Finite Group Databasebeta

← #14099

Subgroups — #14099

Order 256.

1003 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#391C211yesmaximal, characteristic, complemented
3C2×#387C211yesmaximal, characteristic, complemented
4C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C4C211yesmaximal, characteristic
6#1446C211yesmaximal, characteristic, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1439C211yesmaximal, characteristic
9#1439C211yesmaximal, characteristic
10#1437C211yesmaximal, characteristic
11#1437C211yesmaximal, characteristic
12#1436C211yesmaximal, characteristic, complemented
13#1436C211yesmaximal, characteristic, complemented
14#1436C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C2×C22⋊Q8—24no—
19C2×C22⋊Q8—24no—
20C2×C22⋊Q8C418yescomplemented
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2211yescharacteristic
32C22×C42C2211yescharacteristic
33C22×C42C2211yescharacteristic
34C22×C42C2211yescharacteristic
35C22×C42C2211yescharacteristic
36#391—24no—
37#387—24no—
38C4×C4⋊C4—24no—
39C4×C22⋊C4—24no—
40C2×#95C2211yescharacteristic
41C2×#95C2211yescharacteristic
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
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8C2×C414yes—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4C2×C414yes—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C22⋊Q8—232no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C2×C414yes—
99C2×C4⋊C4C2×C414yes—
100C2×C4⋊C4C2×C414yes—
101C2×C4⋊C4D414yes—
102C2×C4⋊C4D414yes—
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—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4C2×C414yes—
119C2×C22⋊C4C2×C414yes—
120C2×C22⋊C4Q814yescomplemented
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
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—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—41no—
149#95—41no—
150#95D414yes—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24C22×C412yes—
164C24C2×Q812yes—
165C24#4112yes—
166C24C2411yescharacteristic
167C2×Q8—28no—
168C2×Q8—28no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
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—28no—
237C22×C4—28no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4C22×C412yes—
258C22×C4C22×C412yes—
259C22×C4C22×C412yes—
260C22×C4C22×C412yes—
261C22×C4C22×C412yes—
262C22×C4C22×C412yes—
263C22×C4C2×D412yes—
264C22×C4C2×D412yes—
265C22×C4C2×D412yes—
266C22×C4C2×D412yes—
267C22×C4C2×D412yes—
268C22×C4C2×D412yes—
269C22×C4C2×D412yes—
270C22×C4C2×D412yes—
271C22×C4C2×D412yes—
272C22×C4C2×Q812yes—
273C22×C4C2×Q812yes—
274C22×C4#4112yes—
275C22×C4#4112yes—
276C22×C4#4112yes—
277C22×C4#4112yes—
278C22×C4#4112yes—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—44no—
283C4⋊C4C22⋊C4116yescomplemented
284C22⋊C4—28no—
285C22⋊C4—28no—
286C22⋊C4—28no—
287C22⋊C4—28no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C42—28no—
291C42—44no—
292C42—44no—
293C42—44no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23C4×Q814yes—
329C23C23×C411yescharacteristic
330C23C22×D411yescharacteristic
331C23C22×D411yescharacteristic
332C23C22×D411yescharacteristic
333C23C22×Q811yescharacteristic
334C23C2×#4111yescharacteristic
335C23C2×#4111yescharacteristic
336C23C2×#4111yescharacteristic
337C23C2×#4111yescharacteristic
338C23C2×#4111yescharacteristic
339C23C2×#4111yescharacteristic
340C23H(5,𝔽2)11yescharacteristic
341C23H(5,𝔽2)11yescharacteristic
342C232−1+411yescharacteristic
343C232−1+411yescharacteristic
344Q8—416no—
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—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—416no—
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×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4C2×C22⋊C414yes—
440C2×C4C2×C22⋊C414yes—
441C2×C4C2×C22⋊C414yes—
442C2×C4C4×D414yes—
443C2×C4C22⋊Q814yescomplemented
444C22—24no—
445C22—24no—
446C22—24no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22C22×C22⋊C411yescharacteristic
460C22C2×C4×D411yescharacteristic
461C22C2×C4×Q811yescharacteristic
462C22C4×#4111yescharacteristic
463C22C4×#4111yescharacteristic
464C22#51811yescharacteristic
465C22#51911yescharacteristic
466C22C2×C22⋊Q811yescharacteristic
467C22#52511yescharacteristic
468C22#52511yescharacteristic
469C22#52511yescharacteristic
470C22#52911yescharacteristic
471C22#53211yescharacteristic
472C22#53211yescharacteristic
473C22#53311yescharacteristic
474C22#53911yescharacteristic
475C22#53911yescharacteristic
476C22#54211yescharacteristic
477C22#54211yescharacteristic
478C22D4211yescharacteristic
479C22D4211yescharacteristic
480C22#54611yescharacteristic
481C22#54611yescharacteristic
482C22#54711yescharacteristic
483C22#54811yescharacteristic
484C22#54811yescharacteristic
485C22#54811yescharacteristic
486C22D4×Q811yescharacteristic
487C22D4×Q811yescharacteristic
488C22#55111yescharacteristic
489C22#55111yescharacteristic
490C22#55211yescharacteristic
491C22#55411yescharacteristic
492C22#55411yescharacteristic
493C22#55611yescharacteristic
494C4—24no—
495C4—24no—
496C4—24no—
497C4—24no—
498C4—24no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—81no—
508C2—44no—
509C2#229711yescharacteristic
510C2C4×C22⋊Q811yescharacteristic
511C2#233211yescharacteristic
512C2D4×C22⋊C411yescharacteristic
513C2Q8×C22⋊C411yescharacteristic
514C2#236211yescharacteristic
515C2#236811yescharacteristic
516C2#244911yescharacteristic
517C2#245211yescharacteristic
518C2#246111yescharacteristic
519C2#246311yescharacteristic
520C2#246511yescharacteristic
521C2#247711yescharacteristic
522C2#255611yescharacteristic
523C2#256011yescharacteristic
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×#391C2semidirect
3. C2×#387C2semidirect
4. C2×C4×C4⋊C4C2semidirect
6. #1446C2semidirect
7. #1441C2semidirect
12. #1436C2semidirect
13. #1436C2semidirect
14. #1436C2semidirect
20. C2×C22⋊Q8C4semidirect
120. C2×C22⋊C4Q8semidirect
283. C4⋊C4C22⋊C4semidirect
443. C2×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#3912
C2×#3872
C2×C4×C4⋊C42
C2×C4×C22⋊C42
#14462
#14412
#14392
#14372
#14362
C22⋊#952