Finite Group Databasebeta

← C3×#60622 #10679229

Subgroups — C3×#60622 #10679229

Order 768.

2178 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×#3503C211yesmaximal, characteristic, complemented
2C3×#3498C211yesmaximal, characteristic
3C3×#3141C213yesmaximal, complemented
4C3×#3138C213yesmaximal, complemented
5C3×#3136C2112yesmaximal, complemented
6C3×#3131C213yesmaximal, complemented
7C2×C3×#486C213yesmaximal, complemented
8C3×#2897C214yesmaximal, complemented
9C3×C4×#131C211yesmaximal, characteristic, complemented
10#60622C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×H(5,𝔽2)C2212yes—
12C2×C3×#131C2212yes—
13C3×Q82—21no—
14C3×#557—21no—
15C3×#554—23no—
16C3×#550—21no—
17C3×#550—21no—
18C3×D4×Q8—23no—
19C3×D42—23no—
20C3×D42—23no—
21C3×#544C2216yescomplemented
22C3×#535C2216yes—
23C3×#533C2213yescomplemented
24C3×#532C2213yes—
25C2×C3×C4⋊Q8C2213yescomplemented
26C2×C3×#127C2213yes—
27C3×#520C2212yes—
28C3×C4×#41C2211yescharacteristic
29C3×#490C22112yescomplemented
30C3×#489C22112yescomplemented
31C3×#488C22112yescomplemented
32C3×#487C2211yescharacteristic, complemented
33C3×#487C2213yescomplemented
34C3×#486C2213yescomplemented
35C3×#486C22112yescomplemented
36C3×#486C2216yescomplemented
37C3×#486C2213yescomplemented
38C3×#418C22112yescomplemented
39C3×#417C22112yescomplemented
40C3×#416C22112yescomplemented
41C2×C3×#102C22112yescomplemented
42C3×#405C2216yescomplemented
43C3×C4×M16C2213yescomplemented
44C2×C3×C4×C8C2213yescomplemented
45#3503C2×C311yescharacteristic, complemented
46#3498C2×C311yescharacteristic
47#3141C2×C313yescomplemented
48#3138C2×C313yescomplemented
49#3136C2×C3112yescomplemented
50#3131C2×C313yescomplemented
51C2×#486C2×C313yescomplemented
52#2897C2×C314yescomplemented
53C4×#131C2×C311yescharacteristic, complemented
54C3×H(5,𝔽2)—216no—
55C2×C3×#41—22no—
56C2×C3×#41C2316yes—
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8—23no—
59C22×C3×D4—26no—
60C22×C3×D4—23no—
61C22×C3×D4—23no—
62C22×C3×D4C2316yes—
63C3×#131—28no—
64C2×C3×M16C2316yes—
65C22×C3×C8C2316yes—
66C3×C4⋊Q8—26no—
67C3×C4⋊Q8—23no—
68C3×C4⋊Q8—23no—
69C3×C4⋊Q8C2311yescharacteristic
70C3×C4⋊Q8C2313yes—
71C3×C4⋊Q8C2316yes—
72C3×C4⋊Q8C2311yescharacteristic
73C3×#127—23no—
74C3×#127—23no—
75C3×#127C2313yes—
76C3×#127C2316yes—
77C3×#125—26no—
78C3×#125C2316yes—
79C3×#124C2316yes—
80C3×C22⋊Q8—23no—
81C3×C22⋊Q8—23no—
82C3×C22⋊Q8—212no—
83C3×#121—23no—
84C3×#121—23no—
85C3×#121—26no—
86C3×#121—26no—
87C3×#120—212no—
88C3×C4×Q8—22no—
89C3×C4×Q8—23no—
90C3×C4×Q8—23no—
91C3×C4×Q8D411yescharacteristic
92C3×C4×Q8D411yescharacteristic
93C3×C4×D4—26no—
94C3×C4×D4—23no—
95C3×C4×D4—23no—
96C3×C4×D4D413yes—
97C3×C4×D4D413yes—
98C3×#117C2316yes—
