Finite Group Databasebeta

← C2×#10707 #353143

Subgroups — C2×#10707 #353143

Order 512.

2618 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C212yesmaximal
2C23×#97C211yesmaximal, characteristic
3C22×#341C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5C22×#339C212yesmaximal
6#10707C218yesmaximal, direct product factor
7C23×M16C2212yes—
8C24×C8C2211yescharacteristic
9C24×C8C2211yescharacteristic
10C22×#117C2212yes—
11C23×C42C2211yescharacteristic
12C2×#405—28no—
13C2×#405—24no—
14C2×#405C4116yescomplemented
15C2×#405C2218yes—
16C22×#97—28no—
17C22×#97—22no—
18C22×#97C418yescomplemented
19C22×#97C2214yes—
20C22×C4×C8—22no—
21C2×#341C418yescomplemented
22C2×#341C2214yes—
23#1735C414yescomplemented
24#1735C2214yes—
25C2×#340C414yescomplemented
26C2×#339C4116yescomplemented
27C2×#339C2218yes—
28C24×C4—21no—
29C24×C4C2311yescharacteristic
30C22×M16—24no—
31C22×M16—22no—
32C22×M16—24no—
33C22×M16—24no—
34C22×M16C2×C418yes—
35C22×M16C2×C418yes—
36C22×M16C2×C418yes—
37C22×M16C2314yes—
38C23×C8—24no—
39C23×C8—22no—
40C23×C8—24no—
41C23×C8C2×C414yes—
42C23×C8C2×C412yes—
43C23×C8C2×C412yes—
44C23×C8C2×C414yes—
45C23×C8C2×C414yes—
46C23×C8C2×C418yes—
47C23×C8D418yes—
48C23×C8D412yes—
49C23×C8D412yes—
50C23×C8Q814yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#117—24no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—22no—
57C2×#117C2×C418yes—
58C2×#117C2×C418yes—
59C2×#117C2×C414yes—
60C2×#117C2314yes—
61C22×C4⋊C4C2×C414yes—
62C22×C22⋊C4—22no—
63C22×C42—24no—
64C22×C42—22no—
65C22×C42—21no—
66C22×C42—22no—
67C22×C42C2×C414yes—
68C22×C42C2×C418yes—
69C22×C42C2312yes—
70#405—232no—
71#405—216no—
72#405C2×C4132yescomplemented
73C2×#97—232no—
74C2×#97—216no—
75C2×#97—216no—
76C2×#97—28no—
77C2×#97C2×C4116yescomplemented
78C2×C4×C8—216no—
79C2×C4×C8—28no—
80#341C2×C4116yescomplemented
81#340C2×C4116yescomplemented
82#339C2×C4132yescomplemented
83C25#4111yescharacteristic
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—28no—
87C23×C4—22no—
88C23×C4—48no—
89C23×C4C22×C414yes—
90C23×C4C22×C412yes—
91C23×C4C22×C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C414yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D414yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×Q812yes—
99C23×C4#4111yescharacteristic
100C23×C4#4114yes—
101C23×C4#4112yes—
102C23×C4C2411yescharacteristic
103C2×M16—28no—
104C2×M16—216no—
105C2×M16—28no—
106C2×M16—28no—
107C2×M16—216no—
108C2×M16—216no—
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16C42132yes—
113C2×M16C22×C418yes—
114C2×M16C22×C418yes—
115C2×M16C22×C418yes—
116C22×C8—28no—
117C22×C8—28no—
118C22×C8—28no—
119C22×C8—216no—
120C22×C8—28no—
121C22×C8—28no—
122C22×C8—28no—
123C22×C8—28no—
124C22×C8—24no—
125C22×C8—24no—
126C22×C8—24no—
127C22×C8C4218yes—
128C22×C8C4218yes—
129C22×C8C42116yes—
130C22×C8C22⋊C418yes—
