Finite Group Databasebeta

← #30874

Subgroups — #30874

Order 256.

539 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486C211yesmaximal, characteristic, complemented
2C2×#459C211yesmaximal, characteristic, complemented
3#2541C211yesmaximal, characteristic, complemented
4#2452C211yesmaximal, characteristic, complemented
5#2067C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#2011C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1916C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×C22⋊Q8—21no—
24C2×#121—21no—
25C2×#121—21no—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28#486—24no—
29#459—24no—
30C2×#107C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C4×C8C2211yescharacteristic
43#396—21no—
44#393—21no—
45#392—21no—
46#392—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#389C2211yescharacteristic
53#385—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—41no—
80C2×D8—41no—
81C2×D8—41no—
82C2×D8—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C4⋊Q8—22no—
88C4⋊Q8—41no—
89#127—22no—
90#127—22no—
91#127—41no—
92#127—41no—
93#127—41no—
94#127—41no—
95C22⋊Q8—44no—
96#121—44no—
97#121—44no—
98C4×D4—22no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
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×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#107—41no—
125C4⋊C8—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—41no—
131#102—41no—
132#102—41no—
133#102—41no—
134#102—41no—
135#102D414yescomplemented
136C4×C8—22no—
137#95—21no—
138#95—21no—
139#95D411yescharacteristic
140#95D411yescharacteristic
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—81no—
148C2×D4—21no—
149C2×D4—21no—
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—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4C2×D412yes—
189C2×D4#4112yes—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—81no—
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2411yescharacteristic
207D8—44no—
208D8—44no—
209D8—44no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8C2×D412yes—
222C4⋊C4—22no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4D814yescomplemented
234C4⋊C4C2×D412yes—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—81no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C23—22no—
252C23—22no—
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—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C22×D411yescharacteristic
280Q8—84no—
281D4—22no—
282D4—22no—
283D4—22no—
284D4—22no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—84no—
298D4—84no—
299D4—84no—
300D4—84no—
301D4—84no—
302D4—84no—
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—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
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—41no—
328C2×C4—41no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4C2×D812yes—
332C2×C4C2×D812yes—
333C2×C4C2×D812yes—
334C2×C4AGL(1,ℤ/8)12yes—
335C2×C4C22×D411yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C2×C42−1+411yescharacteristic
341C8—22no—
342C8—41no—
343C8—41no—
344C8—41no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—81no—
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—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#53511yescharacteristic
374C22D4211yescharacteristic
375C22#54811yescharacteristic
376C22C22×D811yescharacteristic
377C22C2×AGL(1,ℤ/8)11yescharacteristic
378C22#57611yescharacteristic
379C22#57711yescharacteristic
380C4—22no—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4#53711yescharacteristic
391C4#53811yescharacteristic
392C4#54611yescharacteristic
393C4#56211yescharacteristic
394C2—82no—
395C2—82no—
396C2—161no—
397C2—161no—
398C2#267611yescharacteristic
399C2#320511yescharacteristic
400C2#322411yescharacteristic
401C2#326511yescharacteristic
402C2D4×D811yescharacteristic
403C2#335811yescharacteristic
404C2#336611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#486C2semidirect
2. C2×#459C2semidirect
3. #2541C2semidirect
4. #2452C2semidirect
5. #2067C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2023C2semidirect
9. #2011C2semidirect
10. #2011C2semidirect
11. #1970C2semidirect
12. #1916C2semidirect
13. #1879C2semidirect
14. #1879C2semidirect
15. #1862C2semidirect
135. #102D4semidirect
233. C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4862
C2×#4592
#25412
#24522
#20672
#20412
#20232
#20112
#19702
#19162
#18792
#18622