Finite Group Databasebeta

← #30396

Subgroups — #30396

Order 256.

630 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3#2537C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic, complemented
5#2093C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2071C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1869C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic, complemented
20C2×#121—21no—
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic, complemented
25C2×#120—21no—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic, complemented
28#466—24no—
29#447—24no—
30C2×#107C2211yescharacteristic, complemented
31C2×#107C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic, complemented
43#394—21no—
44#394—21no—
45#394—21no—
46#392—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#384C2211yescharacteristic, complemented
50#380—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic, complemented
54#379—21no—
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8—41no—
85C2×D8D414yescomplemented
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93#121—22no—
94#121—22no—
95#121—22no—
96#121—22no—
97#121—41no—
98#121—41no—
99#121—41no—
100#121—41no—
101#121—44no—
102#120—44no—
103C4×D4—41no—
104C4×D4—41no—
105C4×D4—41no—
106C4×D4—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42C2311yescharacteristic
130#107—22no—
131#107—41no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—41no—
137#102—41no—
138#102—41no—
139#102—41no—
140C22⋊C8—22no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143#95—21no—
144#95—21no—
145#95—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—81no—
155C24—81no—
156C24C2×D411yescharacteristic
157C24C2×D411yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4D814yescomplemented
195C2×D4C2×D412yes—
196C2×D4C2×D412yes—
197C2×D4#4112yes—
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—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2411yescharacteristic
223D8—44no—
224D8—44no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—44no—
237C2×C8C2×D412yes—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—81no—
264C22⋊C4—81no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—48no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23C2×D812yes—
308C23C2×D812yes—
309C23C22×D411yescharacteristic
310D4—22no—
311D4—22no—
312D4—22no—
313D4—22no—
314D4—22no—
315D4—22no—
316D4—44no—
317D4—48no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—84no—
325D4—84no—
326D4—84no—
327D4—84no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
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—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—84no—
363C2×C4C2×D812yes—
364C2×C4C22×D411yescharacteristic
365C2×C4C2×#4111yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C2×C4H(5,𝔽2)11yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C8—22no—
371C8—41no—
372C8—41no—
373C8—41no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—816no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#53411yescharacteristic
411C22D4211yescharacteristic
412C22#54611yescharacteristic
413C22C22×D811yescharacteristic
414C22#57511yescharacteristic
415C22#57611yescharacteristic
416C22#57711yescharacteristic
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4#53411yescharacteristic
427C4#53811yescharacteristic
428C4#54611yescharacteristic
429C4#56011yescharacteristic
430C2—42no—
431C2—42no—
432C2—82no—
433C2—82no—
434C2—82no—
435C2#266811yescharacteristic
436C2#318611yescharacteristic
437C2#319811yescharacteristic
438C2#326711yescharacteristic
439C2D4×D811yescharacteristic
440C2#329411yescharacteristic
441C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#466C2semidirect
2. C2×#447C2semidirect
3. #2537C2semidirect
4. #2408C2semidirect
5. #2093C2semidirect
6. #2075C2semidirect
7. #2071C2semidirect
8. #2033C2semidirect
9. #2033C2semidirect
10. #1999C2semidirect
11. #1893C2semidirect
12. #1879C2semidirect
13. #1879C2semidirect
14. #1869C2semidirect
15. #1786C2semidirect
19. C23×C8C22semidirect
24. C2×#121C22semidirect
27. C2×C4×D4C22semidirect
30. C2×#107C22semidirect
31. C2×#107C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
42. C2×C22⋊C8C22semidirect
49. #384C22semidirect
53. #380C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
85. C2×D8D4semidirect
194. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×#4472
#25372
#24082
#20932
#20752
#20712
#20332
#19992
#18932
#18792
#18692
#17862