Finite Group Databasebeta

← #25123

Subgroups — #25123

Order 256.

600 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C2×#417C211yesmaximal, characteristic, complemented
5#2416C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1935C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1853C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121—21no—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#465—24no—
29#449—24no—
30#435—24no—
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34#417—24no—
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic
46#396—21no—
47#394—21no—
48#394—21no—
49#392—21no—
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic
54#386—21no—
55#386—21no—
56#385—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×D4D411yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—41no—
84C2×D8—22no—
85C2×D8—22no—
86C2×D8—41no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—41no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—41no—
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22⋊Q8—22no—
98C22⋊Q8—41no—
99#121—22no—
100#121—22no—
101#121—22no—
102#121—22no—
103#121—41no—
104#121—41no—
105#121—44no—
106C4×D4—22no—
107C4×D4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4D411yescharacteristic
130C2×C22⋊C4D411yescharacteristic
131C2×C42C2311yescharacteristic
132#106—22no—
133#106—41no—
134C4⋊C8—22no—
135#103—22no—
136#103—22no—
137#103D414yescomplemented
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—41no—
143C22⋊C8—22no—
144C22⋊C8D414yescomplemented
145#97—22no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151C24—21no—
152C24—21no—
153C24C2×D411yescharacteristic
154C24C2×D411yescharacteristic
155#41—28no—
156#41—44no—
157#41—44no—
158#41—44no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8C2×D412yes—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—41no—
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—42no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4#4112yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
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—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—81no—
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D412yes—
219C22×C4#4112yes—
220C22×C4C2411yescharacteristic
221SD16—44no—
222D8—44no—
223M16—44no—
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—44no—
234C2×C8C2×D412yes—
235C2×C8C2×D412yes—
236C2×C8C2×D412yes—
237C4⋊C4—22no—
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—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4C2×D412yes—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C42—21no—
264C42—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—22no—
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#12112yes—
292C23AGL(1,ℤ/8)12yes—
293C23C22×D411yescharacteristic
294Q8—22no—
295Q8—22no—
296Q8—41no—
297Q8—41no—
298D4—22no—
299D4—22no—
300D4—44no—
301D4—48no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—84no—
309D4—84no—
310D4—84no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—48no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
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—84no—
350C2×C4#12112yes—
351C2×C4C22×D411yescharacteristic
352C2×C4C22×D411yescharacteristic
353C2×C4C2×#4111yescharacteristic
354C2×C4C2×#4111yescharacteristic
355C2×C4H(5,𝔽2)11yescharacteristic
356C2×C4H(5,𝔽2)11yescharacteristic
357C8—22no—
358C8—22no—
359C8—22no—
360C8—41no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C2×#12111yescharacteristic
390C22D4211yescharacteristic
391C22#54611yescharacteristic
392C22C2×AGL(1,ℤ/8)11yescharacteristic
393C22#57511yescharacteristic
394C22#57711yescharacteristic
395C22#57711yescharacteristic
396C4—22no—
397C4—41no—
398C4—42no—
399C4—42no—
400C4—41no—
401C4—41no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4#52511yescharacteristic
407C4#53811yescharacteristic
408C4D4211yescharacteristic
409C4#54611yescharacteristic
410C2—42no—
411C2—42no—
412C2—82no—
413C2—161no—
414C2#242411yescharacteristic
415C2#305511yescharacteristic
416C2#306011yescharacteristic
417C2#326511yescharacteristic
418C2#327111yescharacteristic
419C2#330611yescharacteristic
420C2#330711yescharacteristic
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×#465C2semidirect
2. C2×#449C2semidirect
3. C2×#435C2semidirect
4. C2×#417C2semidirect
5. #2416C2semidirect
6. #2407C2semidirect
7. #2065C2semidirect
8. #2055C2semidirect
9. #2035C2semidirect
10. #1999C2semidirect
11. #1972C2semidirect
12. #1935C2semidirect
13. #1895C2semidirect
14. #1893C2semidirect
15. #1853C2semidirect
137. #103D4semidirect
144. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C2×#4492
C2×#4352
C2×#4172
#24162
#24072
#20652
#20552
#20352
#19992
#19722
#19352
#18952
#18932
#18532