Finite Group Databasebeta

← #30701

Subgroups — #30701

Order 256.

559 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#480C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3#2541C211yesmaximal, characteristic, complemented
4#2416C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2011C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1999C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1918C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic, complemented
14#1874C211yesmaximal, characteristic, complemented
15#1869C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×#127—21no—
20C2×#127C2211yescharacteristic
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30#480—24no—
31#466—24no—
32C2×#107C2211yescharacteristic
33C2×#107C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45#396—21no—
46#394—21no—
47#392—21no—
48#392—21no—
49#390—21no—
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#386—21no—
53#380C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4D411yescharacteristic
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×D8—22no—
77C2×D8—41no—
78C2×D8—41no—
79C2×D8—41no—
80C2×D8—41no—
81C2×D8—41no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#127—41no—
90#127—41no—
91#127—44no—
92#121—22no—
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—41no—
102#121—41no—
103#121—41no—
104#121—44no—
105C4×D4—41no—
106C4×D4—41no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130#107—22no—
131#107D414yescomplemented
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—22no—
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—
149C24C2×D411yescharacteristic
150C24C2×D411yescharacteristic
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—41no—
170C2×D4—41no—
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—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4D814yescomplemented
196C22×C4—21no—
197C22×C4—21no—
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—42no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2411yescharacteristic
221D8—44no—
222D8—44no—
223C2×C8—21no—
224C2×C8—21no—
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—44no—
235C2×C8C2×D412yes—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4C2×D412yes—
247C4⋊C4C2×D412yes—
248C4⋊C4#4112yes—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C42—21no—
261C42—21no—
262C42—42no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
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—
296C23C2×D812yes—
297C23C2×D812yes—
298C23C22×D411yescharacteristic
299D4—44no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—84no—
313D4—84no—
314D4—84no—
315D4—84no—
316D4—84no—
317D4—84no—
318D4—84no—
319D4—84no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
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—84no—
355C2×C4C2×D812yes—
356C2×C4C22×D411yescharacteristic
357C2×C4C2×#4111yescharacteristic
358C2×C4H(5,𝔽2)11yescharacteristic
359C2×C4H(5,𝔽2)11yescharacteristic
360C2×C4H(5,𝔽2)11yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C8—22no—
363C8—41no—
364C8—41no—
365C8—41no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#53411yescharacteristic
402C22D4211yescharacteristic
403C22#54611yescharacteristic
404C22C22×D811yescharacteristic
405C22#57511yescharacteristic
406C22#57611yescharacteristic
407C22#57711yescharacteristic
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—81no—
416C4—81no—
417C4#53511yescharacteristic
418C4#54011yescharacteristic
419C4#55011yescharacteristic
420C4#56011yescharacteristic
421C2—42no—
422C2—42no—
423C2—161no—
424C2—161no—
425C2—161no—
426C2#267311yescharacteristic
427C2#318611yescharacteristic
428C2#319811yescharacteristic
429C2#327111yescharacteristic
430C2D4×D811yescharacteristic
431C2#329411yescharacteristic
432C2#330911yescharacteristic
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×#480C2semidirect
2. C2×#466C2semidirect
3. #2541C2semidirect
4. #2416C2semidirect
5. #2041C2semidirect
6. #2033C2semidirect
7. #2011C2semidirect
8. #1999C2semidirect
9. #1999C2semidirect
10. #1999C2semidirect
11. #1925C2semidirect
12. #1918C2semidirect
13. #1893C2semidirect
14. #1874C2semidirect
15. #1869C2semidirect
131. #107D4semidirect
195. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C2×#4662
#25412
#24162
#20412
#20332
#20112
#19992
#19252
#19182
#18932
#18742
#18692