99C3×#117C2313yes—
100C2×C3×C4⋊C4—23no—
101C2×C3×C4⋊C4C2316yes—
102C2×C3×C42C2313yes—
103C3×#103C2314yes—
104C3×#103C23112yes—
105C3×#102C23112yes—
106C3×#102C23124yes—
107C3×#102C23112yes—
108C3×#97C2316yes—
109C3×C4×C8C2311yescharacteristic
110C3×C4×C8C2313yes—
111C3×C4×C8C2316yes—
112C2×H(5,𝔽2)C22×C312yes—
113C2×#131C22×C312yes—
114Q82—21no—
115#557—21no—
116#554—23no—
117#550—21no—
118#550—21no—
119D4×Q8—23no—
120D42—23no—
121D42—23no—
122#544C22×C316yescomplemented
123#535C22×C316yes—
124#533C22×C313yescomplemented
125#532C22×C313yes—
126C2×C4⋊Q8C22×C313yescomplemented
127C2×#127C22×C313yes—
128#520C22×C312yes—
129C4×#41C22×C311yescharacteristic
130#490C22×C3112yescomplemented
131#489C22×C3112yescomplemented
132#488C22×C3112yescomplemented
133#487C22×C311yescharacteristic, complemented
134#487C22×C313yescomplemented
135#486C22×C313yescomplemented
136#486C22×C3112yescomplemented
137#486C22×C316yescomplemented
138#486C22×C313yescomplemented
139#418C22×C3112yescomplemented
140#417C22×C3112yescomplemented
141#416C22×C3112yescomplemented
142C2×#102C22×C3112yescomplemented
143#405C22×C316yescomplemented
144C4×M16C22×C313yescomplemented
145C2×C4×C8C22×C313yescomplemented
146C24×C3—26no—
147C3×#41—224no—
148C3×#41—48no—
149C3×#41#4118yes—
150C2×C3×Q8—21no—
151C2×C3×Q8—23no—
152C2×C3×Q8—26no—
153C2×C3×Q8—86no—
154C2×C3×Q8C2×D411yescharacteristic
155C2×C3×Q8C2412yes—
156C2×C3×D4—26no—
157C2×C3×D4—23no—
158C2×C3×D4—23no—
159C2×C3×D4—26no—
160C2×C3×D4—26no—
161C2×C3×D4—26no—
162C2×C3×D4—224no—
163C2×C3×D4—424no—
164C2×C3×D4—86no—
165C2×C3×D4—86no—
166C2×C3×D4C2×D413yes—
167C2×C3×D4C2416yes—
168C22×C3×C4—26no—
169C22×C3×C4—26no—
170C22×C3×C4C2×D413yes—
171C22×C3×C4C2×D413yes—
172C22×C3×C4C2413yes—
173C3×M16—212no—
174C2×C3×C8—212no—
175C2×C3×C8C2412yes—
176C2×C3×C8C2416yes—
177C3×C4⋊C4—23no—
178C3×C4⋊C4—26no—
179C3×C4⋊C4—26no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—26no—
184C3×C4⋊C4—26no—
185C3×C4⋊C4C2×D413yes—
186C3×C4⋊C4C2×D413yes—
187C3×C4⋊C4C2416yes—
188C3×C4⋊C4C2412yes—
189C3×C22⋊C4—26no—
190C3×C22⋊C4—26no—
191C3×C22⋊C4—26no—
192C3×C22⋊C4C2×D416yes—
193C3×C42—26no—
194C3×C42C2×D413yes—
195C3×C42C2×D413yes—
196C3×C42C2411yescharacteristic
197C3×C42C2413yes—
198H(5,𝔽2)—216no—
199C2×#41—22no—
200C2×#41C23×C316yes—
201C2×#41C23×C311yescharacteristic
202C22×Q8—23no—
203C22×D4—26no—
204C22×D4—23no—
205C22×D4—23no—
206C22×D4C23×C316yes—
207#131—28no—
208C2×M16C23×C316yes—
209C22×C8C23×C316yes—
210C4⋊Q8—26no—
211C4⋊Q8—23no—
212C4⋊Q8—23no—
213C4⋊Q8C23×C311yescharacteristic
214C4⋊Q8C23×C313yes—
215C4⋊Q8C23×C316yes—
216C4⋊Q8C23×C311yescharacteristic
217#127—23no—
218#127—23no—
219#127C23×C313yes—
220#127C23×C316yes—
221#125—26no—
222#125C23×C316yes—
