Finite Group Databasebeta

← #24363

Subgroups — #24363

Order 256.

713 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#135C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5#2432C211yesmaximal, characteristic, complemented
6#2341C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1892C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1850C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C2×#135—24no—
19C2×#135—24no—
20C2×#135—24no—
21C22×Q16C2211yescharacteristic
22C22×Q16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×SD16C2211yescharacteristic
25C22×SD16C2211yescharacteristic
26C22×D8C2211yescharacteristic
27C23×C8C2211yescharacteristic
28C2×C4⋊Q8C2211yescharacteristic
29C2×#124C2211yescharacteristic
30C2×C22⋊Q8C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×Q8C2211yescharacteristic
33#464—24no—
34#462—24no—
35#452—24no—
36C2×C4⋊C8C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47#394—21no—
48#394—21no—
49#391C2211yescharacteristic
50#390—21no—
51#390—21no—
52#390C2211yescharacteristic
53#390C2211yescharacteristic
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—22no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—41no—
73C2×#41—41no—
74C2×#41—41no—
75C2×#41—41no—
76C2×#41D414yes—
77C22×Q8C2311yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4—21no—
81C22×D4—21no—
82C22×D4C2311yescharacteristic
83C22×D4C2311yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87#135—232no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16D414yes—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—22no—
99C2×SD16—22no—
100C2×SD16—41no—
101C2×SD16—41no—
102C2×SD16D414yes—
103C2×D8—22no—
104C2×D8—22no—
105C2×D8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—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⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4D411yescharacteristic
142C2×C22⋊C4D411yescharacteristic
143C2×C42—21no—
144C2×C42—21no—
145C2×C42C2311yescharacteristic
146C2×C42C2311yescharacteristic
147C4⋊C8—22no—
148C4⋊C8—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#103—41no—
154#102—22no—
155#102—22no—
156#102—41no—
157C22⋊C8—22no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95D411yescharacteristic
162#95D411yescharacteristic
163C24—21no—
164C24—21no—
165C24C2×D411yescharacteristic
166#41—28no—
167#41—28no—
168#41—44no—
169#41—44no—
170#41—44no—
171#41—44no—
172#41—44no—
173#41—44no—
174C2×Q8—21no—
175C2×Q8—21no—
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×Q8C2×D412yes—
189C2×Q8C2×D412yes—
190C2×Q8C2×D412yes—
191C2×Q8C2×D412yes—
192C2×D4—24no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4C2×D412yes—
209C2×D4C2×D412yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
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—44no—
236SD16—28no—
237SD16—28no—
238SD16—44no—
239D8—28no—
240C2×C8—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—
272C22⋊C4—42no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—42no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
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#12112yes—
303C23C22×D411yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—22no—
307Q8—22no—
308Q8—22no—
309Q8—41no—
310Q8—41no—
311Q8—41no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315Q8#12014yes—
316D4—22no—
317D4—22no—
318D4—22no—
319D4—22no—
320D4—48no—
321D4—48no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4#12112yes—
364C2×C4#13512yescomplemented
365C2×C4C22×D411yescharacteristic
366C2×C4C22×D411yescharacteristic
367C2×C4C22×D411yescharacteristic
368C2×C4C2×#4111yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C2×C4H(5,𝔽2)11yescharacteristic
371C8—22no—
372C8—22no—
373C8—41no—
374C8—41no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22C2×#12111yescharacteristic
397C22D4211yescharacteristic
398C22D4211yescharacteristic
399C22C2×#13511yescharacteristic
400C22#57511yescharacteristic
401C22#57711yescharacteristic
402C22#57811yescharacteristic
403C4—22no—
404C4—24no—
405C4—22no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4C2×#12011yescharacteristic
415C4#53411yescharacteristic
416C4#54611yescharacteristic
417C4#54611yescharacteristic
418C2—42no—
419C2—82no—
420C2—82no—
421C2#241011yescharacteristic
422C2#303511yescharacteristic
423C2#304611yescharacteristic
424C2#326911yescharacteristic
425C2#327111yescharacteristic
426C2#328411yescharacteristic
427C2#328511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#135C2
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×#452C2semidirect
5. #2432C2semidirect
6. #2341C2semidirect
7. #2055C2semidirect
8. #2054C2semidirect
9. #2013C2semidirect
10. #2012C2semidirect
11. #1892C2semidirect
12. #1880C2semidirect
13. #1878C2semidirect
14. #1850C2semidirect
15. #1788C2semidirect
364. C2×C4#135semidirect

Maximal subgroups

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

subgroupindex
C22×#1352
C2×#4642
C2×#4622
C2×#4522
#24322
#23412
#20552
#20542
#20132
#20122
#18922
#18802
#18782
#18502
#17882