Finite Group Databasebeta

← #15614

Subgroups — #15614

Order 256.

959 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#398C211yesmaximal, characteristic, complemented
3C2×#396C211yesmaximal, characteristic
4C2×#395C211yesmaximal, characteristic, complemented
5C2×#389C211yesmaximal, characteristic, complemented
6C2×#385C211yesmaximal, characteristic
7#1446C211yesmaximal, characteristic, complemented
8#1444C211yesmaximal, characteristic, complemented
9#1441C211yesmaximal, characteristic, complemented
10#1441C211yesmaximal, characteristic, complemented
11#1439C211yesmaximal, characteristic
12#1439C211yesmaximal, characteristic
13#1437C211yesmaximal, characteristic
14C4⋊#95C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C2×C22⋊Q8—24no—
19C2×C22⋊Q8—24no—
20C2×C22⋊Q8—24no—
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31C22×C22⋊C4C2211yescharacteristic
32C22×C22⋊C4C2211yescharacteristic
33C22×C22⋊C4C2211yescharacteristic
34C22×C22⋊C4C2211yescharacteristic
35C22×C22⋊C4C2211yescharacteristic
36C22×C22⋊C4C2211yescharacteristic
37C22×C42C2211yescharacteristic
38#398—24no—
39#396—24no—
40#395—24no—
41#389—24no—
42#385—24no—
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
58C2×#95C2211yescharacteristic
59C25C2311yescharacteristic
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—41no—
66C22×Q8—41no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—41no—
73C23×C4—41no—
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
87C23×C4C2311yescharacteristic
88C22⋊Q8—232no—
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—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4D414yes—
117C2×C4⋊C4D414yes—
118C2×C4⋊C4D414yes—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4D414yes—
135C2×C22⋊C4Q814yescomplemented
136C2×C22⋊C4Q814yescomplemented
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—41no—
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—22no—
149#95—22no—
150#95—22no—
151#95—22no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24C2×D412yes—
169C24C2×Q812yes—
170C24C2×Q812yes—
171C24C2411yescharacteristic
172C2×Q8—28no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
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—28no—
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×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—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4C2×D412yes—
267C22×C4C2×D412yes—
268C22×C4C2×D412yes—
269C22×C4C2×D412yes—
270C22×C4C2×D412yes—
271C22×C4C2×D412yes—
272C22×C4C2×D412yes—
273C22×C4C2×D412yes—
274C22×C4C2×D412yes—
275C22×C4C2×D412yes—
276C22×C4C2×D412yes—
277C22×C4C2×Q812yes—
278C22×C4C2×Q812yes—
279C22×C4C2×Q812yes—
280C22×C4C2×Q812yes—
281C22×C4#4112yes—
282C22×C4#4112yes—
283C22×C4#4112yes—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C22⋊C4—28no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C42—44no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23C4⋊Q814yes—
336C23C22×D411yescharacteristic
337C23C22×D411yescharacteristic
338C23C22×D411yescharacteristic
339C23C22×D411yescharacteristic
340C23C22×Q811yescharacteristic
341C23C22×Q811yescharacteristic
342C23C2×#4111yescharacteristic
343C23C2×#4111yescharacteristic
344C23C2×#4111yescharacteristic
345C23H(5,𝔽2)11yescharacteristic
346C23H(5,𝔽2)11yescharacteristic
347C23H(5,𝔽2)11yescharacteristic
348C232−1+411yescharacteristic
349C232−1+411yescharacteristic
350C232−1+411yescharacteristic
351Q8—416no—
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—22no—
397C2×C4—416no—
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×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4#12014yes—
449C2×C4#12114yes—
450C2×C4C22⋊Q814yescomplemented
451C22—24no—
452C22—24no—
453C22—24no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22C2×#12011yescharacteristic
467C22C2×#12111yescharacteristic
468C22C2×C22⋊Q811yescharacteristic
469C22C2×C4⋊Q811yescharacteristic
470C22#53211yescharacteristic
471C22#53311yescharacteristic
472C22#53511yescharacteristic
473C22#53611yescharacteristic
474C22#53711yescharacteristic
475C22#53711yescharacteristic
476C22#53911yescharacteristic
477C22#54111yescharacteristic
478C22#54211yescharacteristic
479C22#54311yescharacteristic
480C22#54411yescharacteristic
481C22D4211yescharacteristic
482C22D4211yescharacteristic
483C22#54611yescharacteristic
484C22#54611yescharacteristic
485C22#54611yescharacteristic
486C22#54611yescharacteristic
487C22#54711yescharacteristic
488C22#54711yescharacteristic
489C22#54811yescharacteristic
490C22D4×Q811yescharacteristic
491C22D4×Q811yescharacteristic
492C22D4×Q811yescharacteristic
493C22D4×Q811yescharacteristic
494C22D4×Q811yescharacteristic
495C22#55211yescharacteristic
496C22#55411yescharacteristic
497C22#55411yescharacteristic
498C22#55411yescharacteristic
499C22#56211yescharacteristic
500C22#56311yescharacteristic
501C4—24no—
502C4—24no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—81no—
514C4—81no—
515C2—44no—
516C2#240811yescharacteristic
517C2#241711yescharacteristic
518C2#242911yescharacteristic
519C2#243011yescharacteristic
520C2#243411yescharacteristic
521C2#244011yescharacteristic
522C2#245211yescharacteristic
523C2#254911yescharacteristic
524C2#255611yescharacteristic
525C2#268111yescharacteristic
526C2#268311yescharacteristic
527C2#269211yescharacteristic
528C2#269811yescharacteristic
529C2#270411yescharacteristic
530C2#272011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#398C2semidirect
4. C2×#395C2semidirect
5. C2×#389C2semidirect
7. #1446C2semidirect
8. #1444C2semidirect
9. #1441C2semidirect
10. #1441C2semidirect
14. C4⋊#95C2semidirect
135. C2×C22⋊C4Q8semidirect
136. C2×C22⋊C4Q8semidirect
450. 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×#3982
C2×#3962
C2×#3952
C2×#3892
C2×#3852
#14462
#14442
#14412
#14392
#14372
C4⋊#952
C22⋊#952