Finite Group Databasebeta

← #513598

Subgroups — #513598

Order 512.

3794 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C213yesmaximal
2#13822C213yesmaximal, complemented
3#13819C213yesmaximal, complemented
4#13814C213yesmaximal, complemented
5#13813C213yesmaximal
6#13812C211yesmaximal, characteristic, complemented
7#13809C211yesmaximal, characteristic, complemented
8#13809C211yesmaximal, characteristic, complemented
9D4×#95C211yesmaximal, characteristic
10#13788C213yesmaximal, complemented
11#13787C213yesmaximal, complemented
12C4×#381C213yesmaximal, complemented
13#13750C211yesmaximal, characteristic, complemented
14#13749C211yesmaximal, characteristic, complemented
15C2×C4×#95C211yesmaximal, characteristic, complemented
16C22×#123C2211yescharacteristic
17C22×C4×D4C2213yes—
18C23×C42C2213yes—
19C4×#123C4148yescomplemented
20C2×#388C2213yes—
21C2×#388C2213yes—
22C2×#387C2213yes—
23C2×#387C2213yes—
24C2×#386C2213yes—
25C2×#385C2213yes—
26C2×#381C2213yes—
27C2×C4×C4⋊C4C2213yes—
28C2×C4×C4⋊C4C2213yes—
29C2×C4×C4⋊C4C2213yes—
30C2×C4×C22⋊C4C2213yes—
31C2×C4×C22⋊C4C2216yes—
32C2×C4×C22⋊C4C2213yes—
33C2×C4×C22⋊C4C2213yes—
34C2×C4×C22⋊C4C2213yes—
35C2×#376C2211yescharacteristic
36C22×#95C2211yescharacteristic
37C22×#95C2213yes—
38C22×#95C2211yescharacteristic
39#1437C2216yes—
40#1436C2216yes—
41#1436C2216yes—
42#1436C2216yes—
43C4⋊#95C2213yes—
44C4⋊#95C2213yes—
45#1434C2216yes—
46#1434C2213yes—
47#1434C2213yes—
48#1434C2213yes—
49#1434C2213yes—
50#1434C2213yes—
51#1434C2213yes—
52C22⋊#95C2213yes—
53C22⋊#95C2213yes—
54C22⋊#95C2213yes—
55C22⋊#95C2213yes—
56C4×#95C2211yescharacteristic
57C4×#95C2213yes—
58C4×#95C2211yescharacteristic
59C4×#95C2213yes—
60C4×#95C2213yes—
61C4×#95C2211yescharacteristic
62C4×#95C2213yes—
63C4×#95C2213yes—
64C4×#95C2211yescharacteristic
65C4×#95C2213yes—
66C4×#95C2216yes—
67C23×D4C2311yescharacteristic
68C24×C4C2313yes—
69C24×C4C2313yes—
70C24×C4C2311yescharacteristic
71C24×C4C2313yes—
72C2×#123C2×C4124yes—
73C2×C4×D4—412no—
74C2×C4×D4C2×C4124yes—
75C22×C4⋊C4C2313yes—
76C22×C4⋊C4C2313yes—
77C22×C4⋊C4C2313yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4C2313yes—
81C22×C22⋊C4C2313yes—
82C22×C22⋊C4C2313yes—
83C22×C22⋊C4C2313yes—
84C22×C22⋊C4C2313yes—
85C22×C22⋊C4C2311yescharacteristic
86C22×C22⋊C4C2311yescharacteristic
87C22×C22⋊C4C2311yescharacteristic
88C22×C42—412no—
89C22×C42C2×C4124yescomplemented
90C22×C42C2313yes—
91C22×C42C2316yes—
92C22×C42C2316yes—
93C22×C42C2313yes—
94C22×C42C2313yes—
95C22×C42C2313yes—
96C22×C42C2313yes—
97C22×C42C2313yes—
98C22×C42C2313yes—
99#388C2×C4124yescomplemented
100#388C2×C4124yescomplemented
101#387C2×C4124yescomplemented
102#387C2×C4124yescomplemented
103#386C2×C4124yes—
