Finite Group Databasebeta

← C3×#60129 #10678736

Subgroups — C3×#60129 #10678736

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic
2C3×#2925C216yesmaximal, complemented
3C3×#2924C216yesmaximal, complemented
4C3×#2923C213yesmaximal, complemented
5C3×#2922C213yesmaximal, complemented
6C3×#2921C213yesmaximal, complemented
7C2×C3×#433C213yesmaximal, complemented
8C3×#2906C212yesmaximal, complemented
9C3×#2880C212yesmaximal
10C3×#2875C211yesmaximal, characteristic, complemented
11C3×C4×#131C211yesmaximal, characteristic, complemented
12#60129C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#131C2212yes—
15C2×C3×#131C2212yes—
16C3×#520C416yes—
17C3×#520C412yes—
18C3×#519C412yes—
19C3×#518C416yes—
20C3×C4×#41C2211yescharacteristic
21C3×C4×#41C2211yescharacteristic
22C3×C4×#41C2213yes—
23C3×C4×#41C2213yes—
24C2×C3×#117C2213yes—
25C2×C3×#117C2213yes—
26C3×#433C2211yescharacteristic
27C3×#433C2213yes—
28C3×#433C2216yes—
29C3×#433C2216yes—
30C3×#433C2213yescomplemented
31C3×#433C2213yescomplemented
32C3×#433C2216yescomplemented
33C3×#433C2216yescomplemented
34C3×#433C2216yescomplemented
35C3×#432C22112yescomplemented
36C3×#431C22112yescomplemented
37C3×#424C2216yescomplemented
38C3×#424C2216yescomplemented
39C3×#423C2216yescomplemented
40C2×C3×C4⋊C8C2216yescomplemented
41C3×#408C2216yes—
42C3×#408C2216yes—
43C3×#407C2216yes—
44C2×C3×C22⋊C8C2216yes—
45C3×#405C2216yescomplemented
46C3×#405C2216yescomplemented
47C3×C4×M16C2213yescomplemented
48C3×C4×M16C2213yescomplemented
49C2×C3×#97C2213yescomplemented
50C2×C3×C4×C8C2213yescomplemented
51#3428C2×C311yescharacteristic
52#2925C2×C316yescomplemented
53#2924C2×C316yescomplemented
54#2923C2×C313yescomplemented
55#2922C2×C313yescomplemented
56#2921C2×C313yescomplemented
57C2×#433C2×C313yescomplemented
58#2906C2×C312yescomplemented
59#2880C2×C312yes—
60#2875C2×C311yescharacteristic, complemented
61C4×#131C2×C311yescharacteristic, complemented
62C2×C3×#41—28no—
63C2×C3×#41C2×C411yescharacteristic
64C2×C3×#41C2×C413yes—
65C2×C3×#41C2313yes—
66C2×C3×#41C2311yescharacteristic
67C22×C3×Q8C2×C411yescharacteristic
68C22×C3×D4C2×C413yes—
69C23×C3×C4C2313yes—
70C3×#131—28no—
71C3×#131—28no—
72C2×C3×M16C2316yes—
73C2×C3×M16C2316yes—
74C22×C3×C8C2316yes—
75C22×C3×C8C2316yes—
76C3×C4×Q8C2×C416yes—
77C3×C4×Q8C2×C416yes—
78C3×C4×Q8C2×C412yes—
79C3×C4×Q8C2×C412yes—
80C3×C4×D4C2×C4112yes—
81C3×C4×D4C2×C416yes—
82C3×C4×D4C2×C4112yes—
83C3×C4×D4C2×C416yes—
84C3×C4×D4C2×C416yes—
85C3×C4×D4C2×C416yes—
86C3×#117C2×C416yes—
87C3×#117C2×C416yes—
88C3×#117C2×C416yes—
89C3×#117C2×C416yes—
90C3×#117C2313yes—
91C3×#117C2313yes—
92C3×#117C2311yescharacteristic
93C3×#117C2313yes—
94C3×#117C2313yes—
95C3×#117C2313yes—
96C2×C3×C4⋊C4C2×C416yes—
97C2×C3×C4⋊C4C2×C416yes—
98C2×C3×C22⋊C4C2×C416yes—
99C2×C3×C22⋊C4C2×C416yes—
100C2×C3×C42C2313yes—
101C2×C3×C42C2313yes—
102C2×C3×C42C2316yes—
103C3×C4⋊C8C2312yes—
104C3×C4⋊C8C2316yes—
105C3×C4⋊C8C2316yes—
