Finite Group Databasebeta

← C2×#21313 #6372186

Subgroups — C2×#21313 #6372186

Order 512.

4762 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic
2C22×#423C211yesmaximal, characteristic
3C2×C4×#117C211yesmaximal, characteristic
4#21313C2116yesmaximal, direct product factor
5C2×#1988C214yesmaximal, complemented
6C2×#1987C214yesmaximal, complemented
7C2×#1736C214yesmaximal
8C24×Q8—21no—
9C23×M16C2212yes—
10C22×C4⋊Q8—28no—
11C22×C4⋊Q8—21no—
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8—24no—
14C22×C4⋊Q8C4116yescomplemented
15C22×C4⋊Q8C2212yes—
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2218yes—
18C22×#117C2212yes—
19C23×C4⋊C4—21no—
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22C2×#423—28no—
23C2×#423C2218yes—
24C22×C4⋊C8C2212yes—
25C22×C4⋊C8C2212yes—
26C4×#117—24no—
27C4×#117C2218yes—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32C2×#376—21no—
33C2×C43—21no—
34#1988C22132yescomplemented
35#1987C22132yescomplemented
36#1736C22132yes—
37C2×#340C22116yes—
38C23×Q8—216no—
39C23×Q8—28no—
40C23×Q8—22no—
41C23×Q8—22no—
42C23×Q8—416no—
43C24×C4—22no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—28no—
48C22×M16—216no—
49C22×M16C2318yes—
50C23×C8C2314yes—
51C2×C4⋊Q8—216no—
52C2×C4⋊Q8—216no—
53C2×C4⋊Q8—264no—
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—28no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—28no—
59C2×C4⋊Q8—28no—
60C2×C4⋊Q8—232no—
61C2×C4⋊Q8C2×C4116yes—
62C2×C4⋊Q8C2×C4116yes—
63C2×C4⋊Q8C2×C4164yescomplemented
64C2×C4⋊Q8C2318yes—
65C2×C4⋊Q8C2318yes—
66C2×#117—24no—
67C2×#117—28no—
68C2×#117Q8116yescomplemented
69C2×#117C2318yes—
70C22×C4⋊C4—216no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4C2×C4116yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2318yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4—22no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42C2×C418yes—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D414yes—
89C22×C42D418yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2314yes—
93#423—232no—
94C2×C4⋊C8—28no—
95C2×C4⋊C8—28no—
96C2×C4⋊C8C2318yes—
97C2×C4⋊C8C2318yes—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4C2318yes—
101C4×C4⋊C4C2318yes—
102C4×C22⋊C4—28no—
103C4×C22⋊C4—28no—
104#376—28no—
105C2×#95—22no—
106C2×#95—22no—
107C43—28no—
108#340C23164yes—
109C25C2×D411yescharacteristic
110C22×Q8—216no—
111C22×Q8—216no—
112C22×Q8—264no—
113C22×Q8—28no—
114C22×Q8—28no—
115C22×Q8—416no—
116C22×Q8—464no—
117C22×Q8—4128no—
118C22×Q8—416no—
119C23×C4—28no—
120C23×C4—28no—
121C23×C4—216no—
122C23×C4—28no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—416no—
126C23×C4C22⋊C418yes—
127C23×C4C22⋊C418yes—
128C23×C4C22×C414yes—
129C23×C4C2×D414yes—
130C23×C4C2×D414yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D414yes—
138C23×C4C2×Q818yes—
139C23×C4#4118yes—
140C23×C4C2412yes—
141C23×C4C2411yescharacteristic
142C2×M16—216no—
143C2×M16—232no—
144C2×M16—432no—
145C22×C8—28no—
146C22×C8—216no—
147C22×C8C2418yes—
148C4⋊Q8—232no—
149C4⋊Q8—232no—
150C4⋊Q8—264no—
151C4⋊Q8—264no—
152C4⋊Q8—216no—
153C4⋊Q8—216no—
154C4⋊Q8C22×C4132yes—
155C4⋊Q8C22×C4132yes—
156#117—232no—
157#117—216no—
158#117C2×Q8132yescomplemented
159C2×C4⋊C4—216no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—264no—
162C2×C4⋊C4—232no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—216no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4C22×C4116yes—
173C2×C4⋊C4C22×C4132yes—
174C2×C4⋊C4C2×Q8116yescomplemented
175C2×C4⋊C4C2418yes—
176C2×C4⋊C4C2418yes—
177C2×C22⋊C4—216no—
178C2×C22⋊C4—28no—
179C2×C42—216no—
180C2×C42—28no—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—48no—
184C2×C42—48no—
185C2×C42C22⋊C418yes—
186C2×C42C22⋊C4132yes—
187C2×C42C22⋊C418yes—
188C2×C42C22×C414yes—
189C2×C42C22×C414yes—
190C2×C42C22×C4116yes—
191C2×C42C2×D414yes—
192C2×C42C2×D414yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D414yes—
196C2×C42C2×D414yes—
197C2×C42C2×D4116yes—
198C2×C42C2412yes—
