2—22no—
504C2#1519111yescharacteristic
505C2C2×#202711yescharacteristic
506C2C2×#202811yescharacteristic
507C2#2582911yescharacteristic
508C2#2583011yescharacteristic
509C2#2587511yescharacteristic
510C2#2587611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
61. C2×C4⋊Q8Q8semidirect
132. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#213242
#213132
#147142
#113922
#112762
#108572
#108392
C4×#3392