Finite Group Databasebeta

← C32×#3069 #11770118

Subgroups — C32×#3069 #11770118

Order 1152.

1338 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#575C211yesmaximal, characteristic
2C2×C32×#137C211yesmaximal, characteristic, complemented
3C2×C32×#137C211yesmaximal, characteristic, complemented
4C32×#568C211yesmaximal, characteristic, complemented
5C32×#473C214yesmaximal, complemented
6C32×#471C214yesmaximal, complemented
7C32×#430C211yesmaximal, characteristic, complemented
8C32×#413C211yesmaximal, characteristic, complemented
9C2×C32×#101C211yesmaximal, characteristic, complemented
10C3×#3069C314yesmaximal, direct product factor
11C2×C32×#41C2211yescharacteristic
12C2×C32×#41C2211yescharacteristic
13C22×C32×Q8C2211yescharacteristic
14C32×#137—22no—
15C32×#137—21no—
16C32×#137—22no—
17C32×#137—22no—
18C32×#137C2214yes—
19C32×#137C2212yes—
20C32×AGL(1,ℤ/8)—21no—
21C32×AGL(1,ℤ/8)C2212yes—
22C32×#135—22no—
23C2×C32×Q16—21no—
24C2×C32×Q16C2212yescomplemented
25C2×C32×SD16—21no—
26C2×C32×SD16C2212yescomplemented
27C32×#131—22no—
28C32×#131C2214yes—
29C2×C32×M16—22no—
30C2×C32×M16C2211yescharacteristic
31C2×C32×M16C2211yescharacteristic
32C2×C32×M16C2211yescharacteristic
33C2×C32×M16C2211yescharacteristic
34C32×#108C2214yescomplemented
35C32×#101C2214yescomplemented
36C32×#101C2212yescomplemented
37C32×#100C2212yescomplemented
38C3×#575C2×C314yes—
39C2×C3×#137C2×C314yescomplemented
40C2×C3×#137C2×C314yescomplemented
41C3×#568C2×C314yescomplemented
42C3×#473C2×C3116yescomplemented
43C3×#471C2×C3116yescomplemented
44C3×#430C2×C314yescomplemented
45C3×#413C2×C314yescomplemented
46C2×C3×#101C2×C314yescomplemented
47C32×#41—21no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C32×#41—21no—
52C32×#41—21no—
53C32×#41D412yes—
54C32×#41C2312yes—
55C2×C32×Q8—21no—
56C2×C32×Q8—21no—
57C2×C32×Q8—21no—
58C2×C32×Q8—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8D411yescharacteristic
61C2×C32×Q8C2312yes—
62C2×C32×Q8C2311yescharacteristic
63C2×C32×D4—21no—
64C2×C32×D4—21no—
65C2×C32×D4D411yescharacteristic
66C2×C32×D4C2311yescharacteristic
67C22×C32×C4—21no—
68C22×C32×C4—21no—
69C22×C32×C4—21no—
70C22×C32×C4C2311yescharacteristic
71C32×Q16—24no—
72C32×Q16—22no—
73C32×Q16—24no—
74C32×SD16—22no—
75C32×SD16—24no—
76C32×SD16—22no—
77C32×SD16—24no—
78C32×D8—22no—
79C32×M16—21no—
80C32×M16—21no—
81C32×M16—21no—
82C32×M16—24no—
83C32×M16D412yes—
84C32×M16D412yes—
85C32×M16C2312yes—
86C32×M16C2312yes—
87C32×M16C2312yes—
88C2×C32×C8—21no—
89C2×C32×C8—21no—
90C2×C32×C8—21no—
91C2×C32×C8—24no—
92C2×C32×C8C2312yes—
93#3069C3211yescharacteristic, direct product factor, Sylow
94C2×C3×#41C22×C314yes—
95C2×C3×#41C22×C314yes—
96C22×C3×Q8C22×C314yes—
97C3×#137—28no—
98C3×#137—24no—
99C3×#137—28no—
100C3×#137—28no—
101C3×#137C22×C3116yes—
102C3×#137C22×C318yes—
103C3×AGL(1,ℤ/8)—24no—
104C3×AGL(1,ℤ/8)C22×C318yes—
105C3×#135—28no—
106C2×C3×Q16—24no—
107C2×C3×Q16C22×C318yescomplemented
108C2×C3×SD16—24no—
109C2×C3×SD16C22×C318yescomplemented
110C3×#131—28no—
111C3×#131C22×C3116yes—
112C2×C3×M16—28no—
113C2×C3×M16C22×C314yes—
114C2×C3×M16C22×C314yes—
115C2×C3×M16C22×C314yes—
116C2×C3×M16C22×C314yes—
117C3×#108C22×C3116yescomplemented
118C3×#101C22×C3116yescomplemented
119C3×#101C22×C318yescomplemented
120C3×#100C22×C318yescomplemented
121C23×C32—21no—
122C23×C32—21no—
123C23×C32#4111yescharacteristic
124C32×Q8—21no—
125C32×Q8—22no—
126C32×Q8—22no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—41no—
130C32×Q8—41no—
131C32×Q8C2×D412yes—
132C32×D4—22no—
133C32×D4—22no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—41no—
138C32×D4—41no—
