Finite Group Databasebeta

← C2×#29738 #6433033

Subgroups — C2×#29738 #6433033

Order 512.

3378 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#466C211yesmaximal, characteristic, complemented
2C22×#408C211yesmaximal, characteristic, complemented
3C2×#2615C212yesmaximal, complemented
4#29738C2116yesmaximal, direct product factor
5C2×#2076C212yesmaximal, complemented
6C2×#2033C214yesmaximal, complemented
7C2×#1999C212yesmaximal, complemented
8C2×#1895C212yesmaximal, complemented
9C2×#1808C211yesmaximal, characteristic, complemented
10C23×#41C2211yescharacteristic
11C23×D8C2211yescharacteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2212yes—
15C22×#121C2214yes—
16C2×#466—28no—
17C2×#466C2218yescomplemented
18C22×#107C2211yescharacteristic
19C22×#107C2212yes—
20C22×#102C2212yes—
21C22×#102C2214yes—
22C2×#408—28no—
23C2×#408C2218yescomplemented
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26#2615—28no—
27#2615C22116yescomplemented
28C2×#393—22no—
29C2×#392—24no—
30C2×#392—22no—
31C2×#380C2214yes—
32C2×#380C2212yes—
33#2076C22116yescomplemented
34#2033C22132yescomplemented
35#1999C22116yescomplemented
36#1895C22116yescomplemented
37#1808C2218yescomplemented
38C2×#340C2212yes—
39C2×#340C2214yes—
40C22×#41—24no—
41C22×#41—24no—
42C22×#41—44no—
43C22×#41—44no—
44C22×#41D414yes—
45C22×#41C2314yes—
46C23×Q8D411yescharacteristic
47C23×D4—24no—
48C23×D4—22no—
49C23×D4D412yes—
50C23×D4D411yescharacteristic
51C23×D4C2312yes—
52C24×C4C2312yes—
53C24×C4C2311yescharacteristic
54C22×D8—24no—
55C22×D8—44no—
56C22×D8C2314yes—
57C22×M16—24no—
58C22×M16—44no—
59C22×M16C2314yes—
60C23×C8—44no—
61C23×C8D418yescomplemented
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2314yes—
66C2×#121—28no—
67C2×#121—24no—
68C2×#121—28no—
69C2×#121—28no—
70C2×#121—48no—
71C2×#121—48no—
72C2×#121C2318yes—
73C2×#121C23116yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—22no—
80#466—232no—
81C2×#107—24no—
82C2×#107—44no—
83C2×#107C2314yes—
84C2×#107C2318yes—
85C2×#102—28no—
86C2×#102—48no—
87C2×#102C2318yes—
88C2×#102C23116yes—
89#408—232no—
90C2×C22⋊C8—28no—
91C2×C22⋊C8—28no—
92C2×C22⋊C8C2318yes—
93C2×C22⋊C8C2318yes—
94#393—216no—
95#392—232no—
96#392—216no—
97#380—28no—
98#380—24no—
99#380C2318yes—
100#380C23116yes—
101C2×#95—24no—
102C2×#95—22no—
103#340C2318yes—
104#340C23116yes—
105C25—22no—
106C25C2×D412yes—
107C25C2×D411yescharacteristic
108C2×#41—216no—
109C2×#41—28no—
110C2×#41—28no—
111C2×#41—416no—
112C2×#41—48no—
113C2×#41—48no—
114C2×#41—416no—
115C2×#41—416no—
116C2×#41—432no—
117C22×Q8—24no—
118C22×Q8—44no—
119C22×Q8C2×D414yes—
120C22×D4—28no—
121C22×D4—216no—
122C22×D4—28no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—24no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—44no—
130C22×D4—48no—
131C22×D4—416no—
132C22×D4—48no—
133C22×D4—88no—
134C22×D4—88no—
135C22×D4—88no—
136C22×D4C2×D418yes—
137C22×D4C2×D414yes—
138C22×D4C2414yes—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—28no—
144C23×C4—28no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4C2×D414yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D414yes—
152C23×C4#4114yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155C23×C4C2411yescharacteristic
156C2×D8—28no—
157C2×D8—48no—
158C2×D8—416no—
159C2×M16—28no—
160C2×M16—48no—
161C2×M16—416no—
162C22×C8—24no—
163C22×C8—24no—
164C22×C8—24no—
165C22×C8—28no—
166C22×C8—416no—
167C22×C8—48no—
168C22×C8C2×D414yes—
169C22×C8C2×D414yes—
170C22×C8C2×D4116yescomplemented
171C22×C8C2414yes—
172C22×C8C2412yes—
173C22×C8C2412yes—
174#121—216no—
175#121—216no—
176#121—416no—
177#121—416no—
178#121—432no—
179#121—416no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4C2414yes—
187C2×C4⋊C4C2418yes—
188C2×C22⋊C4—216no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—216no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—416no—
193C2×C22⋊C4—48no—
194C2×C22⋊C4—84no—
195C2×C22⋊C4—88no—
196#107—28no—
197#107—48no—
198#102—216no—
199#102—416no—
200C22⋊C8—216no—
201C22⋊C8—216no—
202#95—216no—
203#95—28no—
204C24—24no—
205C24—22no—
206C24—24no—
207C24—22no—
208C24—24no—
