Finite Group Databasebeta

← C2×#7716 #161678

Subgroups — C2×#7716 #161678

Order 512.

1076 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2C2×#1839C211yesmaximal, characteristic, complemented
3C2×#1831C211yesmaximal, characteristic, complemented
4#7716C218yesmaximal, direct product factor
5C2×#1272C212yesmaximal, complemented
6C2×#1270C212yesmaximal
7C22×#125C411yescharacteristic
8C22×#124C411yescharacteristic
9C22×C4⋊C8—21no—
10#2861—22no—
11#2861—22no—
12#2861C414yes—
13#2861C2214yes—
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#390C411yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#388—21no—
19C2×#388—21no—
20C2×#388—21no—
21C2×#388—21no—
22C2×C43C2211yescharacteristic
23#1839—22no—
24#1839C2214yescomplemented
25#1831—22no—
26#1831C2214yescomplemented
27C2×#336—21no—
28C2×#336—21no—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1272C2218yescomplemented
33#1270C2218yes—
34C23×Q8C2×C411yescharacteristic
35C23×D4C2×C411yescharacteristic
36C23×C8—22no—
37C23×C8—22no—
38C2×#125—22no—
39C2×#125—44no—
40C2×#125C2×C414yes—
41C2×#124—22no—
42C2×#124—44no—
43C2×#124C2×C414yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—24no—
62C2×C4⋊C8—44no—
63#391—22no—
64#391—21no—
65#391—21no—
66#391C2×C412yes—
67#391C2×C414yes—
68#391C2312yes—
69#390—22no—
70#390—21no—
71#390—21no—
72#390C2×C412yes—
73#390C2×C414yes—
74#390C2312yes—
75#388—24no—
76#388—24no—
77#388—24no—
78#388—24no—
79#388—24no—
80#388—24no—
81#388—24no—
82#388—24no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C43C2×C412yes—
88C43D412yes—
89C43D412yes—
90C43C2312yes—
91#336—22no—
92#336—24no—
93#336—22no—
94#336—24no—
95#336—24no—
96#336C2314yes—
97#336C2314yes—
98C25—21no—
99C22×Q8—21no—
100C22×Q8—24no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8C22×C412yes—
104C22×D4—21no—
105C22×D4—24no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4C22×C412yes—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—44no—
122C22×C8—44no—
123C22×C8—44no—
124C22×C8—44no—
125C22×C8—44no—
126C22×C8—44no—
127#125—48no—
128#125—48no—
129#124—48no—
130#124—48no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C412yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42C22⋊C412yes—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C412yes—
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2411yescharacteristic
180C4⋊C8—48no—
181#95—24no—
182#95—24no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187C24—22no—
188C24—42no—
189C24—42no—
190C24—42no—
191C24—42no—
192C24—42no—
193C24—42no—
194C24—42no—
195C24C2×C22⋊C411yescharacteristic
196C2×Q8—24no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—88no—
203C2×D4—24no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—88no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4#10014yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4#13114yes—
248C22×C4#13514yes—
249C22×C4AGL(1,ℤ/8)12yes—
250C22×C4#13712yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—88no—
261C2×C8—88no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—44no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—48no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C42—24no—
278C42—24no—
279C42—22no—
280C42—22no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—42no—
287C42—42no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—42no—
292C42—42no—
293C42—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23#40812yes—
309C23#41012yes—
310C23C2×#10012yes—
311C23#41612yes—
312C23#41712yes—
313C23#42112yes—
314C23#42112yes—
315C23C22×C22⋊C411yescharacteristic
316Q8—84no—
317Q8—84no—
318D4—84no—
319D4—84no—
320C2×C4—28no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4#40812yes—
364C2×C4C2×#10014yes—
365C2×C4#41612yes—
366C2×C4#41712yes—
367C2×C4C2×#13112yes—
368C2×C4C2×#13512yes—
369C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
370C2×C4C2×#13711yescharacteristic
371C8—84no—
372C8—84no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—168no—
381C22#146914yes—
382C22#148214yes—
383C22#149514yes—
384C22#151214yes—
385C22#152114yes—
386C22#153614yes—
387C22#155114yes—
388C22C2×#40811yescharacteristic
389C22C2×#41011yescharacteristic
390C22C22×#10011yescharacteristic
391C22C2×#41611yescharacteristic
392C22C2×#41711yescharacteristic
393C22C2×#42111yescharacteristic
394C22C2×#42111yescharacteristic
395C4—24no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—82no—
407C4—82no—
408C4—82no—
409C2—162no—
410C2#771618yesdirect product factor
411C2C2×#146911yescharacteristic
412C2C2×#148211yescharacteristic
413C2C2×#149511yescharacteristic
414C2C2×#151211yescharacteristic
415C2C2×#152111yescharacteristic
416C2C2×#153611yescharacteristic
417C2C2×#155111yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1839C2semidirect
3. C2×#1831C2semidirect
4. #7716C2direct (N × H)
5. C2×#1272C2semidirect
24. #1839C22semidirect
26. #1831C22semidirect
32. #1272C22semidirect
410. C2#7716direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28612
C2×#18392
C2×#18312
#77162
C2×#12722
C2×#12702