Finite Group Databasebeta

← #14200

Subgroups — #14200

Order 256.

843 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#396C211yesmaximal, characteristic
2C2×#389C211yesmaximal, characteristic, complemented
3C2×#388C211yesmaximal, characteristic
4C2×#388C211yesmaximal, characteristic
5C2×#388C211yesmaximal, characteristic
6C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
7C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1441C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic
11#1437C211yesmaximal, characteristic
12#1437C211yesmaximal, characteristic
13#1436C211yesmaximal, characteristic, complemented
14#1436C211yesmaximal, characteristic, complemented
15#1434C211yesmaximal, characteristic, complemented
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C42C2211yescharacteristic
31C22×C42C2211yescharacteristic
32C22×C42C2211yescharacteristic
33C22×C42C2211yescharacteristic
34C22×C42C2211yescharacteristic
35#396C418yescomplemented
36#389—24no—
37#388—24no—
38#388—24no—
39#388—24no—
40C4×C4⋊C4—24no—
41C4×C4⋊C4—24no—
42C2×#95C2211yescharacteristic
43C2×#95C2211yescharacteristic
44C2×#95C2211yescharacteristic
45C2×#95C2211yescharacteristic
46C2×#95C2211yescharacteristic
47C2×#95C2211yescharacteristic
48C2×#95C2211yescharacteristic
49C2×#95C2211yescharacteristic
50C2×#95C2211yescharacteristic
51C2×#95C2211yescharacteristic
52C2×#95C2211yescharacteristic
53C2×#95C2211yescharacteristic
54C2×#95C2211yescharacteristic
55C2×#95C2211yescharacteristic
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C25C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4C2×C414yes—
91C2×C4⋊C4C2×C414yes—
92C2×C4⋊C4C2×C414yes—
93C2×C4⋊C4D414yes—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4C2×C414yes—
113C2×C22⋊C4C2×C414yes—
114C2×C22⋊C4C2×C414yes—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42D414yes—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—22no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—41no—
144#95C2×C414yes—
145#95D414yes—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24C22×C412yes—
161C24C2411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4C22×C412yes—
239C22×C4C22×C412yes—
240C22×C4C22×C412yes—
241C22×C4C22×C412yes—
242C22×C4C22×C412yes—
243C22×C4C22×C412yes—
244C22×C4C2×D412yes—
245C22×C4C2×D412yes—
246C22×C4C2×D412yes—
247C22×C4C2×D412yes—
248C22×C4C2×D412yes—
249C22×C4C2×D412yes—
250C22×C4C2×D412yes—
251C22×C4C2×D412yes—
252C22×C4C2×D412yes—
253C22×C4#4112yes—
254C22×C4#4112yes—
255C22×C4#4112yes—
256C22×C4#4112yes—
257C22×C4#4112yes—
258C22×C4#4112yes—
259C22×C4#4112yes—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—28no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C42—28no—
275C42—28no—
276C42—44no—
277C42—44no—
278C42—44no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C23×C411yescharacteristic
315C23C22×D411yescharacteristic
316C23C22×D411yescharacteristic
317C23C22×D411yescharacteristic
318C23C2×#4111yescharacteristic
319C23C2×#4111yescharacteristic
320C23C2×#4111yescharacteristic
321C23C2×#4111yescharacteristic
322C23C2×#4111yescharacteristic
323C23C2×#4111yescharacteristic
324C23C2×#4111yescharacteristic
325C23H(5,𝔽2)11yescharacteristic
326C232−1+411yescharacteristic
327C232−1+411yescharacteristic
328C232−1+411yescharacteristic
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4C4×D414yes—
423C2×C4C4×D414yes—
424C2×C4C4×D414yes—
425C2×C4#12114yes—
426C2×C4#12114yes—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22C2×C4×D411yescharacteristic
443C22C2×C4×D411yescharacteristic
444C22C2×C4×D411yescharacteristic
445C22C4×#4111yescharacteristic
446C22C4×#4111yescharacteristic
447C22#51911yescharacteristic
448C22#52011yescharacteristic
449C22C2×#12111yescharacteristic
450C22C2×#12111yescharacteristic
451C22#52511yescharacteristic
452C22#52511yescharacteristic
453C22#52911yescharacteristic
454C22#52911yescharacteristic
455C22#53211yescharacteristic
456C22#53211yescharacteristic
457C22#53611yescharacteristic
458C22#53711yescharacteristic
459C22#54111yescharacteristic
460C22#54211yescharacteristic
461C22D4211yescharacteristic
462C22#54711yescharacteristic
463C22#54711yescharacteristic
464C22#54711yescharacteristic
465C22#54811yescharacteristic
466C22#54811yescharacteristic
467C22#54811yescharacteristic
468C22#55111yescharacteristic
469C22#55211yescharacteristic
470C22#55211yescharacteristic
471C22#55211yescharacteristic
472C22#55311yescharacteristic
473C22#55311yescharacteristic
474C22#55611yescharacteristic
475C22#55611yescharacteristic
476C22#55611yescharacteristic
477C4—24no—
478C4—24no—
479C4—24no—
480C4—24no—
481C4—24no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—81no—
490C4—81no—
491C2—82no—
492C2C4×#12111yescharacteristic
493C2C4×#12111yescharacteristic
494C2#232311yescharacteristic
495C2#236111yescharacteristic
496C2#236211yescharacteristic
497C2#236211yescharacteristic
498C2#237311yescharacteristic
499C2#241311yescharacteristic
500C2#241611yescharacteristic
501C2#246111yescharacteristic
502C2#246911yescharacteristic
503C2#251911yescharacteristic
504C2#252511yescharacteristic
505C2#254511yescharacteristic
506C2#260111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#389C2semidirect
6. C2×C4×C4⋊C4C2semidirect
7. C2×C4×C4⋊C4C2semidirect
8. #1441C2semidirect
9. #1441C2semidirect
13. #1436C2semidirect
14. #1436C2semidirect
15. #1434C2semidirect
35. #396C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3962
C2×#3892
C2×#3882
C2×C4×C4⋊C42
#14412
#14392
#14372
#14362
#14342