104#385C2×C4124yescomplemented
105#381C2×C4124yescomplemented
106C4×C4⋊C4—212no—
107C4×C4⋊C4C2×C4124yescomplemented
108C4×C4⋊C4C2×C4124yescomplemented
109C4×C22⋊C4—212no—
110C4×C22⋊C4—412no—
111C4×C22⋊C4C2×C4124yescomplemented
112C4×C22⋊C4C2×C4124yescomplemented
113C4×C22⋊C4C2×C4124yescomplemented
114#376—24no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—212no—
118C2×#95C2313yes—
119C2×#95C2313yes—
120C2×#95C2313yes—
121C2×#95C2313yes—
122C2×#95C2313yes—
123C2×#95C2313yes—
124C2×#95C2316yes—
125C2×#95C2316yes—
126C2×#95C2316yes—
127C2×#95C2316yes—
128C2×#95C2316yes—
129C2×#95C2313yes—
130C2×#95C2313yes—
131C2×#95C2313yes—
132C2×#95C2313yes—
133C2×#95C2313yes—
134C2×#95C2313yes—
135C2×#95C2313yes—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2311yescharacteristic
140C2×#95C2313yes—
141C25C2411yescharacteristic
142C25C2411yescharacteristic
143C22×D4—26no—
144C22×D4—22no—
145C22×D4C22×C4112yes—
146C23×C4—26no—
147C23×C4—22no—
148C23×C4—212no—
149C23×C4—212no—
150C23×C4—26no—
151C23×C4—26no—
152C23×C4—212no—
153C23×C4—26no—
154C23×C4—26no—
155C23×C4—43no—
156C23×C4—43no—
157C23×C4—46no—
158C23×C4—46no—
159C23×C4—46no—
160C23×C4C22×C4112yes—
161C23×C4C22×C4112yes—
162C23×C4C22×C4112yes—
163C23×C4C22×C4112yes—
164C23×C4C2413yes—
165C23×C4C2413yes—
166C23×C4C2413yes—
167C23×C4C2413yes—
168C23×C4C2413yes—
169C23×C4C2413yes—
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2413yes—
176C23×C4C2413yes—
177#123C42164yescomplemented
178C4×D4—448no—
179C2×C4⋊C4—212no—
180C2×C4⋊C4—43no—
181C2×C4⋊C4—43no—
182C2×C4⋊C4—46no—
183C2×C4⋊C4C22×C4112yes—
184C2×C4⋊C4C22×C4112yes—
185C2×C4⋊C4C22×C4112yes—
186C2×C4⋊C4C22×C4112yes—
187C2×C4⋊C4C22×C4112yes—
188C2×C22⋊C4—212no—
189C2×C22⋊C4—26no—
190C2×C22⋊C4—26no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—43no—
193C2×C22⋊C4—43no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4C22×C4112yes—
196C2×C22⋊C4C22×C4112yes—
197C2×C22⋊C4C22×C4112yes—
198C2×C22⋊C4C22×C4112yes—
199C2×C22⋊C4C22×C4112yes—
200C2×C22⋊C4C22×C4112yes—
201C2×C22⋊C4C22×C4112yes—
202C2×C22⋊C4C22×C4112yes—
203C2×C42—26no—
204C2×C42—26no—
205C2×C42—212no—
206C2×C42—212no—
207C2×C42—46no—
208C2×C42—448no—
209C2×C42—46no—
210C2×C42—46no—
211C2×C42—46no—
212C2×C42—46no—
213C2×C42—46no—
214C2×C42—46no—
215C2×C42—46no—
216C2×C42—46no—
217C2×C42—46no—
218C2×C42C22×C4112yes—
219C2×C42C22×C4112yes—
220C2×C42C22×C4112yes—
221C2×C42C22×C4112yes—
222C2×C42C22×C4112yes—
223C2×C42C22×C4112yes—
224C2×C42C22×C4112yes—
225#95—26no—
226#95—26no—
227#95—26no—
228#95—212no—
229#95—26no—
230#95—26no—
231#95—26no—
232#95—46no—
233#95—46no—
234#95—46no—
235#95—46no—
236#95C22×C4112yes—
237#95C22×C4112yes—
238#95C22×C4112yes—