131C22×C8C22⋊C4116yes—
132C22×C8C22⋊C418yes—
133C22×C8C22⋊C4116yes—
134C22×C8C4⋊C418yes—
135C22×C8C4⋊C4132yes—
136C22×C8C4⋊C418yes—
137C22×C8C22×C414yes—
138C22×C8C22×C414yes—
139C22×C8C22×C414yes—
140C22×C8C22×C414yes—
141C22×C8C22×C418yes—
142C22×C8C2×D414yes—
143C22×C8C2×D418yes—
144C22×C8C2×Q814yes—
145#117—28no—
146#117—28no—
147#117—28no—
148#117—28no—
149#117—28no—
150#117—24no—
151#117—24no—
152#117C42116yes—
153#117C22×C418yes—
154#117C22×C418yes—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4C42116yes—
158C2×C4⋊C4C22×C418yes—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C42—28no—
164C2×C42—28no—
165C2×C42—24no—
166C2×C42—28no—
167C2×C42—24no—
168C2×C42—216no—
169C2×C42—28no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42C42116yes—
174C2×C42C22×C418yes—
175C2×C42C22×C414yes—
176#97—232no—
177#97—232no—
178C4×C8—232no—
179C24—28no—
180C24#11714yes—
181C24#11712yes—
182C24C4×D411yescharacteristic
183C24C4×D414yes—
184C24C4×D411yescharacteristic
185C24C4×Q812yes—
186C24C2×#4111yescharacteristic
187C22×C4—24no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—216no—
191C22×C4—216no—
192C22×C4—48no—
193C22×C4—432no—
194C22×C4—416no—
195C22×C4—416no—
196C22×C4C2×C4214yes—
197C22×C4C2×C4212yes—
198C22×C4C2×C4212yes—
199C22×C4C2×C4218yes—
200C22×C4C2×C4214yes—
201C22×C4C2×C4214yes—
202C22×C4C2×C4214yes—
203C22×C4C2×C22⋊C414yes—
204C22×C4C2×C22⋊C414yes—
205C22×C4C2×C22⋊C414yes—
206C22×C4C2×C4⋊C418yes—
207C22×C4C2×C4⋊C412yes—
208C22×C4C2×C4⋊C412yes—
209C22×C4#11714yes—
210C22×C4#11718yes—
211C22×C4#11712yes—
212C22×C4#11714yes—
213C22×C4C4×D412yes—
214C22×C4C4×D418yes—
215C22×C4C4×D412yes—
216C22×C4C4×D414yes—
217C22×C4C4×D418yes—
218C22×C4C4×D414yes—
219C22×C4C4×D414yes—
220C22×C4C4×D414yes—
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D414yes—
223C22×C4C4×D411yescharacteristic
224C22×C4C4×Q812yes—
225C22×C4C4×Q818yes—
226C22×C4C4×Q812yes—
227C22×C4C4×Q812yes—
228C22×C4C23×C411yescharacteristic
229C22×C4C23×C412yes—
230C22×C4C23×C411yescharacteristic
231C22×C4C23×C412yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D412yes—
235C22×C4C22×Q811yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C22×C4C2×#4112yes—
238M16—28no—
239M16—28no—
240M16—28no—
241M16—232no—
242M16C2×C42116yes—
243C2×C8—28no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—28no—
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—28no—
250C2×C8—216no—
251C2×C8—216no—
252C2×C8—232no—
253C2×C8#95132yes—
254C2×C8#95132yes—
255C2×C8C2×C4218yes—
256C2×C8C2×C4218yes—
257C2×C8C2×C22⋊C418yes—
258C2×C8C2×C22⋊C418yes—
259C2×C8C2×C22⋊C418yes—
260C2×C8C2×C4⋊C418yes—
261C2×C8C2×C4⋊C4116yes—
262C4⋊C4—28no—
263C4⋊C4—216no—
264C4⋊C4C2×C42116yes—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—216no—
268C42—28no—
269C42—24no—
270C42—28no—
271C42—28no—
272C42—216no—
273C42—216no—
274C42C2×C4218yes—
