Finite Group Databasebeta

← C2×#29670 #6435003

Subgroups — C2×#29670 #6435003

Order 512.

4088 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#540C211yesmaximal, characteristic, complemented
2C22×#459C212yesmaximal, complemented
3C2×#2619C211yesmaximal, characteristic, complemented
4#29670C2116yesmaximal, direct product factor
5C2×#2025C212yesmaximal, complemented
6C2×#2011C212yesmaximal, complemented
7C2×#1999C214yesmaximal, complemented
8C2×#1804C212yesmaximal, complemented
9C2×#1798C211yesmaximal, characteristic, complemented
10C23×D8C2214yes—
11C2×#540—216no—
12C2×#540—24no—
13C2×#540C2218yescomplemented
14C22×#127C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#125C2211yescharacteristic
17C22×#123—22no—
18C22×#121—22no—
19C22×#121C2212yes—
20C22×#121C2212yes—
21C22×C4×D4C2212yes—
22C22×#117C2211yescharacteristic
23C2×#459—216no—
24C2×#459C22116yescomplemented
25C22×C4⋊C8C2212yes—
26C22×#102C2214yes—
27C22×#102C2214yes—
28C22×C22⋊C8C2212yes—
29#2619—24no—
30#2619C2218yescomplemented
31C2×#394—22no—
32C2×#392—22no—
33C2×#390C2212yes—
34C2×#382C2211yescharacteristic
35#2025C22116yescomplemented
36#2011C22116yescomplemented
37#1999C22132yescomplemented
38#1804C22116yescomplemented
39#1798C2218yescomplemented
40C2×#340C2214yes—
41C2×#336C2212yes—
42C23×D4—22no—
43C23×D4—22no—
44C23×D4—22no—
45C23×D4—22no—
46C23×D4C2312yes—
47C23×D4C2312yes—
48C24×C4—22no—
49C24×C4C2311yescharacteristic
50C22×D8—216no—
51C22×D8—48no—
52C22×D8—48no—
53C22×D8C23116yes—
54C23×C8C2314yes—
55#540—264no—
56#540—264no—
57C2×#127—28no—
58C2×#127—22no—
59C2×#127—28no—
60C2×#127—22no—
61C2×#127C2314yes—
62C2×#127C2314yes—
63C2×#125—28no—
64C2×#125—22no—
65C2×#125C2314yes—
66C2×#123—232no—
67C2×#123—216no—
68C2×#121—232no—
69C2×#121—28no—
70C2×#121—24no—
71C2×#121—24no—
72C2×#121—28no—
73C2×#121—216no—
74C2×#121—48no—
75C2×#121C2318yes—
76C2×#121C2318yes—
77C2×C4×D4—24no—
78C2×C4×D4—28no—
79C2×C4×D4D4116yescomplemented
80C2×C4×D4C2318yes—
81C2×#117—28no—
82C2×#117D414yes—
83C2×#117C2314yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4D412yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#459—264no—
96C2×C4⋊C8—28no—
97C2×C4⋊C8C2318yes—
98C2×#102—216no—
99C2×#102—48no—
100C2×#102C23116yes—
101C2×#102C23116yes—
102C2×C22⋊C8—44no—
103C2×C22⋊C8C2318yes—
104#394—216no—
105#392—216no—
106#390—24no—
107#390C2318yes—
108#382—22no—
109#382C2314yes—
110C2×#95—22no—
111#340C23116yes—
112#336C2318yes—
113C25—22no—
114C25—22no—
115C25C2×D411yescharacteristic
116C22×D4—216no—
117C22×D4—216no—
118C22×D4—28no—
119C22×D4—28no—
120C22×D4—22no—
121C22×D4—24no—
122C22×D4—28no—
123C22×D4—28no—
124C22×D4—22no—
125C22×D4—28no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—28no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—48no—
134C22×D4—48no—
135C22×D4—88no—
136C22×D4—88no—
137C22×D4—88no—
138C22×D4—88no—
139C22×D4C2×D418yes—
140C22×D4C2414yes—
141C22×D4C2414yes—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—216no—
148C23×C4—28no—
149C23×C4—22no—
150C23×C4—88no—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2412yes—
156C23×C4C2411yescharacteristic
157C2×D8—232no—
158C2×D8—416no—
159C2×D8—416no—
160C2×D8—464no—
161C22×C8—28no—
162C22×C8—28no—
163C22×C8—28no—
164C22×C8C2418yes—
165#127—216no—
166#127—216no—
167#127—216no—
168#127—48no—
169#125—216no—
170#125—216no—
171#123—2128no—
172#121—232no—
173#121—2128no—
174#121—216no—
175#121—216no—
176#121—416no—
177#121—416no—
178C4×D4—232no—
179C4×D4—216no—
180C4×D4C2×D4132yescomplemented
181#117—216no—
182#117—216no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C2×D418yes—
191C2×C4⋊C4C2×D418yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C22⋊C4—216no—
195C2×C22⋊C4—216no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—216no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—88no—
203C2×C22⋊C4C2×D418yes—
204C2×C42—24no—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D418yes—
208C2×C42C2412yes—
209C2×C42C2412yes—
210C4⋊C8—216no—
211#102—232no—
212#102—416no—