239#95C22×C4112yes—
240#95C22×C4112yes—
241#95C22×C4112yes—
242#95C22×C4112yes—
243#95C22×C4112yes—
244#95C22×C4112yes—
245#95C22×C4112yes—
246#95#4114yes—
247#95#41112yes—
248C24—23no—
249C24—21no—
250C24—21no—
251C24—23no—
252C24—23no—
253C24—21no—
254C24—23no—
255C24—21no—
256C24—21no—
257C24—23no—
258C24C23×C416yes—
259C24C23×C416yes—
260C24H(5,𝔽2)16yes—
261C242−1+412yes—
262C24C2511yescharacteristic
263C2×D4—224no—
264C2×D4—224no—
265C2×D4C2×C42116yes—
266C22×C4—224no—
267C22×C4—224no—
268C22×C4—21no—
269C22×C4—23no—
270C22×C4—23no—
271C22×C4—23no—
272C22×C4—26no—
273C22×C4—26no—
274C22×C4—23no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—23no—
279C22×C4—23no—
280C22×C4—23no—
281C22×C4—26no—
282C22×C4—21no—
283C22×C4—224no—
284C22×C4—224no—
285C22×C4—23no—
286C22×C4—26no—
287C22×C4—23no—
288C22×C4—23no—
289C22×C4—23no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—23no—
293C22×C4—23no—
294C22×C4—23no—
295C22×C4—23no—
296C22×C4—26no—
297C22×C4—26no—
298C22×C4—26no—
299C22×C4—26no—
300C22×C4—23no—
301C22×C4—23no—
302C22×C4—23no—
303C22×C4—23no—
304C22×C4—26no—
305C22×C4—26no—
306C22×C4—26no—
307C22×C4—23no—
308C22×C4—23no—
309C22×C4—23no—
310C22×C4—23no—
311C22×C4—224no—
312C22×C4—224no—
313C22×C4—224no—
314C22×C4—412no—
315C22×C4—412no—
316C22×C4—412no—
317C22×C4—412no—
318C22×C4—424no—
319C22×C4—424no—
320C22×C4—424no—
321C22×C4—424no—
322C22×C4—812no—
323C22×C4—812no—
324C22×C4C2×C42116yes—
325C22×C4C23×C416yes—
326C22×C4C23×C416yes—
327C22×C4C23×C416yes—
328C22×C4C23×C416yes—
329C22×C4C23×C416yes—
330C22×C4C23×C416yes—
331C22×C4C23×C416yes—
332C22×C4C23×C416yes—
333C22×C4C23×C416yes—
334C22×C4C23×C416yes—
335C22×C4C23×C416yes—
336C22×C4C23×C416yes—
337C22×C4C23×C416yes—
338C22×C4C2×#4116yes—
339C22×C4C2×#4116yes—
340C22×C4C2×#41112yes—
341C4⋊C4—224no—
342C4⋊C4—412no—
343C4⋊C4—412no—
344C4⋊C4C2×C42116yes—
345C4⋊C4C2×C42116yes—
346C22⋊C4—224no—
347C22⋊C4—224no—
348C22⋊C4—224no—
349C22⋊C4—412no—
350C22⋊C4—412no—
351C22⋊C4C2×C42116yes—
352C22⋊C4C2×C42116yes—
353C22⋊C4C2×C42116yes—
354C42—224no—
355C42—412no—
356C42—412no—
357C42—412no—
358C42—412no—
359C42—412no—
360C42—412no—
361C42—812no—
362C42—812no—
363C23—26no—
364C23—26no—
365C23—26no—
366C23—26no—
367C23—26no—
368C23—212no—
369C23—26no—
370C23—26no—
371C23—22no—
372C23—26no—
373C23—26no—
374C23—26no—
375C23—26no—
376C23—26no—
377C23—26no—
378C23—43no—
379C23—46no—
380C23—43no—
381C23—43no—
382C23—41no—
383C23C22×C4214yes—
384C23C22×C4214yes—
385C23#518112yes—
386C23#520112yes—
387C23C24×C413yes—
388C23C22×#4113yes—
389C23C22×#4111yescharacteristic
390C23C2×H(5,𝔽2)13yes—
391C23C2×2−1+411yescharacteristic
392C23#58511yescharacteristic
393C23#58513yes—
394D4—232no—
