Finite Group Databasebeta

← #22444

Subgroups — #22444

Order 256.

717 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
1C22×#137C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4Q8×C22⋊C4C211yesmaximal, characteristic, complemented
5#2339C211yesmaximal, characteristic
6#1971C211yesmaximal, characteristic, complemented
7#1969C211yesmaximal, characteristic, complemented
8#1878C211yesmaximal, characteristic, complemented
9#1878C211yesmaximal, characteristic, complemented
10#1877C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1809C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic
14#1787C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C2×#137—24no—
19C2×#137—24no—
20C2×#137C418yescomplemented
21C22×Q16C2211yescharacteristic
22C22×Q16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×SD16C2211yescharacteristic
25C22×M16C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30#440—24no—
31C4×SD16—24no—
32C2×#107C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#391—21no—
46#391—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#386—21no—
52#386—21no—
53#381—21no—
54#380—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57C4×C22⋊C4—21no—
58C4×C22⋊C4C2211yescharacteristic
59C2×#95—21no—
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#336C2211yescharacteristic
63#336C2211yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41C2×C414yes—
70C22×Q8—21no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2×C414yescomplemented
77C22×Q8C2311yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4—21no—
81C22×D4C2311yescharacteristic
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87#137—232no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16C2×C414yescomplemented
93C2×Q16C2×C414yescomplemented
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16C2×C414yes—
97C2×SD16C2×C414yes—
98C2×SD16D414yes—
99C2×SD16D414yes—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16C2×C414yescomplemented
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C4×Q8—22no—
108C4×Q8—22no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—41no—
114C4×D4—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#107—22no—
132#106—22no—
133#103—22no—
134#103—22no—
135#103—41no—
136#103—41no—
137#103—41no—
138#103—41no—
139#102—22no—
140#102—22no—
141C22⋊C8—22no—
142#97—22no—
143C4×C8—22no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95D411yescharacteristic
155#95D411yescharacteristic
156C24—21no—
157C24C2×D411yescharacteristic
158#41—28no—
159#41—44no—
160#41—44no—
161#41—44no—
162C2×Q8—24no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—28no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8C22×C412yes—
183C2×Q8C22×C412yes—
184C2×Q8C22×C412yes—
185C2×Q8C2×D412yes—
186C2×Q8C2×D412yes—
187C2×D4—24no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4C22×C412yes—
194C2×D4C2×D412yes—
195C2×D4C2×D412yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4C22×C412yes—
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4#4112yes—
226C22×C4C2411yescharacteristic
227Q16—28no—
228Q16—28no—
229SD16—28no—
230SD16C22⋊C4116yescomplemented
231M16—28no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8C22×C412yes—
241C2×C8C22×C412yes—
242C2×C8C2×D412yes—
243C2×C8C2×D412yes—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—22no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—42no—
265C42—42no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C4×D412yes—
280C23C22×D411yescharacteristic
281Q8—22no—
282Q8—22no—
283Q8—22no—
284Q8—22no—
285Q8—22no—
286Q8—22no—
287Q8—22no—
288Q8—48no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8C2×C22⋊C414yes—
294D4—22no—
295D4—22no—
296D4—22no—
297D4—48no—
298D4C2×C22⋊C414yes—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—41no—
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—84no—
340C2×C4—84no—
341C2×C4C4×D412yes—
342C2×C4#13712yescomplemented
343C2×C4C23×C411yescharacteristic
344C2×C4C22×D411yescharacteristic
345C2×C4C22×D411yescharacteristic
346C2×C4C2×#4111yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C2×C4H(5,𝔽2)11yescharacteristic
349C8—22no—
350C8—41no—
351C8—41no—
352C8C2×C22⋊C414yes—
353C22—22no—
354C22—24no—
355C22—22no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22C2×C4×D411yescharacteristic
368C22D4211yescharacteristic
369C22D4211yescharacteristic
370C22C2×#13711yescharacteristic
371C22#57511yescharacteristic
372C22#57711yescharacteristic
373C22#57811yescharacteristic
374C4—22no—
375C4—24no—
376C4—22no—
377C4—42no—
378C4—42no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4C22×C22⋊C411yescharacteristic
387C4#51811yescharacteristic
388C4#54611yescharacteristic
389C4#54611yescharacteristic
390C2—42no—
391C2—44no—
392C2D4×C22⋊C411yescharacteristic
393C2C4×#13711yescharacteristic
394C2#294711yescharacteristic
395C2#326611yescharacteristic
396C2#326911yescharacteristic
397C2#327511yescharacteristic
398C2#327611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×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
2. C2×#440C2semidirect
3. C2×C4×SD16C2semidirect
4. Q8×C22⋊C4C2semidirect
6. #1971C2semidirect
7. #1969C2semidirect
8. #1878C2semidirect
9. #1878C2semidirect
10. #1877C2semidirect
11. #1877C2semidirect
12. #1809C2semidirect
14. #1787C2semidirect
15. #1753C2semidirect
20. C2×#137C4semidirect
76. C22×Q8C2×C4semidirect
92. C2×Q16C2×C4semidirect
93. C2×Q16C2×C4semidirect
102. C2×M16C2×C4semidirect
230. SD16C22⋊C4semidirect
342. C2×C4#137semidirect

Maximal subgroups

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

subgroupindex
C22×#1372
C2×#4402
C2×C4×SD162
Q8×C22⋊C42
#23392
#19712
#19692
#18782
#18772
#18092
#17882
#17872
#17532