223#124C23×C316yes—
224C22⋊Q8—23no—
225C22⋊Q8—23no—
226C22⋊Q8—212no—
227#121—23no—
228#121—23no—
229#121—26no—
230#121—26no—
231#120—212no—
232C4×Q8—22no—
233C4×Q8—23no—
234C4×Q8—23no—
235C4×Q8C3×D411yescharacteristic
236C4×Q8C3×D411yescharacteristic
237C4×D4—26no—
238C4×D4—23no—
239C4×D4—23no—
240C4×D4C3×D413yes—
241C4×D4C3×D413yes—
242#117C23×C316yes—
243#117C23×C313yes—
244C2×C4⋊C4—23no—
245C2×C4⋊C4C23×C316yes—
246C2×C42C23×C313yes—
247#103C23×C314yes—
248#103C23×C3112yes—
249#102C23×C3112yes—
250#102C23×C3124yes—
251#102C23×C3112yes—
252#97C23×C316yes—
253C4×C8C23×C311yescharacteristic
254C4×C8C23×C313yes—
255C4×C8C23×C316yes—
256C23×C3—26no—
257C23×C3—412no—
258C23×C3—86no—
259C23×C3—86no—
260C23×C3C22×D413yes—
261C3×Q8—24no—
262C3×Q8—46no—
263C3×Q8—42no—
264C3×Q8—86no—
265C3×Q8#12414yes—
266C3×D4—212no—
267C3×D4—412no—
268C3×D4—412no—
269C3×D4—46no—
270C3×D4—46no—
271C3×D4—86no—
272C3×D4—86no—
273C3×D4#124112yes—
274C2×C3×C4—26no—
275C2×C3×C4—26no—
276C2×C3×C4—22no—
277C2×C3×C4—26no—
278C2×C3×C4—22no—
279C2×C3×C4—412no—
280C2×C3×C4—86no—
281C2×C3×C4C22×D413yes—
282C2×C3×C4C22×D413yes—
283C2×C3×C4C22×D413yes—
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#41112yes—
287C2×C3×C4C2511yescharacteristic
288C3×C8—22no—
289C3×C8—26no—
290C24—26no—
291#41—224no—
292#41—48no—
293#41C3×#4118yes—
294C2×Q8—21no—
295C2×Q8—23no—
296C2×Q8—26no—
297C2×Q8—86no—
298C2×Q8C2×C3×D411yescharacteristic
299C2×Q8C24×C312yes—
300C2×D4—26no—
301C2×D4—23no—
302C2×D4—23no—
303C2×D4—26no—
304C2×D4—26no—
305C2×D4—26no—
306C2×D4—224no—
307C2×D4—424no—
308C2×D4—86no—
309C2×D4—86no—
310C2×D4C2×C3×D413yes—
311C2×D4C24×C316yes—
312C22×C4—26no—
313C22×C4—26no—
314C22×C4C2×C3×D413yes—
315C22×C4C2×C3×D413yes—
316C22×C4C24×C313yes—
317M16—212no—
318C2×C8—212no—
319C2×C8C24×C312yes—
320C2×C8C24×C316yes—
321C4⋊C4—23no—
322C4⋊C4—26no—
323C4⋊C4—26no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—26no—
328C4⋊C4—26no—
329C4⋊C4C2×C3×D413yes—
330C4⋊C4C2×C3×D413yes—
331C4⋊C4C24×C316yes—
332C4⋊C4C24×C312yes—
333C22⋊C4—26no—
334C22⋊C4—26no—
335C22⋊C4—26no—
336C22⋊C4C2×C3×D416yes—
337C42—26no—
338C42C2×C3×D413yes—
339C42C2×C3×D413yes—
340C42C24×C311yescharacteristic
341C42C24×C313yes—
342C22×C3—23no—
343C22×C3—23no—
344C22×C3—46no—
345C22×C3—46no—
346C22×C3—46no—
347C22×C3—1612no—
348C22×C3C2×#12416yes—
349C22×C3C23×D411yescharacteristic
350C3×C4—23no—
351C3×C4—23no—
352C3×C4—42no—
353C3×C4—46no—
354C3×C4—42no—
355C3×C4C2×#12416yes—
356C3×C4#57612yes—
357C3×C4#57712yes—
358C3×C4C22×#4112yes—
359C23—26no—
360C23—412no—
361C23—86no—
362C23—86no—
363C23C22×C3×D413yes—
364Q8—24no—
365Q8—46no—
366Q8—42no—
367Q8—86no—
368Q8C3×#12414yes—
369D4—212no—
370D4—412no—
371D4—46no—
