Finite Group Databasebeta

← #23525

Subgroups — #23525

Order 256.

666 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#575C211yesmaximal, characteristic
2C2×#430C211yesmaximal, characteristic, complemented
3C2×#421C211yesmaximal, characteristic, complemented
4#2896C211yesmaximal, characteristic, complemented
5C2×#413C211yesmaximal, characteristic, complemented
6C2×#413C211yesmaximal, characteristic, complemented
7#2049C214yesmaximal, complemented
8#2048C214yesmaximal, complemented
9C2×#344C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C22×#41C2211yescharacteristic
12#575—22no—
13#575—22no—
14#575—22no—
15#575—22no—
16C2×#137—21no—
17C2×#137C2212yes—
18C2×AGL(1,ℤ/8)—21no—
19C2×AGL(1,ℤ/8)C2212yes—
20C2×#135—22no—
21C22×M16C2211yescharacteristic
22C22×M16C2211yescharacteristic
23C22×M16C2211yescharacteristic
24C22×M16C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#430—22no—
27#430C2214yescomplemented
28C2×#108—22no—
29#421—22no—
30#421C2214yes—
31C2×#104—22no—
32#419C2212yescomplemented
33#418C2212yescomplemented
34#417—22no—
35#416—22no—
36#413—22no—
37#413—22no—
38#413—22no—
39#413C2214yes—
40C2×#101—21no—
41C2×#101C2212yescomplemented
42C2×#100—21no—
43C2×#100C2212yescomplemented
44#344C2214yescomplemented
45C2×#41—21no—
46C2×#41—21no—
47C2×#41—21no—
48C2×#41—21no—
49C2×#41—21no—
50C2×#41—21no—
51C2×#41—21no—
52C2×#41—21no—
53C2×#41—21no—
54C2×#41—21no—
55C2×#41—21no—
56C2×#41—21no—
57C2×#41—21no—
58C2×#41—21no—
59C2×#41—21no—
60C2×#41—21no—
61C2×#41—21no—
62C2×#41—21no—
63C2×#41—21no—
64C2×#41—21no—
65C2×#41D412yes—
66C2×#41D412yes—
67C2×#41D412yes—
68C2×#41C2312yes—
69C22×Q8D411yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78#137—24no—
79#137—24no—
80#137—24no—
81#137—24no—
82AGL(1,ℤ/8)—24no—
83AGL(1,ℤ/8)—24no—
84AGL(1,ℤ/8)—24no—
85AGL(1,ℤ/8)—24no—
86#135—28no—
87#135—28no—
88C2×Q16—22no—
89C2×SD16—22no—
90C2×SD16—22no—
91C2×D8—22no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—21no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—21no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16D412yes—
109C2×M16D412yes—
110C2×M16D412yes—
111C2×M16D412yes—
112C2×M16C2312yes—
113C2×M16C2312yes—
114C2×M16C2312yes—
115C2×M16C2312yes—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120#117—21no—
121#117—21no—
122#117—21no—
123#117C2312yes—
124C2×C4⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C42—21no—
127#108—28no—
128#104—28no—
129#103—24no—
130#102—24no—
131#101—24no—
132#101—24no—
133#100—24no—
134#100—24no—
135C24—21no—
136C24—21no—
137C24#4111yescharacteristic
138#41—22no—
139#41—22no—
140#41—22no—
141#41—22no—
142#41—22no—
143#41—22no—
144#41—22no—
145#41—22no—
146#41—22no—
147#41—22no—
148#41—22no—
149#41—22no—
150#41—22no—
151#41—22no—
152#41—22no—
153#41—22no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161#41—41no—
162#41—41no—
163#41—41no—
164#41—41no—
165#41—41no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8C2×D412yes—
181C2×Q8C2×D412yes—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×D412yes—
205C2×D4C2×D412yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4#4111yescharacteristic
230C22×C4#4111yescharacteristic
231C22×C4#4111yescharacteristic
232C22×C4#4111yescharacteristic
233C22×C4#4111yescharacteristic