213C22⋊C8—48no—
214#95—28no—
215C24—22no—
216C24—28no—
217C24—22no—
218C24—22no—
219C24—28no—
220C24—24no—
221C24—24no—
222C24—44no—
223C24—44no—
224C24—48no—
225C24—48no—
226C24—44no—
227C24—44no—
228C24—44no—
229C24—48no—
230C24—48no—
231C24C22×D412yes—
232C24C22×D411yescharacteristic
233C2×D4—232no—
234C2×D4—232no—
235C2×D4—28no—
236C2×D4—216no—
237C2×D4—216no—
238C2×D4—24no—
239C2×D4—28no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—28no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—416no—
248C2×D4—416no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—48no—
252C2×D4—416no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—416no—
257C2×D4—44no—
258C2×D4—816no—
259C2×D4—816no—
260C2×D4—832no—
261C2×D4—816no—
262C2×D4—816no—
263C2×D4—832no—
264C2×D4—832no—
265C2×D4—832no—
266C2×D4C22×D418yes—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—232no—
283C22×C4—24no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—832no—
290C22×C4—816no—
291C22×C4#12018yes—
292C22×C4C22×D414yes—
293C22×C4C22×D412yes—
294C22×C4C22×D414yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D414yes—
297C22×C4H(5,𝔽2)12yes—
298C22×C4H(5,𝔽2)14yes—
299C22×C42−1+412yes—
300C22×C4C2511yescharacteristic
301D8—464no—
302C2×C8—28no—
303C2×C8—28no—
304C2×C8—28no—
305C2×C8—416no—
306C4⋊C4—232no—
307C4⋊C4—216no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—28no—
311C4⋊C4—44no—
312C4⋊C4C22×D418yes—
313C22⋊C4—232no—
314C22⋊C4—232no—
315C22⋊C4—216no—
316C22⋊C4—416no—
317C22⋊C4—816no—
318C42—24no—
319C42—24no—
320C42#12018yes—
321C42C22×D418yes—
322C23—22no—
323C23—28no—
324C23—22no—
325C23—22no—
326C23—28no—
327C23—44no—
328C23—416no—
329C23—44no—
330C23—44no—
331C23—48no—
332C23—48no—
333C23—416no—
334C23—48no—
335C23—416no—
336C23—48no—
337C23—48no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—48no—
342C23—48no—
343C23—816no—
344C23—816no—
345C23—88no—
346C23—88no—
347C23—88no—
348C23—816no—
349C23—816no—
350C23C2×#12012yes—
351C23#53512yes—
352C23#53712yes—
353C23#57618yes—
354C23C23×D411yescharacteristic
355D4—28no—
356D4—28no—
357D4—416no—
358D4—44no—
359D4—48no—
360D4—44no—
361D4—48no—
362D4—832no—
363D4—832no—
364D4—832no—
365D4—832no—
366D4—88no—
367C2×C4—216no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—28no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—416no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4—832no—
385C2×C4—88no—
386C2×C4C2×#12018yes—
387C2×C4C2×#12012yes—
388C2×C4#53512yes—
389C2×C4#53518yes—
390C2×C4#53712yes—
391C2×C4#53718yes—
392C2×C4C23×D412yes—
393C2×C4C2×H(5,𝔽2)11yescharacteristic
394C2×C4C2×H(5,𝔽2)12yes—
395C2×C4C2×2−1+411yescharacteristic
396C8—48no—
397C22—42no—
398C22—48no—
399C22—44no—
400C22—48no—
401C22—44no—
402C22—48no—
403C22—44no—
404C22—88no—
405C22—88no—
406C22—832no—
407C22—88no—
408C22—84no—
409C22—84no—
410C22—84no—
411C22—88no—
412C22—88no—
413C22—1616no—
414C22#261514yes—
415C22#300818yes—
416C22#322018yes—
417C22#325918yes—
418C22C22×#12011yescharacteristic
419C22C2×#53511yescharacteristic
420C22C2×#53711yescharacteristic
421C22C2×#57614yes—
422C4—28no—
423C4—42no—
424C4—44no—
425C4—48no—
426C4—84no—
427C4#261514yes—
428C4C2×#53512yes—
429C4C2×#53712yes—
430C2—82no—
431C2—88no—
432C2—164no—
433C2#2967018yesdirect product factor
434C2C2×#261511yescharacteristic
435C2C2×#300812yes—
436C2C2×#322012yes—
437C2C2×#325912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#540C2semidirect
2. C22×#459C2semidirect
3. C2×#2619C2semidirect
4. #29670C2direct (N × H)
5. C2×#2025C2semidirect
6. C2×#2011C2semidirect
7. C2×#1999C2semidirect
8. C2×#1804C2semidirect
9. C2×#1798C2semidirect
13. C2×#540C22semidirect
24. C2×#459C22semidirect
30. #2619C22semidirect
35. #2025C22semidirect
36. #2011C22semidirect
37. #1999C22semidirect
38. #1804C22semidirect
39. #1798C22semidirect
79. C2×C4×D4D4semidirect
180. C4×D4C2×D4semidirect
433. C2#29670direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5402
C22×#4592
C2×#26192
#296702
C2×#20252
C2×#20112
C2×#19992
C2×#18042
C2×#17982