Finite Group Databasebeta

← #359229

Subgroups — #359229

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2C2×#2061C211yesmaximal, characteristic
3C4×#428C212yesmaximal
4#13779C211yesmaximal, characteristic
5#11392C211yesmaximal, characteristic
6#11342C211yesmaximal, characteristic
7#11340C212yesmaximal
8#11217C212yesmaximal
9#11211C211yesmaximal, characteristic
10C4×#343C211yesmaximal, characteristic
11#10701C212yesmaximal
12C2×#544C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×C4×M16C2212yes—
16#2331—22no—
17#2330—22no—
18#2329—22no—
19#2317C2212yes—
20#2316C2211yescharacteristic
21C4×C4⋊Q8—22no—
22C4×#125—22no—
23C4×C22⋊Q8—22no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C4×#117C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2061C418yescomplemented
31#1776—22no—
32#1776—22no—
33C4×#107—22no—
34C4×#106—22no—
35C2×#343C2212yes—
36C2×#343C2211yescharacteristic
37C2×#343C2211yescharacteristic
38#1736C2212yes—
39#1736C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44#1436—22no—
45#1436—22no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2211yescharacteristic
48C22×M16C2312yes—
49C22×M16C2312yes—
50#544—24no—
51#544—22no—
52#544C2×C414yes—
53C2×C4⋊Q8—22no—
54C2×#125—22no—
55C2×C22⋊Q8—22no—
56C2×C4×Q8—22no—
57C2×#117C2312yes—
58C2×#117C2312yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#428—24no—
70#428—24no—
71#428—24no—
72#428C2×C418yes—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77C4×M16—24no—
78C4×M16Q818yescomplemented
79C2×#97—22no—
80C2×C4×C8—22no—
81#391—24no—
82#391—22no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#387—24no—
90#387—24no—
91#387—24no—
92#385—24no—
93#384—22no—
94#384—22no—
95#384—22no—
96#382—22no—
97#382—22no—
98#382C2×C414yes—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112#376—22no—
113#376—22no—
114#376Q814yescomplemented
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D412yes—
122C43—22no—
123#343—24no—
124#343—22no—
125#343C2×C414yes—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#339—24no—
131#339—24no—
132#339—24no—
133#339C2×C418yes—
134C22×Q8—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16C22×C414yes—
153C2×M16C2×Q814yes—
154C2×M16C2×Q814yes—
155C2×M16#4114yes—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—28no—
159C4⋊Q8—28no—
160#125—28no—
161#125—28no—
162C22⋊Q8—28no—
163C22⋊Q8—28no—
164C4×Q8—84no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117C22×C414yes—
172#117#4114yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4#4112yes—
201C2×C4⋊C4#4112yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—82no—
220C2×C42—82no—
221C2×C42C22×C412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×Q812yes—
225C2×C42C2×Q812yes—
226C2×C42C2×Q814yes—
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#107—28no—
231#107—28no—
232#106—28no—
233#106—28no—
234#97—28no—
235C4×C8—28no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—28no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—24no—
246#95—24no—
247#95—41no—
248#95—41no—
249#95—82no—
250#95—82no—
251C24C22×D411yescharacteristic
252C2×Q8—28no—
253C2×Q8—82no—
254C2×Q8—82no—
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—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C4×D414yes—
297C22×C4#12011yescharacteristic
298C22×C4#12012yes—
299C22×C4#12011yescharacteristic
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4C4⋊Q812yes—
303C22×C4C4⋊Q812yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D412yes—
306C22×C4C22×Q812yes—
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4112yes—
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314M16—24no—
315M16—24no—
316M16—24no—
317M16—24no—
318M16—24no—
319M16—42no—
320M16—42no—
321M16C4×Q818yescomplemented
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—28no—
329C4⋊C4—216no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—84no—
345C4⋊C4—82no—
346C4⋊C4—88no—
347C4⋊C4—82no—
348C4⋊C4—84no—
349C4⋊C4C4×D414yes—
350C4⋊C4C4×D414yes—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C42—24no—
356C42—22no—
357C42—24no—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—44no—
362C42—44no—
363C42—44no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—82no—
368C42—82no—
369C42—82no—
370C42—84no—
371C42C4×Q814yescomplemented
372C42C22⋊Q814yes—
373C42C22⋊Q814yes—
374C42C4⋊Q814yes—
375C42C4⋊Q814yes—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23C2×C4×D411yescharacteristic
381C23C2×#12011yescharacteristic
382C23C2×C22⋊Q811yescharacteristic
383C23#52511yescharacteristic
384C23#52511yescharacteristic
385C23C2×C4⋊Q811yescharacteristic
386C23#53211yescharacteristic
387Q8—88no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—42no—
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—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4#39512yes—
445C2×C4#39512yes—
446C2×C4#39514yes—
447C2×C4C2×C4×D412yes—
448C2×C4C2×C4×Q811yescharacteristic
449C2×C4C2×C4×Q812yes—
450C2×C4C4×#4111yescharacteristic
451C2×C4C2×C22⋊Q812yes—
452C2×C4#52512yes—
453C2×C4#52512yes—
454C2×C4#52911yescharacteristic
455C2×C4#52912yes—
456C2×C4C2×C4⋊Q812yes—
457C2×C4#53212yes—
458C2×C4#53312yes—
459C2×C4#53312yes—
460C2×C4#53312yes—
461C2×C4#53311yescharacteristic
462C2×C4#53312yes—
463C8—28no—
464C8—44no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22C4×#12011yescharacteristic
478C22C4×C22⋊Q811yescharacteristic
479C22C4×C4⋊Q811yescharacteristic
480C22C2×#39511yescharacteristic
481C22#238811yescharacteristic
482C22#239611yescharacteristic
483C22#239911yescharacteristic
484C22#301911yescharacteristic
485C22#302011yescharacteristic
486C22#302111yescharacteristic
487C22#302211yescharacteristic
488C4—22no—
489C4—22no—
490C4—24no—
491C4—44no—
492C4—44no—
493C4—44no—
494C4—44no—
495C4—82no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4C4×C22⋊Q812yes—
502C4C4×C4⋊Q812yes—
503C4#239612yes—
504C4#239912yes—
505C2—21no—
506C2—21no—
507C2—21no—
508C2—21no—
509C2C4×#39511yescharacteristic
510C2#2147411yescharacteristic
511C2#2147511yescharacteristic
512C2#2356111yescharacteristic
513C2#2356211yescharacteristic
514C2#2359111yescharacteristic
515C2#2359411yescharacteristic
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
30. #2061C4semidirect
78. C4×M16Q8semidirect
114. #376Q8semidirect
321. M16C4×Q8semidirect
371. C42C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
C2×#20612
C4×#4282
#137792
#113922
#113422
#113402
#112172
#112112
C4×#3432
#107012