234C22×C4C2411yescharacteristic
235Q16—28no—
236SD16—28no—
237SD16—28no—
238D8—28no—
239M16—22no—
240M16—22no—
241M16—22no—
242M16—22no—
243M16—22no—
244M16—22no—
245M16—22no—
246M16—22no—
247M16—22no—
248M16—22no—
249M16—41no—
250M16—41no—
251M16—41no—
252M16—41no—
253M16C2×D414yes—
254M16C2×D414yes—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C22⋊C4—22no—
270C22⋊C4—41no—
271C42—22no—
272C42—41no—
273C23—22no—
274C23—22no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—81no—
280C23—82no—
281C23—82no—
282C23#12111yescharacteristic
283C23#12111yescharacteristic
284C23#12311yescharacteristic
285C23#12311yescharacteristic
286C23#12411yescharacteristic
287C23#12411yescharacteristic
288C23C2×#4111yescharacteristic
289Q8—22no—
290Q8—22no—
291Q8—22no—
292Q8—22no—
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297D4—22no—
298D4—22no—
299D4—22no—
300D4—22no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—42no—
305D4—42no—
306D4—42no—
307D4—42no—
308D4—42no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4#12011yescharacteristic
330C2×C4#12011yescharacteristic
331C2×C4#12011yescharacteristic
332C2×C4#12011yescharacteristic
333C2×C4#12111yescharacteristic
334C2×C4#12111yescharacteristic
335C2×C4#12111yescharacteristic
336C2×C4#12111yescharacteristic
337C2×C4#12111yescharacteristic
338C2×C4#12111yescharacteristic
339C2×C4#12111yescharacteristic
340C2×C4#12111yescharacteristic
341C2×C4#12111yescharacteristic
342C2×C4#12111yescharacteristic
343C2×C4#12311yescharacteristic
344C2×C4#12311yescharacteristic
345C2×C4#12311yescharacteristic
346C2×C4#12311yescharacteristic
347C2×C4#12311yescharacteristic
348C2×C4#12311yescharacteristic
349C2×C4#12411yescharacteristic
350C2×C4#12411yescharacteristic
351C2×C4C22×D411yescharacteristic
352C2×C4C22×D411yescharacteristic
353C2×C4C22×D411yescharacteristic
354C2×C4C22×D411yescharacteristic
355C2×C4C2×#4111yescharacteristic
356C2×C4C2×#4111yescharacteristic
357C8—24no—
358C8—24no—
359C8—42no—
360C8—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22#39411yescharacteristic
377C22#39411yescharacteristic
378C22#39411yescharacteristic
379C22#39411yescharacteristic
380C22C2×#12111yescharacteristic
381C22C2×#12311yescharacteristic
382C22C2×#12411yescharacteristic
383C4—42no—
384C4—42no—
385C4—82no—
386C4#39411yescharacteristic
387C4#39411yescharacteristic
388C4#39411yescharacteristic
389C4#39411yescharacteristic
390C4C2×#12011yescharacteristic
391C4C2×#12111yescharacteristic
392C4C2×#12111yescharacteristic
393C4C2×#12311yescharacteristic
394C2—21no—
395C2—21no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—82no—
402C2—82no—
403C2C2×#39411yescharacteristic
subgroupG/H
centerC4#394
commutator, Frattini, cosocleC22×C4C24
socleC2C2×#394
prime radicalC2×#575C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#430C2semidirect
3. C2×#421C2semidirect
4. #2896C2semidirect
5. C2×#413C2semidirect
6. C2×#413C2semidirect
7. #2049C2semidirect
8. #2048C2semidirect
9. C2×#344C2semidirect
27. #430C22semidirect
32. #419C22semidirect
33. #418C22semidirect
41. C2×#101C22semidirect
43. C2×#100C22semidirect
44. #344C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5752
C2×#4302
C2×#4212
#28962
C2×#4132
#20492
#20482
C2×#3442