139C32×D4C2×D412yes—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—21no—
144C2×C32×C4—22no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4#4111yescharacteristic
154C2×C32×C4C2411yescharacteristic
155C32×C8—22no—
156C32×C8—22no—
157C32×C8—22no—
158C32×C8C2×D414yes—
159#575C2×C3211yescharacteristic
160C2×#137C2×C3211yescharacteristic, complemented
161C2×#137C2×C3211yescharacteristic, complemented
162#568C2×C3211yescharacteristic, complemented
163#473C2×C3214yescomplemented
164#471C2×C3214yescomplemented
165#430C2×C3211yescharacteristic, complemented
166#413C2×C3211yescharacteristic, complemented
167C2×#101C2×C3211yescharacteristic, complemented
168C3×#41—24no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—24no—
172C3×#41—24no—
173C3×#41—24no—
174C3×#41C3×D418yes—
175C3×#41C23×C318yes—
176C2×C3×Q8—24no—
177C2×C3×Q8—24no—
178C2×C3×Q8—24no—
179C2×C3×Q8—24no—
180C2×C3×Q8—24no—
181C2×C3×Q8C3×D414yes—
182C2×C3×Q8C23×C318yes—
183C2×C3×Q8C23×C314yes—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4C3×D414yes—
187C2×C3×D4C23×C314yes—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4C23×C314yes—
192C3×Q16—216no—
193C3×Q16—28no—
194C3×Q16—216no—
195C3×SD16—28no—
196C3×SD16—216no—
197C3×SD16—28no—
198C3×SD16—216no—
199C3×D8—28no—
200C3×M16—24no—
201C3×M16—24no—
202C3×M16—24no—
203C3×M16—216no—
204C3×M16C3×D418yes—
205C3×M16C3×D418yes—
206C3×M16C23×C318yes—
207C3×M16C23×C318yes—
208C3×M16C23×C318yes—
209C2×C3×C8—24no—
210C2×C3×C8—24no—
211C2×C3×C8—24no—
212C2×C3×C8—216no—
213C2×C3×C8C23×C318yes—
214C22×C32—22no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—81no—
218C22×C32—81no—
219C22×C32#12111yescharacteristic
220C22×C32#12111yescharacteristic
221C22×C32C2×#4111yescharacteristic
222C32×C4—22no—
223C32×C4—41no—
224C32×C4—42no—
225C32×C4#12111yescharacteristic
226C32×C4#12111yescharacteristic
227C32×C4C22×D411yescharacteristic
228C32×C4C22×D411yescharacteristic
229C2×#41C22×C3211yescharacteristic
230C2×#41C22×C3211yescharacteristic
231C22×Q8C22×C3211yescharacteristic
232#137—22no—
233#137—21no—
234#137—22no—
235#137—22no—
236#137C22×C3214yes—
237#137C22×C3212yes—
238AGL(1,ℤ/8)—21no—
239AGL(1,ℤ/8)C22×C3212yes—
240#135—22no—
241C2×Q16—21no—
242C2×Q16C22×C3212yescomplemented
243C2×SD16—21no—
244C2×SD16C22×C3212yescomplemented
245#131—22no—
246#131C22×C3214yes—
247C2×M16—22no—
248C2×M16C22×C3211yescharacteristic
249C2×M16C22×C3211yescharacteristic
250C2×M16C22×C3211yescharacteristic
251C2×M16C22×C3211yescharacteristic
252#108C22×C3214yescomplemented
253#101C22×C3214yescomplemented
254#101C22×C3212yescomplemented
255#100C22×C3212yescomplemented
256C23×C3—24no—
257C23×C3—24no—
258C23×C3C3×#4114yes—
259C3×Q8—24no—
260C3×Q8—28no—
261C3×Q8—28no—
262C3×Q8—24no—
263C3×Q8—24no—
264C3×Q8—44no—
265C3×Q8—44no—
266C3×Q8C2×C3×D418yes—
267C3×D4—28no—
268C3×D4—28no—
269C3×D4—24no—
270C3×D4—24no—
271C3×D4—24no—
272C3×D4—44no—
273C3×D4—44no—
274C3×D4C2×C3×D418yes—
275C2×C3×C4—24no—
276C2×C3×C4—28no—
277C2×C3×C4—28no—
278C2×C3×C4—24no—
279C2×C3×C4—28no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4C2×C3×D414yes—
285C2×C3×C4C2×C3×D414yes—
286C2×C3×C4C2×C3×D414yes—
287C2×C3×C4C2×C3×D414yes—
288C2×C3×C4C3×#4114yes—
289C2×C3×C4C24×C314yes—
290C3×C8—28no—
291C3×C8—28no—
292C3×C8—28no—
293C3×C8C2×C3×D4116yes—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—42no—
298C2×C32—81no—
299C2×C32C2×#12111yescharacteristic
300#41—21no—
301#41—21no—
302#41—21no—
303#41—21no—
304#41—21no—
305#41—21no—
306#41C32×D412yes—
307#41C23×C3212yes—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C32×D411yescharacteristic
314C2×Q8C23×C3212yes—
315C2×Q8C23×C3211yescharacteristic
316C2×D4—21no—
