Finite Group Databasebeta

← #365511

Subgroups — #365511

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#11339C214yesmaximal
7#10899C214yesmaximal
8#10851C211yesmaximal, characteristic
9C4×#343C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#428C2214yes—
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2370—22no—
15#2369—21no—
16#2365—21no—
17#2355—21no—
18#2355—22no—
19Q8×C4⋊C4—21no—
20#2329—22no—
21#2317C2212yes—
22C4×C22⋊Q8—22no—
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2212yes—
26#2274C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1992—22no—
30#1990—21no—
31#1990—21no—
32#1986C418yescomplemented
33#1847—22no—
34#1845—21no—
35#1844—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1736C2214yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41#1442—22no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125D412yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×#117C2312yes—
59C2×#117C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42D412yes—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#428—28no—
70#428—44no—
71C2×#107—24no—
72C2×#106—24no—
73C4×M16—22no—
74C4×M16—22no—
75C4×M16—22no—
76C4×M16C2×C414yes—
77C2×#97—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#391—24no—
82#391—22no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—22no—
87#387—24no—
88#387—24no—
89#387—24no—
90#385—24no—
91#384—22no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384D411yescharacteristic
97#384D411yescharacteristic
98#382—22no—
99#382—44no—
100#382D412yes—
101#380—21no—
102#380—21no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C414yes—
117C4×C22⋊C4—24no—
118C2×#95—21no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95D412yes—
122#343—24no—
123#343Q818yescomplemented
124#340—24no—
125#340—24no—
126#339—44no—
127#339C2×C4116yes—
128C22×Q8—21no—
129C22×Q8—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16C22×C414yes—
146C2×M16C2×Q814yes—
147C2×M16C2×Q814yes—
148C2×M16#4114yes—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8C4⋊C414yes—
154C4⋊Q8C4⋊C414yes—
155#125—24no—
156#125C4⋊C418yes—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—84no—
166#117—22no—
167#117—22no—
168#117—24no—
169#117—24no—
170#117C22×C414yes—
171#117C2×Q814yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q814yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42C22×C414yes—
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228#107—48no—
229#106—48no—
230#97—24no—
231#97—42no—
232C4×C8—24no—
233C4×C8—42no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—48no—
244#95—84no—
245C24C22×D411yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—81no—
251C2×Q8—81no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4C4×D412yes—
282C22×C4C4×D412yes—
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q812yes—
289C22×C4C4⋊Q814yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q811yescharacteristic
294C22×C4C22×Q812yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4C2×#4111yescharacteristic
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+412yes—
300M16—24no—
301M16—24no—
302M16—24no—
303M16—24no—
304M16—24no—
305M16—42no—
306M16—42no—
307M16C4×Q818yescomplemented
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—28no—
319C4⋊C4—28no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—48no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4—82no—
339C4⋊C4C2×C4⋊C414yes—
340C4⋊C4C2×C4⋊C414yes—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C42—24no—
346C42—24no—
347C42—28no—
348C42—44no—
349C42—44no—
350C42—44no—
351C42—82no—
352C42—84no—
353C42—84no—
354C42C2×C4⋊C412yes—
355C42C2×C4⋊C412yes—
356C42C4×D412yes—
357C42C4×D412yes—
358C42C22⋊Q812yes—
359C42C22⋊Q812yes—
360C42#12312yes—
361C42#12312yes—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23C2×C4×D411yescharacteristic
367C23C2×#12011yescharacteristic
368C23C2×C22⋊Q811yescharacteristic
369C23C2×C4⋊Q811yescharacteristic
370C23#53211yescharacteristic
371C23#53511yescharacteristic
372C23#53611yescharacteristic
373Q8—48no—
374Q8—84no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
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—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#38512yes—
420C2×C4#38512yes—
421C2×C4#38512yes—
422C2×C4#38512yes—
423C2×C4#45412yes—
424C2×C4#45512yes—
425C2×C4C22×C4⋊C411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×C4×Q812yes—
428C2×C4#52012yes—
429C2×C4C2×C22⋊Q811yescharacteristic
430C2×C4C2×#12311yescharacteristic
431C2×C4#53312yes—
432C2×C4#54212yes—
433C2×C4#54612yes—
434C2×C4D4×Q812yes—
435C2×C4D4×Q812yes—
436C2×C4#55012yes—
437C2×C4#55212yes—
438C2×C4#55412yes—
439C2×C4#55712yes—
440C2×C4Q8212yes—
441C8—28no—
442C8—44no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22C2×#38511yescharacteristic
456C22C4×C4⋊Q811yescharacteristic
457C22#232611yescharacteristic
458C22#243111yescharacteristic
459C22#243411yescharacteristic
460C22#256511yescharacteristic
461C22#256711yescharacteristic
462C22C2×#45511yescharacteristic
463C22C2×#45411yescharacteristic
464C22#302111yescharacteristic
465C22#302211yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—44no—
470C4—44no—
471C4—44no—
472C4—84no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—162no—
477C4#186014yes—
478C4Q8×C4⋊C412yes—
479C4#236012yes—
480C4#242312yes—
481C4#245112yes—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2#1397611yescharacteristic
487C2C2×#186011yescharacteristic
488C2#2086511yescharacteristic
489C2#2562111yescharacteristic
490C2#2562411yescharacteristic
491C2#2582311yescharacteristic
492C2#2582611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×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
32. #1986C4semidirect
123. #343Q8semidirect
307. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342132
#213222
C2×#19862
#208002
#139112
#113392
#108992
#108512
C4×#3432