199C2×C42C2412yes—
200C4⋊C8—216no—
201C4⋊C8—216no—
202#95—28no—
203#95—28no—
204C24—24no—
205C24—24no—
206C24C2×C22⋊C414yes—
207C24C22⋊Q818yes—
208C24C22×D412yes—
209C24C22×D411yescharacteristic
210C2×Q8—232no—
211C2×Q8—232no—
212C2×Q8—464no—
213C2×Q8—432no—
214C2×Q8—4256no—
215C2×Q8—432no—
216C2×Q8—464no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—216no—
220C22×C4—216no—
221C22×C4—232no—
222C22×C4—232no—
223C22×C4—232no—
224C22×C4—216no—
225C22×C4—216no—
226C22×C4—28no—
227C22×C4—28no—
228C22×C4—464no—
229C22×C4—432no—
230C22×C4—48no—
231C22×C4—416no—
232C22×C4—48no—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C4116yes—
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C22⋊C4116yes—
237C22×C4C2×C22⋊C418yes—
238C22×C4C2×C22⋊C414yes—
239C22×C4C2×C22⋊C414yes—
240C22×C4C2×C22⋊C414yes—
241C22×C4C4×Q8132yes—
242C22×C4C22⋊Q818yes—
243C22×C4C22⋊Q8116yes—
244C22×C4#124116yes—
245C22×C4C4⋊Q8116yes—
246C22×C4AGL(1,ℤ/8)18yes—
247C22×C4#13718yes—
248C22×C4C23×C414yes—
249C22×C4C23×C412yes—
250C22×C4C22×D412yes—
251C22×C4C22×D412yes—
252C22×C4C22×D412yes—
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D412yes—
255C22×C4C22×D412yes—
256C22×C4C22×D412yes—
257C22×C4C22×D414yes—
258C22×C4C22×Q814yes—
259C22×C4C22×Q818yes—
260C22×C4C2×#4114yes—
261C22×C4C2×#4118yes—
262C22×C4C2511yescharacteristic
263M16—432no—
264C2×C8—28no—
265C2×C8—216no—
266C2×C8—416no—
267C4⋊C4—232no—
268C4⋊C4—232no—
269C4⋊C4—216no—
270C4⋊C4—232no—
271C4⋊C4—216no—
272C4⋊C4—28no—
273C4⋊C4—28no—
274C4⋊C4C23×C4116yes—
275C4⋊C4C22×Q8116yescomplemented
276C22⋊C4—232no—
277C42—232no—
278C42—416no—
279C42—416no—
280C42C2×C22⋊C4116yes—
281C42C2×C22⋊C418yes—
282C42C2×C22⋊C418yes—
283C42C2×C22⋊C4116yes—
284C42C23×C414yes—
285C42C23×C414yes—
286C42C22×D414yes—
287C42C22×D414yes—
288C23—232no—
289C23—28no—
290C23—28no—
291C23—24no—
292C23—24no—
293C23#391116yes—
294C23C22×C22⋊C414yes—
295C23C22×C22⋊C412yes—
296C23C2×C22⋊Q818yes—
297C23C2×C22⋊Q814yes—
298C23C2×AGL(1,ℤ/8)14yes—
299C23C2×#13714yes—
300C23C23×D411yescharacteristic
301Q8—464no—
302Q8—464no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—232no—
308C2×C4—232no—
309C2×C4—216no—
310C2×C4—216no—
311C2×C4—464no—
312C2×C4—432no—
313C2×C4—432no—
314C2×C4—48no—
315C2×C4—416no—
316C2×C4—48no—
317C2×C4#391132yes—
318C2×C4#391116yes—
319C2×C4#418116yes—
320C2×C4#419116yes—
321C2×C4C22×C22⋊C414yes—
322C2×C4C22×C22⋊C414yes—
323C2×C4C22×C22⋊C414yes—
324C2×C4C22×C22⋊C414yes—
325C2×C4C22×C22⋊C414yes—
326C2×C4C22×C22⋊C412yes—
327C2×C4C2×C4×Q8116yes—
328C2×C4C2×C4×Q818yes—
329C2×C4C2×C22⋊Q818yes—
330C2×C4C2×C22⋊Q814yes—
331C2×C4C2×#12414yes—
332C2×C4C2×#12418yes—
333C2×C4C2×C4⋊Q818yes—
334C2×C4C2×C4⋊Q814yes—
335C2×C4C2×AGL(1,ℤ/8)18yes—
336C2×C4C2×#13718yes—
337C2×C4C24×C411yescharacteristic
338C2×C4C23×D411yescharacteristic
339C2×C4C23×Q812yes—
340C2×C4C22×#4112yes—
341C8—48no—
342C22—232no—
343C22—216no—
344C22—24no—
345C22—24no—
346C22C2×#39118yes—
347C22C2×#39114yes—
348C22C2×#41814yes—
349C22C2×#41914yes—
350C22#3079116yes—
351C22C23×C22⋊C411yescharacteristic
352C22C22×C22⋊Q812yes—
353C22C22×AGL(1,ℤ/8)12yes—
354C22C22×#13712yes—
355C4—24no—
356C4—24no—
357C4—416no—
358C4—416no—
359C4C2×#39118yes—
360C4C2×#39114yes—
361C4C2×#41818yes—
362C4C2×#41918yes—
363C4C22×C4×Q812yes—
364C4C22×#12411yescharacteristic
365C4C22×C4⋊Q811yescharacteristic
366C2—28no—
367C2#2131318yesdirect product factor
368C2C22×#39111yescharacteristic
369C2C22×#41811yescharacteristic
370C2C22×#41911yescharacteristic
371C2C2×#307914yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21313C2direct (N × H)
5. C2×#1988C2semidirect
6. C2×#1987C2semidirect
14. C22×C4⋊Q8C4semidirect
34. #1988C22semidirect
35. #1987C22semidirect
63. C2×C4⋊Q8C2×C4semidirect
68. C2×#117Q8semidirect
158. #117C2×Q8semidirect
174. C2×C4⋊C4C2×Q8semidirect
275. C4⋊C4C22×Q8semidirect
367. C2#21313direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C22×#4232
C2×C4×#1172
#213132
C2×#19882
C2×#19872
C2×#17362