275C23—24no—
276C23—24no—
277C23—216no—
278C23—216no—
279C23—28no—
280C23—28no—
281C23#37614yes—
282C23C4×C22⋊C414yes—
283C23C4×C22⋊C414yes—
284C23C4×C22⋊C414yes—
285C23C4×C4⋊C418yes—
286C23C4×C4⋊C412yes—
287C23C4×C4⋊C412yes—
288C23#44418yes—
289C23C2×#11712yes—
290C23C2×#11711yescharacteristic
291C23C2×C4×D412yes—
292C23C2×C4×D411yescharacteristic
293C23C2×C4×Q811yescharacteristic
294C2×C4—28no—
295C2×C4—216no—
296C2×C4—416no—
297C2×C4—432no—
298C2×C4—48no—
299C2×C4—416no—
300C2×C4—432no—
301C2×C4C4314yes—
302C2×C4C4314yes—
303C2×C4C2×#9514yes—
304C2×C4C2×#9514yes—
305C2×C4#37614yes—
306C2×C4C4×C22⋊C414yes—
307C2×C4C4×C22⋊C414yes—
308C2×C4C4×C22⋊C414yes—
309C2×C4C4×C22⋊C414yes—
310C2×C4C4×C22⋊C414yes—
311C2×C4C4×C22⋊C418yes—
312C2×C4C4×C22⋊C418yes—
313C2×C4C4×C4⋊C418yes—
314C2×C4C4×C4⋊C412yes—
315C2×C4C4×C4⋊C412yes—
316C2×C4C4×C4⋊C414yes—
317C2×C4C4×C4⋊C418yes—
318C2×C4C4×C4⋊C414yes—
319C2×C4C4×C4⋊C418yes—
320C2×C4C22×C4211yescharacteristic
321C2×C4C22×C4211yescharacteristic
322C2×C4C22×C4212yes—
323C2×C4C22×C4211yescharacteristic
324C2×C4C22×C4212yes—
325C2×C4C22×C22⋊C411yescharacteristic
326C2×C4C22×C22⋊C412yes—
327C2×C4C22×C4⋊C412yes—
328C2×C4C22×C4⋊C411yescharacteristic
329C2×C4C2×#11711yescharacteristic
330C2×C4C2×#11712yes—
331C2×C4C2×C4×D411yescharacteristic
332C2×C4C2×C4×D412yes—
333C2×C4C2×C4×D412yes—
334C2×C4C2×C4×D412yes—
335C2×C4C2×C4×D412yes—
336C2×C4C2×C4×Q812yes—
337C2×C4C2×C4×Q811yescharacteristic
338C8—28no—
339C8—216no—
340C8C2×#95116yes—
341C22—24no—
342C22—24no—
343C22—28no—
344C22—28no—
345C22—28no—
346C22—216no—
347C22—22no—
348C22—22no—
349C22—24no—
350C22C4×#9514yes—
351C22C4×#9514yes—
352C22#1765116yes—
353C22#177318yes—
354C22C2×#37611yescharacteristic
355C22C2×C4×C22⋊C411yescharacteristic
356C22C2×C4×C22⋊C412yes—
357C22C2×C4×C4⋊C412yes—
358C22C2×C4×C4⋊C411yescharacteristic
359C22C2×#44414yes—
360C4—48no—
361C4—416no—
362C4C4×#9514yes—
363C4C4×#9514yes—
364C4C2×C4311yescharacteristic
365C4C22×#9511yescharacteristic
366C4C2×C4×C22⋊C411yescharacteristic
367C4C2×C4×C22⋊C412yes—
368C4C2×C4×C4⋊C411yescharacteristic
369C4C2×C4×C4⋊C412yes—
370C2—21no—
371C2—24no—
372C2—21no—
373C2—22no—
374C2#1070718yesdirect product factor
375C2C2×C4×#9511yescharacteristic
376C2C2×#176514yes—
377C2C2×#177312yes—
subgroupG/H
centerC23×C4C2×D4
commutatorC4C2×C43
Frattini, cosocleC23×C4C24
socleC24C4×D4
prime radicalC25#41
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10707C2direct (N × H)
14. C2×#405C4semidirect
18. C22×#97C4semidirect
21. C2×#341C4semidirect
23. #1735C4semidirect
25. C2×#340C4semidirect
26. C2×#339C4semidirect
72. #405C2×C4semidirect
77. C2×#97C2×C4semidirect
80. #341C2×C4semidirect
81. #340C2×C4semidirect
82. #339C2×C4semidirect
374. C2#10707direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4052
C23×#972
C22×#3412
C2×#17352
C22×#3392
#107072