106C3×C4⋊C8C2316yes—
107C3×C4⋊C8C23112yes—
108C3×C22⋊C8C2312yes—
109C3×C22⋊C8C2316yes—
110C3×C22⋊C8C2316yes—
111C3×C22⋊C8C2316yes—
112C3×C22⋊C8C23112yes—
113C3×#97C2311yescharacteristic
114C3×#97C2313yes—
115C3×#97C2316yes—
116C3×#97C2316yes—
117C3×C4×C8C2311yescharacteristic
118C3×C4×C8C2313yes—
119C3×C4×C8C2316yes—
120C3×C4×C8C2316yes—
121C22×#41C22×C311yescharacteristic
122C2×#131C22×C312yes—
123C2×#131C22×C312yes—
124#520C3×C416yes—
125#520C3×C412yes—
126#519C3×C412yes—
127#518C3×C416yes—
128C4×#41C22×C311yescharacteristic
129C4×#41C22×C311yescharacteristic
130C4×#41C22×C313yes—
131C4×#41C22×C313yes—
132C2×#117C22×C313yes—
133C2×#117C22×C313yes—
134#433C22×C311yescharacteristic
135#433C22×C313yes—
136#433C22×C316yes—
137#433C22×C316yes—
138#433C22×C313yescomplemented
139#433C22×C313yescomplemented
140#433C22×C316yescomplemented
141#433C22×C316yescomplemented
142#433C22×C316yescomplemented
143#432C22×C3112yescomplemented
144#431C22×C3112yescomplemented
145#424C22×C316yescomplemented
146#424C22×C316yescomplemented
147#423C22×C316yescomplemented
148C2×C4⋊C8C22×C316yescomplemented
149#408C22×C316yes—
150#408C22×C316yes—
151#407C22×C316yes—
152C2×C22⋊C8C22×C316yes—
153#405C22×C316yescomplemented
154#405C22×C316yescomplemented
155C4×M16C22×C313yescomplemented
156C4×M16C22×C313yescomplemented
157C2×#97C22×C313yescomplemented
158C2×C4×C8C22×C313yescomplemented
159C24×C3C22×C413yes—
160C3×#41—24no—
161C3×#41—212no—
162C3×#41—212no—
163C3×#41#4118yes—
164C2×C3×Q8—24no—
165C2×C3×Q8C22×C413yes—
166C2×C3×Q8C22×C411yescharacteristic
167C2×C3×D4—212no—
168C2×C3×D4C22×C416yes—
169C2×C3×D4C22×C413yes—
170C2×C3×D4C22×C413yes—
171C22×C3×C4—212no—
172C22×C3×C4C22×C413yes—
173C22×C3×C4C22×C413yes—
174C22×C3×C4C22×C413yes—
175C22×C3×C4C22×C4112yes—
176C22×C3×C4C22×C416yes—
177C22×C3×C4C22×C416yes—
178C22×C3×C4C2413yes—
179C22×C3×C4C2411yescharacteristic
180C22×C3×C4C2413yes—
181C3×M16—212no—
182C3×M16—212no—
183C2×C3×C8—212no—
184C2×C3×C8—212no—
185C2×C3×C8C2412yes—
186C2×C3×C8C2416yes—
187C2×C3×C8C2416yes—
188C2×C3×C8C2412yes—
189C3×C4⋊C4C22×C412yes—
190C3×C4⋊C4C22×C416yes—
191C3×C4⋊C4C22×C416yes—
192C3×C4⋊C4C22×C416yes—
193C3×C4⋊C4C22×C416yes—
194C3×C4⋊C4C22×C416yes—
195C3×C22⋊C4C22×C412yes—
196C3×C22⋊C4C22×C416yes—
197C3×C22⋊C4C22×C416yes—
198C3×C22⋊C4C22×C416yes—
199C3×C22⋊C4C22×C416yes—
200C3×C22⋊C4C22×C416yes—
201C3×C42C22×C4112yes—
202C3×C42C22×C416yes—
203C3×C42C22×C416yes—
204C3×C42C2411yescharacteristic
205C3×C42C2413yes—
206C3×C42C2411yescharacteristic
207C3×C42C2413yes—
208C2×#41—28no—
209C2×#41C2×C3×C411yescharacteristic
210C2×#41C2×C3×C413yes—
211C2×#41C23×C313yes—
212C2×#41C23×C311yescharacteristic
213C22×Q8C2×C3×C411yescharacteristic
214C22×D4C2×C3×C413yes—
215C23×C4C23×C313yes—
216#131—28no—
217#131—28no—
218C2×M16C23×C316yes—
219C2×M16C23×C316yes—
220C22×C8C23×C316yes—
221C22×C8C23×C316yes—
222C4×Q8C2×C3×C416yes—
223C4×Q8C2×C3×C416yes—
224C4×Q8C2×C3×C412yes—
