Finite Group Databasebeta

← #24512

Subgroups — #24512

Order 256.

549 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#447C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic, complemented
3#2624C211yesmaximal, characteristic, complemented
4#2309C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#1972C211yesmaximal, characteristic, complemented
10#1970C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1934C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#447—24no—
23#433—24no—
24C2×#107C2211yescharacteristic, complemented
25C2×#106C2211yescharacteristic, complemented
26C2×C4⋊C8C2211yescharacteristic, complemented
27C2×C4⋊C8C2211yescharacteristic, complemented
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic, complemented
35C2×#102C2211yescharacteristic, complemented
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic, complemented
39C2×C4×C8C2211yescharacteristic, complemented
40#399—21no—
41#397—21no—
42#394—21no—
43#394—21no—
44#390—21no—
45#390—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic, complemented
51#390C2211yescharacteristic, complemented
52#389C2211yescharacteristic, complemented
53#389C2211yescharacteristic, complemented
54#380C2211yescharacteristic
55#379—21no—
56#379—21no—
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—41no—
79C2×D8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#121—22no—
85#121—22no—
86#117—22no—
87#117—22no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—41no—
120#106—41no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—41no—
128#102—41no—
129#102—41no—
130#102—41no—
131C22⋊C8—22no—
132C22⋊C8D414yes—
133#97—22no—
134C4×C8—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—24no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—42no—
143C24—42no—
144C24—81no—
145C24—81no—
146C24—81no—
147C24—81no—
148C24C2×D411yescharacteristic
149C2×D4—24no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
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—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4#4112yes—
176C2×D4#4112yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D412yes—
192C22×C4#4112yes—
193C22×C4C2411yescharacteristic
194D8—44no—
195D8—44no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8C2×D412yes—
207C2×C8C2×D412yes—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—48no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—84no—
273C23—84no—
274C23—84no—
275C23—84no—
276C23—84no—
277C23—84no—
278C23#12112yes—
279C23C22×D411yescharacteristic
280D4—22no—
281D4—22no—
282D4—22no—
283D4—22no—
284D4—48no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4#12112yes—
328C2×C4C22×D411yescharacteristic
329C2×C4C2×#4111yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C8—22no—
335C8—22no—
336C8—41no—
337C8—41no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—816no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#12111yescharacteristic
368C22#52511yescharacteristic
369C22#53511yescharacteristic
370C22#57611yescharacteristic
371C22#57611yescharacteristic
372C22#57611yescharacteristic
373C22#57711yescharacteristic
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4#54611yescharacteristic
385C4#54611yescharacteristic
386C4#55111yescharacteristic
387C4#55111yescharacteristic
388C2—42no—
389C2—82no—
390C2—82no—
391C2—161no—
392C2#241111yescharacteristic
393C2#305811yescharacteristic
394C2#306211yescharacteristic
395C2#311811yescharacteristic
396C2#312211yescharacteristic
397C2#322011yescharacteristic
398C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#433C2semidirect
3. #2624C2semidirect
4. #2309C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2054C2semidirect
8. #2054C2semidirect
9. #1972C2semidirect
10. #1970C2semidirect
11. #1936C2semidirect
12. #1934C2semidirect
13. #1875C2semidirect
14. #1854C2semidirect
15. #1786C2semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
26. C2×C4⋊C8C22semidirect
27. C2×C4⋊C8C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
38. C2×#97C22semidirect
39. C2×C4×C8C22semidirect
50. #390C22semidirect
51. #390C22semidirect
52. #389C22semidirect
53. #389C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C2×#4332
#26242
#23092
#20672
#20652
#20542
#19722
#19702
#19362
#19342
#18752
#18542
#17862