372D4—46no—
373D4—412no—
374D4—86no—
375D4—86no—
376D4C3×#124112yes—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—22no—
380C2×C4—26no—
381C2×C4—22no—
382C2×C4—412no—
383C2×C4—86no—
384C2×C4C22×C3×D413yes—
385C2×C4C22×C3×D413yes—
386C2×C4C22×C3×D411yescharacteristic
387C2×C4C22×C3×D411yescharacteristic
388C2×C4C22×C3×D413yes—
389C2×C4C2×C3×#41112yes—
390C2×C4C25×C311yescharacteristic
391C8—22no—
392C8—26no—
393C2×C3—26no—
394C2×C3—86no—
395C2×C3C22×#12411yescharacteristic
396C2×C3C2×#57611yescharacteristic
397C2×C3C2×#57711yescharacteristic
398C22—23no—
399C22—23no—
400C22—46no—
401C22—46no—
402C22—46no—
403C22—1612no—
404C22C2×C3×#12416yes—
405C22C23×C3×D411yescharacteristic
406C4—23no—
407C4—23no—
408C4—42no—
409C4—46no—
410C4—42no—
411C4C2×C3×#12416yes—
412C4C3×#57612yes—
413C4C3×#57712yes—
414C4C22×C3×#4112yes—
415C3#6062211yescharacteristic, direct product factor, Sylow
416C2—26no—
417C2—86no—
418C2C22×C3×#12411yescharacteristic
419C2C2×C3×#57611yescharacteristic
420C2C2×C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3498C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3503C2semidirect
3. C3×#3141C2semidirect
4. C3×#3138C2semidirect
5. C3×#3136C2semidirect
6. C3×#3131C2semidirect
7. C2×C3×#486C2semidirect
8. C3×#2897C2semidirect
9. C3×C4×#131C2semidirect
10. #60622C3direct (N × H)
21. C3×#544C22semidirect
23. C3×#533C22semidirect
25. C2×C3×C4⋊Q8C22semidirect
29. C3×#490C22semidirect
30. C3×#489C22semidirect
31. C3×#488C22semidirect
32. C3×#487C22semidirect
33. C3×#487C22semidirect
34. C3×#486C22semidirect
35. C3×#486C22semidirect
36. C3×#486C22semidirect
37. C3×#486C22semidirect
38. C3×#418C22semidirect
39. C3×#417C22semidirect
40. C3×#416C22semidirect
41. C2×C3×#102C22semidirect
42. C3×#405C22semidirect
43. C3×C4×M16C22semidirect
44. C2×C3×C4×C8C22semidirect
45. #3503C2×C3semidirect
47. #3141C2×C3semidirect
48. #3138C2×C3semidirect
49. #3136C2×C3semidirect
50. #3131C2×C3semidirect
51. C2×#486C2×C3semidirect
52. #2897C2×C3semidirect
53. C4×#131C2×C3semidirect
122. #544C22×C3semidirect
124. #533C22×C3semidirect
126. C2×C4⋊Q8C22×C3semidirect
130. #490C22×C3semidirect
131. #489C22×C3semidirect
132. #488C22×C3semidirect
133. #487C22×C3semidirect
134. #487C22×C3semidirect
135. #486C22×C3semidirect
136. #486C22×C3semidirect
137. #486C22×C3semidirect
138. #486C22×C3semidirect
139. #418C22×C3semidirect
140. #417C22×C3semidirect
141. #416C22×C3semidirect
142. C2×#102C22×C3semidirect
143. #405C22×C3semidirect
144. C4×M16C22×C3semidirect
145. C2×C4×C8C22×C3semidirect
415. C3#60622direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×#34982
C3×#31412
C3×#31382
C3×#31362
C3×#31312
C2×C3×#4862
C3×#28972
C3×C4×#1312
#606223