225C4×Q8C2×C3×C412yes—
226C4×D4C2×C3×C4112yes—
227C4×D4C2×C3×C416yes—
228C4×D4C2×C3×C4112yes—
229C4×D4C2×C3×C416yes—
230C4×D4C2×C3×C416yes—
231C4×D4C2×C3×C416yes—
232#117C2×C3×C416yes—
233#117C2×C3×C416yes—
234#117C2×C3×C416yes—
235#117C2×C3×C416yes—
236#117C23×C311yescharacteristic
237#117C23×C313yes—
238#117C23×C313yes—
239#117C23×C313yes—
240#117C23×C313yes—
241#117C23×C313yes—
242C2×C4⋊C4C2×C3×C416yes—
243C2×C4⋊C4C2×C3×C416yes—
244C2×C22⋊C4C2×C3×C416yes—
245C2×C22⋊C4C2×C3×C416yes—
246C2×C42C23×C313yes—
247C2×C42C23×C313yes—
248C2×C42C23×C316yes—
249C4⋊C8C23×C312yes—
250C4⋊C8C23×C316yes—
251C4⋊C8C23×C316yes—
252C4⋊C8C23×C316yes—
253C4⋊C8C23×C3112yes—
254C22⋊C8C23×C312yes—
255C22⋊C8C23×C316yes—
256C22⋊C8C23×C316yes—
257C22⋊C8C23×C316yes—
258C22⋊C8C23×C3112yes—
259#97C23×C311yescharacteristic
260#97C23×C313yes—
261#97C23×C316yes—
262#97C23×C316yes—
263C4×C8C23×C311yescharacteristic
264C4×C8C23×C313yes—
265C4×C8C23×C316yes—
266C4×C8C23×C316yes—
267C23×C3—26no—
268C23×C3—46no—
269C23×C3C23×C411yescharacteristic
270C23×C3C23×C413yes—
271C23×C3C23×C413yes—
272C3×Q8—26no—
273C3×Q8#11714yes—
274C3×D4—212no—
275C3×D4—26no—
276C3×D4#117112yes—
277C2×C3×C4—26no—
278C2×C3×C4—26no—
279C2×C3×C4—26no—
280C2×C3×C4—22no—
281C2×C3×C4—26no—
282C2×C3×C4—46no—
283C2×C3×C4—46no—
284C2×C3×C4—412no—
285C2×C3×C4C23×C413yes—
286C2×C3×C4C23×C416yes—
287C2×C3×C4C23×C412yes—
288C2×C3×C4C23×C416yes—
289C2×C3×C4C23×C412yes—
290C2×C3×C4C23×C411yescharacteristic
291C2×C3×C4C23×C413yes—
292C2×C3×C4C2×#41112yes—
293C2×C3×C4C2511yescharacteristic
294C3×C8—22no—
295C3×C8—22no—
296C3×C8—26no—
297C3×C8—26no—
298C24C22×C3×C413yes—
299#41—24no—
300#41—212no—
301#41—212no—
302#41C3×#4118yes—
303C2×Q8—24no—
304C2×Q8C22×C3×C413yes—
305C2×Q8C22×C3×C411yescharacteristic
306C2×D4—212no—
307C2×D4C22×C3×C416yes—
308C2×D4C22×C3×C413yes—
309C2×D4C22×C3×C413yes—
310C22×C4—212no—
311C22×C4C22×C3×C413yes—
312C22×C4C22×C3×C413yes—
313C22×C4C22×C3×C413yes—
314C22×C4C22×C3×C4112yes—
315C22×C4C22×C3×C416yes—
316C22×C4C22×C3×C416yes—
317C22×C4C24×C313yes—
318C22×C4C24×C311yescharacteristic
319C22×C4C24×C313yes—
320M16—212no—
321M16—212no—
322C2×C8—212no—
323C2×C8—212no—
324C2×C8C24×C312yes—
325C2×C8C24×C316yes—
326C2×C8C24×C316yes—
327C2×C8C24×C312yes—
328C4⋊C4C22×C3×C412yes—
329C4⋊C4C22×C3×C416yes—
330C4⋊C4C22×C3×C416yes—
331C4⋊C4C22×C3×C416yes—
332C4⋊C4C22×C3×C416yes—
333C4⋊C4C22×C3×C416yes—
334C22⋊C4C22×C3×C412yes—
335C22⋊C4C22×C3×C416yes—
336C22⋊C4C22×C3×C416yes—
337C22⋊C4C22×C3×C416yes—
338C22⋊C4C22×C3×C416yes—
339C22⋊C4C22×C3×C416yes—
340C42C22×C3×C4112yes—
341C42C22×C3×C416yes—
342C42C22×C3×C416yes—
343C42C24×C311yescharacteristic
344C42C24×C313yes—
345C42C24×C311yescharacteristic
346C42C24×C313yes—
347C22×C3—26no—
348C22×C3—21no—
349C22×C3—23no—
350C22×C3—26no—
351C22×C3—22no—
352C22×C3—86no—
353C22×C3C2×#11716yes—
354C22×C3C24×C411yescharacteristic
355C3×C4—21no—
356C3×C4—23no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—26no—