317C2×D4—21no—
318C2×D4C32×D411yescharacteristic
319C2×D4C23×C3211yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4C23×C3211yescharacteristic
324Q16—24no—
325Q16—22no—
326Q16—24no—
327SD16—22no—
328SD16—24no—
329SD16—22no—
330SD16—24no—
331D8—22no—
332M16—21no—
333M16—21no—
334M16—21no—
335M16—24no—
336M16C32×D412yes—
337M16C32×D412yes—
338M16C23×C3212yes—
339M16C23×C3212yes—
340M16C23×C3212yes—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—24no—
345C2×C8C23×C3212yes—
346C22×C3—28no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—84no—
350C22×C3—84no—
351C22×C3C3×#12114yes—
352C22×C3C3×#12114yes—
353C22×C3C2×C3×#4114yes—
354C3×C4—28no—
355C3×C4—44no—
356C3×C4—48no—
357C3×C4C3×#12114yes—
358C3×C4C3×#12114yes—
359C3×C4C22×C3×D414yes—
360C3×C4C22×C3×D414yes—
361C32#306911yescharacteristic, direct product factor, Sylow
362C23—21no—
363C23—21no—
364C23C32×#4111yescharacteristic
365Q8—21no—
366Q8—22no—
367Q8—22no—
368Q8—21no—
369Q8—21no—
370Q8—41no—
371Q8—41no—
372Q8C2×C32×D412yes—
373D4—22no—
374D4—22no—
375D4—21no—
376D4—21no—
377D4—21no—
378D4—41no—
379D4—41no—
380D4C2×C32×D412yes—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4C2×C32×D411yescharacteristic
391C2×C4C2×C32×D411yescharacteristic
392C2×C4C2×C32×D411yescharacteristic
393C2×C4C2×C32×D411yescharacteristic
394C2×C4C32×#4111yescharacteristic
395C2×C4C24×C3211yescharacteristic
396C8—22no—
397C8—22no—
398C8—22no—
399C8C2×C32×D414yes—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—48no—
404C2×C3—84no—
405C2×C3C2×C3×#12114yes—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22C32×#12111yescharacteristic
412C22C32×#12111yescharacteristic
413C22C2×C32×#4111yescharacteristic
414C4—22no—
415C4—41no—
416C4—42no—
417C4C32×#12111yescharacteristic
418C4C32×#12111yescharacteristic
419C4C22×C32×D411yescharacteristic
420C4C22×C32×D411yescharacteristic
421C3C3×#306914yesdirect product factor
422C2—21no—
423C2—21no—
424C2—21no—
425C2—42no—
426C2—81no—
427C2C2×C32×#12111yescharacteristic
subgroupG/H
center, socleC2×C32C2×#121
commutator, Frattini, cosocleC2×C4C24×C32
prime radicalC32×#575C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C32 ◁ C22×C32×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C32×#137C2semidirect
3. C2×C32×#137C2semidirect
4. C32×#568C2semidirect
5. C32×#473C2semidirect
6. C32×#471C2semidirect
7. C32×#430C2semidirect
8. C32×#413C2semidirect
9. C2×C32×#101C2semidirect
10. C3×#3069C3direct (N × H)
24. C2×C32×Q16C22semidirect
26. C2×C32×SD16C22semidirect
34. C32×#108C22semidirect
35. C32×#101C22semidirect
36. C32×#101C22semidirect
37. C32×#100C22semidirect
39. C2×C3×#137C2×C3semidirect
40. C2×C3×#137C2×C3semidirect
41. C3×#568C2×C3semidirect
42. C3×#473C2×C3semidirect
43. C3×#471C2×C3semidirect
44. C3×#430C2×C3semidirect
45. C3×#413C2×C3semidirect
46. C2×C3×#101C2×C3semidirect
93. #3069C32direct (N × H)
107. C2×C3×Q16C22×C3semidirect
109. C2×C3×SD16C22×C3semidirect
117. C3×#108C22×C3semidirect
118. C3×#101C22×C3semidirect
119. C3×#101C22×C3semidirect
120. C3×#100C22×C3semidirect
160. C2×#137C2×C32semidirect
161. C2×#137C2×C32semidirect
162. #568C2×C32semidirect
163. #473C2×C32semidirect
164. #471C2×C32semidirect
165. #430C2×C32semidirect
166. #413C2×C32semidirect
167. C2×#101C2×C32semidirect
242. C2×Q16C22×C32semidirect
244. C2×SD16C22×C32semidirect
252. #108C22×C32semidirect
253. #101C22×C32semidirect
254. #101C22×C32semidirect
255. #100C22×C32semidirect
361. C32#3069direct (N × H)
421. C3C3×#3069direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5752
C2×C32×#1372
C32×#5682
C32×#4732
C32×#4712
C32×#4302
C32×#4132
C2×C32×#1012
C3×#30693