209C24—28no—
210C24—28no—
211C24—24no—
212C24—24no—
213C24—24no—
214C24—44no—
215C24—48no—
216C24—44no—
217C24—48no—
218C24—44no—
219C24#12114yes—
220C24C22×D414yes—
221C24C22×D412yes—
222C24C22×D411yescharacteristic
223#41—464no—
224#41—416no—
225#41—416no—
226#41—432no—
227C2×Q8—28no—
228C2×Q8—48no—
229C2×Q8—416no—
230C2×D4—216no—
231C2×D4—28no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—28no—
235C2×D4—28no—
236C2×D4—48no—
237C2×D4—416no—
238C2×D4—416no—
239C2×D4—432no—
240C2×D4—432no—
241C2×D4—416no—
242C2×D4—44no—
243C2×D4—48no—
244C2×D4—48no—
245C2×D4—48no—
246C2×D4—416no—
247C2×D4—816no—
248C2×D4—816no—
249C2×D4—816no—
250C2×D4—832no—
251C2×D4—864no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—216no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—216no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—416no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4#12114yes—
278C22×C4#12118yes—
279C22×C4C22×D414yes—
280C22×C4C22×D412yes—
281C22×C4C22×D414yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C2×#4114yes—
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)14yes—
287C22×C4H(5,𝔽2)14yes—
288C22×C4C2511yescharacteristic
289D8—416no—
290M16—416no—
291C2×C8—28no—
292C2×C8—28no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—28no—
297C2×C8—416no—
298C2×C8—48no—
299C2×C8C22×D414yes—
300C2×C8C22×D414yes—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C22⋊C4—432no—
308C22⋊C4—416no—
309C22⋊C4—88no—
310C22⋊C4—816no—
311C23—28no—
312C23—24no—
313C23—28no—
314C23—28no—
315C23—22no—
316C23—24no—
317C23—22no—
318C23—24no—
319C23—28no—
320C23—28no—
321C23—24no—
322C23—24no—
323C23—48no—
324C23—48no—
325C23—48no—
326C23—416no—
327C23—416no—
328C23—48no—
329C23—48no—
330C23—44no—
331C23—48no—
332C23—48no—
333C23—44no—
334C23—44no—
335C23—88no—
336C23—816no—
337C23—816no—
338C23—88no—
339C23—88no—
340C23C2×#12114yes—
341C23C2×#12112yes—
342C23#53414yes—
343C23#57612yes—
344C23#57614yes—
345C23#57812yes—
346C23C23×D411yescharacteristic
347Q8—416no—
348D4—416no—
349D4—432no—
350D4—44no—
351D4—44no—
352D4—48no—
353D4—832no—
354D4—864no—
355D4—88no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—216no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—48no—
365C2×C4—416no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—816no—
375C2×C4—88no—
376C2×C4C2×#12114yes—
377C2×C4C2×#12112yes—
378C2×C4#53414yes—
379C2×C4#53414yes—
380C2×C4#53514yes—
381C2×C4#53818yes—
382C2×C4C23×D411yescharacteristic
383C2×C4C22×#4111yescharacteristic
384C2×C4C2×H(5,𝔽2)12yes—
385C2×C4C2×H(5,𝔽2)12yes—
386C8—24no—
387C8—24no—
388C8—44no—
389C22—24no—
390C22—24no—
391C22—24no—
392C22—416no—
393C22—44no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—48no—
398C22—48no—
399C22—44no—
400C22—816no—
401C22—84no—
402C22—84no—
403C22—88no—
404C22—88no—
405C22—84no—
406C22—1616no—
407C22#261714yes—
408C22#305814yes—
409C22#306114yes—
410C22#319918yes—
411C22#320318yes—
412C22C22×#12111yescharacteristic
413C22C2×#53412yes—
414C22C2×#57612yes—
415C22C2×#57611yescharacteristic
416C22C2×#57811yescharacteristic
417C4—24no—
418C4—44no—
419C4—84no—
420C4—88no—
421C4#261714yes—
422C4C2×#53411yescharacteristic
423C4C2×#53511yescharacteristic
424C4C2×#53812yes—
425C2—44no—
426C2—84no—
427C2—164no—
428C2#2973818yesdirect product factor
429C2C2×#261711yescharacteristic
430C2C2×#305811yescharacteristic
431C2C2×#306111yescharacteristic
432C2C2×#319912yes—
433C2C2×#320312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#466C2semidirect
2. C22×#408C2semidirect
3. C2×#2615C2semidirect
4. #29738C2direct (N × H)
5. C2×#2076C2semidirect
6. C2×#2033C2semidirect
7. C2×#1999C2semidirect
8. C2×#1895C2semidirect
9. C2×#1808C2semidirect
17. C2×#466C22semidirect
23. C2×#408C22semidirect
27. #2615C22semidirect
33. #2076C22semidirect
34. #2033C22semidirect
35. #1999C22semidirect
36. #1895C22semidirect
37. #1808C22semidirect
61. C23×C8D4semidirect
170. C22×C8C2×D4semidirect
428. C2#29738direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4662
C22×#4082
C2×#26152
#297382
C2×#20762
C2×#20332
C2×#19992
C2×#18952
C2×#18082