Finite Group Databasebeta

← #25153

Subgroups — #25153

Order 256.

593 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#460C211yesmaximal, characteristic, complemented
2C2×#448C211yesmaximal, characteristic, complemented
3C2×#448C211yesmaximal, characteristic, complemented
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5#2428C211yesmaximal, characteristic
6Q8×C22⋊C4C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic, complemented
11#1968C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1865C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#121—21no—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28#460—24no—
29#448—24no—
30#448—24no—
31C4×SD16—24no—
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#394—21no—
47#392—21no—
48#391—21no—
49#391—21no—
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic
54#386—21no—
55#380—21no—
56#380C2211yescharacteristic
57C4×C22⋊C4—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—21no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C2311yescharacteristic
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—22no—
89C2×SD16—22no—
90C2×SD16—41no—
91C2×SD16—41no—
92C2×SD16D414yescomplemented
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97#127—22no—
98#127—41no—
99#121—22no—
100#121—22no—
101#121—41no—
102#121—41no—
103#121—44no—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×D4—22no—
109C4×D4—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
124C2×C22⋊C4D411yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129#106—22no—
130#106—41no—
131C4⋊C8—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103D414yescomplemented
136#102—22no—
137#102—22no—
138#102—22no—
139#102—41no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C4×C8—22no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95D411yescharacteristic
150#95D411yescharacteristic
151C24—21no—
152C24—21no—
153C24C2×D411yescharacteristic
154C2×Q8—24no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8C2×D412yes—
170C2×Q8C2×D412yes—
171C2×Q8#4112yes—
172C2×Q8#4112yes—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4C2×D412yes—
197C22×C4—21no—
198C22×C4—24no—
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—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—81no—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2411yescharacteristic
221SD16—28no—
222SD16—44no—
223SD16—44no—
224SD16—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×C8C2×D412yes—
236C2×C8C2×D412yes—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4C2×D412yes—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—81no—
255C22⋊C4SD1614yescomplemented
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—42no—
261C42—42no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23C2×SD1612yes—
283C23C22×D411yescharacteristic
284Q8—22no—
285Q8—22no—
286Q8—22no—
287Q8—22no—
288Q8—22no—
289Q8—48no—
290Q8—41no—
291Q8—41no—
292Q8#12114yescomplemented
293D4—22no—
294D4—22no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—84no—
302D4—84no—
303D4—84no—
304D4—84no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—48no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
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—84no—
342C2×C4C2×SD1612yes—
343C2×C4C2×SD1612yes—
344C2×C4C22×D411yescharacteristic
345C2×C4C22×D411yescharacteristic
346C2×C4C2×#4111yescharacteristic
347C2×C4C2×#4111yescharacteristic
348C2×C4H(5,𝔽2)11yescharacteristic
349C2×C4H(5,𝔽2)11yescharacteristic
350C8—22no—
351C8—22no—
352C8—41no—
353C8—41no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22#52511yescharacteristic
376C22D4211yescharacteristic
377C22D4211yescharacteristic
378C22C22×SD1611yescharacteristic
379C22#57511yescharacteristic
380C22#57711yescharacteristic
381C22#57711yescharacteristic
382C4—24no—
383C4—22no—
384C4—22no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—42no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4C2×#12111yescharacteristic
394C4#53811yescharacteristic
395C4#54611yescharacteristic
396C4#54611yescharacteristic
397C2—42no—
398C2—82no—
399C2—161no—
400C2#242411yescharacteristic
401C2#310311yescharacteristic
402C2#312611yescharacteristic
403C2#327111yescharacteristic
404C2#327511yescharacteristic
405C2D4×SD1611yescharacteristic
406C2D4×SD1611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#460C2semidirect
2. C2×#448C2semidirect
3. C2×#448C2semidirect
4. C2×C4×SD16C2semidirect
6. Q8×C22⋊C4C2semidirect
7. #2066C2semidirect
8. #2034C2semidirect
9. #2034C2semidirect
10. #2014C2semidirect
11. #1968C2semidirect
12. #1923C2semidirect
13. #1880C2semidirect
14. #1877C2semidirect
15. #1865C2semidirect
92. C2×SD16D4semidirect
135. #103D4semidirect
255. C22⋊C4SD16semidirect
292. Q8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#4602
C2×#4482
C2×C4×SD162
#24282
Q8×C22⋊C42
#20662
#20342
#20142
#19682
#19232
#18802
#18772
#18652