395C2×C4—26no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—26no—
399C2×C4—26no—
400C2×C4—26no—
401C2×C4—26no—
402C2×C4—26no—
403C2×C4—26no—
404C2×C4—26no—
405C2×C4—26no—
406C2×C4—26no—
407C2×C4—26no—
408C2×C4—26no—
409C2×C4—212no—
410C2×C4—212no—
411C2×C4—212no—
412C2×C4—212no—
413C2×C4—232no—
414C2×C4—26no—
415C2×C4—26no—
416C2×C4—26no—
417C2×C4—26no—
418C2×C4—26no—
419C2×C4—26no—
420C2×C4—26no—
421C2×C4—26no—
422C2×C4—26no—
423C2×C4—26no—
424C2×C4—26no—
425C2×C4—26no—
426C2×C4—26no—
427C2×C4—26no—
428C2×C4—26no—
429C2×C4—26no—
430C2×C4—26no—
431C2×C4—26no—
432C2×C4—448no—
433C2×C4—448no—
434C2×C4—43no—
435C2×C4—46no—
436C2×C4—46no—
437C2×C4—46no—
438C2×C4—46no—
439C2×C4—43no—
440C2×C4—46no—
441C2×C4—46no—
442C2×C4—46no—
443C2×C4—46no—
444C2×C4—46no—
445C2×C4—46no—
446C2×C4—43no—
447C2×C4—43no—
448C2×C4—824no—
449C2×C4C22×C4214yes—
450C2×C4C22×C4214yes—
451C2×C4C22×C4214yes—
452C2×C4C22×C4214yes—
453C2×C4C22×C4214yes—
454C2×C4C4×#41112yes—
455C2×C4C4×#41112yes—
456C22—212no—
457C22—212no—
458C22—212no—
459C22—212no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—46no—
466C22—46no—
467C22—46no—
468C22—46no—
469C22#227518yes—
470C22C23×C4211yescharacteristic
471C22C2×C4×#4113yes—
472C22C2×C4×#4113yes—
473C22C2×#51813yes—
474C22C2×#52013yes—
475C22#342813yes—
476C22#342813yes—
477C22#351313yes—
478C22#351316yes—
479C22#351313yes—
480C22#351613yes—
481C22#351611yescharacteristic
482C4—212no—
483C4—212no—
484C4—24no—
485C4—24no—
486C4—24no—
487C4—24no—
488C4—24no—
489C4—46no—
490C4—46no—
491C4—46no—
492C4—46no—
493C4—46no—
494C4—46no—
495C2—28no—
496C2—44no—
497C2C42×#4111yescharacteristic
498C2C2×#227511yescharacteristic
499C2#3399911yescharacteristic
500C2#3422813yes—
501C2#3423013yes—
502C2#3423813yes—
503C2#3424113yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC22C23×C42
prime radicalC23×D4C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13822C2semidirect
3. #13819C2semidirect
4. #13814C2semidirect
6. #13812C2semidirect
7. #13809C2semidirect
8. #13809C2semidirect
10. #13788C2semidirect
11. #13787C2semidirect
12. C4×#381C2semidirect
13. #13750C2semidirect
14. #13749C2semidirect
15. C2×C4×#95C2semidirect
19. C4×#123C4semidirect
89. C22×C42C2×C4semidirect
99. #388C2×C4semidirect
100. #388C2×C4semidirect
101. #387C2×C4semidirect
102. #387C2×C4semidirect
104. #385C2×C4semidirect
105. #381C2×C4semidirect
107. C4×C4⋊C4C2×C4semidirect
108. C4×C4⋊C4C2×C4semidirect
111. C4×C22⋊C4C2×C4semidirect
112. C4×C22⋊C4C2×C4semidirect
113. C4×C22⋊C4C2×C4semidirect
177. #123C42semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
#138222
#138192
#138142
#138132
#138122
#138092
D4×#952
#137882
#137872
C4×#3812
#137502
#137492
C2×C4×#952