360C3×C4—26no—
361C3×C4C2×#11716yes—
362C3×C4C22×#4112yes—
363C23—26no—
364C23—46no—
365C23C23×C3×C411yescharacteristic
366C23C23×C3×C413yes—
367C23C23×C3×C413yes—
368Q8—26no—
369Q8C3×#11714yes—
370D4—212no—
371D4—26no—
372D4C3×#117112yes—
373C2×C4—26no—
374C2×C4—22no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—46no—
379C2×C4—46no—
380C2×C4—412no—
381C2×C4C23×C3×C413yes—
382C2×C4C23×C3×C416yes—
383C2×C4C23×C3×C412yes—
384C2×C4C23×C3×C416yes—
385C2×C4C23×C3×C412yes—
386C2×C4C23×C3×C411yescharacteristic
387C2×C4C23×C3×C413yes—
388C2×C4C2×C3×#41112yes—
389C2×C4C25×C311yescharacteristic
390C8—22no—
391C8—22no—
392C8—26no—
393C8—26no—
394C2×C3—26no—
395C2×C3—43no—
396C2×C3—41no—
397C2×C3C22×#11711yescharacteristic
398C2×C3#357312yes—
399C22—26no—
400C22—21no—
401C22—23no—
402C22—26no—
403C22—22no—
404C22—86no—
405C22C2×C3×#11716yes—
406C22C24×C3×C411yescharacteristic
407C4—21no—
408C4—23no—
409C4—22no—
410C4—22no—
411C4—26no—
412C4—26no—
413C4C2×C3×#11716yes—
414C4C22×C3×#4112yes—
415C3#6012911yescharacteristic, direct product factor, Sylow
416C2—26no—
417C2—41no—
418C2—43no—
419C2C22×C3×#11711yescharacteristic
420C2C3×#357312yes—
subgroupG/H
centerC2×C3×C4C25
commutatorC22C24×C3×C4
Frattini, cosocleC2×C4C25×C3
socleC22×C3C24×C4
prime radicalC22×C3×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2925C2semidirect
3. C3×#2924C2semidirect
4. C3×#2923C2semidirect
5. C3×#2922C2semidirect
6. C3×#2921C2semidirect
7. C2×C3×#433C2semidirect
8. C3×#2906C2semidirect
10. C3×#2875C2semidirect
11. C3×C4×#131C2semidirect
12. #60129C3direct (N × H)
30. C3×#433C22semidirect
31. C3×#433C22semidirect
32. C3×#433C22semidirect
33. C3×#433C22semidirect
34. C3×#433C22semidirect
35. C3×#432C22semidirect
36. C3×#431C22semidirect
37. C3×#424C22semidirect
38. C3×#424C22semidirect
39. C3×#423C22semidirect
40. C2×C3×C4⋊C8C22semidirect
45. C3×#405C22semidirect
46. C3×#405C22semidirect
47. C3×C4×M16C22semidirect
48. C3×C4×M16C22semidirect
49. C2×C3×#97C22semidirect
50. C2×C3×C4×C8C22semidirect
52. #2925C2×C3semidirect
53. #2924C2×C3semidirect
54. #2923C2×C3semidirect
55. #2922C2×C3semidirect
56. #2921C2×C3semidirect
57. C2×#433C2×C3semidirect
58. #2906C2×C3semidirect
60. #2875C2×C3semidirect
61. C4×#131C2×C3semidirect
138. #433C22×C3semidirect
139. #433C22×C3semidirect
140. #433C22×C3semidirect
141. #433C22×C3semidirect
142. #433C22×C3semidirect
143. #432C22×C3semidirect
144. #431C22×C3semidirect
145. #424C22×C3semidirect
146. #424C22×C3semidirect
147. #423C22×C3semidirect
148. C2×C4⋊C8C22×C3semidirect
153. #405C22×C3semidirect
154. #405C22×C3semidirect
155. C4×M16C22×C3semidirect
156. C4×M16C22×C3semidirect
157. C2×#97C22×C3semidirect
158. C2×C4×C8C22×C3semidirect
415. C3#60129direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34282
C3×#29252
C3×#29242
C3×#29232
C3×#29222
C3×#29212
C2×C3×#4332
C3×#29062
C3×#28802
C3×#28752
C3×C4×#1312
#601293