Finite Group Databasebeta

← #24369

Subgroups — #24369

Order 256.

694 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#137C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5#2432C211yesmaximal, characteristic
6Q8×C22⋊C4C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1851C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C2×#137—24no—
19C2×#137—24no—
20C2×#137—24no—
21C22×Q16C2211yescharacteristic
22C22×Q16C2211yescharacteristic
23C22×Q16C2211yescharacteristic
24C22×Q16C2211yescharacteristic
25C22×SD16C2211yescharacteristic
26C22×SD16C2211yescharacteristic
27C22×M16C2211yescharacteristic
28C2×C4⋊Q8C2211yescharacteristic
29C2×#124C2211yescharacteristic
30C2×C22⋊Q8C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×Q8C2211yescharacteristic
33#464—24no—
34#462—24no—
35#451—24no—
36C2×C4⋊C8C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47#394—21no—
48#394—21no—
49#391—21no—
50#391—21no—
51#391C2211yescharacteristic
52#391C2211yescharacteristic
53#391C2211yescharacteristic
54#390C2211yescharacteristic
55#385—21no—
56#385—21no—
57#380—21no—
58#380C2211yescharacteristic
59C4×C22⋊C4—21no—
60C4×C22⋊C4C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#336C2211yescharacteristic
64#336C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—41no—
70C2×#41—41no—
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8D414yes—
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×Q8C2311yescharacteristic
82C22×D4—21no—
83C22×D4C2311yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87#137—232no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—41no—
98C2×Q16—41no—
99C2×Q16D414yes—
100C2×SD16—22no—
101C2×SD16—22no—
102C2×SD16—22no—
103C2×SD16—22no—
104C2×SD16—22no—
105C2×SD16D414yes—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C4⋊Q8—22no—
114C4⋊Q8—41no—
115#124—22no—
116#124—41no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—41no—
122C4×Q8—41no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4D411yescharacteristic
138C2×C22⋊C4D411yescharacteristic
139C2×C42—21no—
140C2×C42—21no—
141C2×C42C2311yescharacteristic
142C2×C42C2311yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—22no—
145#103—22no—
146#103—22no—
147#103—22no—
148#103—22no—
149#103—22no—
150#103—41no—
151#102—22no—
152#102—41no—
153C22⋊C8—22no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95D411yescharacteristic
162#95D411yescharacteristic
163C24—21no—
164C24C2×D411yescharacteristic
165#41—28no—
166#41—44no—
167#41—44no—
168#41—44no—
169C2×Q8—24no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—28no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8C2×D412yes—
196C2×Q8C2×D412yes—
197C2×Q8C2×D412yes—
198C2×Q8C2×D412yes—
199C2×Q8C2×D412yes—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4C2×D412yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D412yes—
231C22×C4C2×D411yescharacteristic
232C22×C4#4112yes—
233C22×C4C2411yescharacteristic
234Q16—28no—
235Q16—28no—
236Q16—44no—
237Q16—44no—
238SD16—28no—
239SD16—28no—
240M16—28no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8C2×D412yes—
252C2×C8C2×D412yes—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C22⋊C4—22no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—42no—
277C42—42no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#12112yes—
292C23C22×D411yescharacteristic
293Q8—22no—
294Q8—22no—
295Q8—22no—
296Q8—22no—
297Q8—22no—
298Q8—22no—
299Q8—22no—
300Q8—48no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308Q8—41no—
309Q8#12014yes—
310D4—22no—
311D4—22no—
312D4—48no—
313D4—41no—
314D4—41no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4#12112yes—
356C2×C4#13712yescomplemented
357C2×C4C22×D411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C2×#4111yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8—41no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22C2×#12111yescharacteristic
382C22D4211yescharacteristic
383C22D4211yescharacteristic
384C22C2×#13711yescharacteristic
385C22#57511yescharacteristic
386C22#57811yescharacteristic
387C22#57811yescharacteristic
388C4—22no—
389C4—24no—
390C4—22no—
391C4—42no—
392C4—42no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4C2×#12011yescharacteristic
401C4#53411yescharacteristic
402C4#54611yescharacteristic
403C4#54611yescharacteristic
404C2—42no—
405C2—82no—
406C2#241011yescharacteristic
407C2#303711yescharacteristic
408C2#304411yescharacteristic
409C2#326911yescharacteristic
410C2#327011yescharacteristic
411C2#327611yescharacteristic
412C2#327711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#464C2semidirect
3. C2×#462C2semidirect
4. C2×#451C2semidirect
6. Q8×C22⋊C4C2semidirect
7. #2056C2semidirect
8. #2055C2semidirect
9. #2016C2semidirect
10. #2015C2semidirect
11. #1894C2semidirect
12. #1880C2semidirect
13. #1878C2semidirect
14. #1851C2semidirect
356. C2×C4#137semidirect

Maximal subgroups

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

subgroupindex
C22×#1372
C2×#4642
C2×#4622
C2×#4512
#24322
Q8×C22⋊C42
#20562
#20552
#20162
#20152
#18942
#18802
#18782
#18512
#17882