Finite Group Databasebeta

← C3×#33754 #10652361

Subgroups — C3×#33754 #10652361

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3050C211yesmaximal, characteristic, complemented
2C2×C3×#466C211yesmaximal, characteristic, complemented
3C3×#2236C211yesmaximal, characteristic, complemented
4C3×#2234C212yesmaximal, complemented
5C3×#2233C211yesmaximal, characteristic, complemented
6C3×#2217C212yesmaximal, complemented
7C3×#2215C212yesmaximal, complemented
8C3×#2187C211yesmaximal, characteristic, complemented
9C3×#2186C212yesmaximal, complemented
10C3×#2184C211yesmaximal, characteristic, complemented
11C2×C3×C22⋊C16C211yesmaximal, characteristic, complemented
12#33754C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#135C2211yescharacteristic, complemented
14C22×C3×D8C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C3×#525—21no—
17C2×C3×#121—21no—
18C2×C3×SD32C2212yescomplemented
19C2×C3×D16C2212yescomplemented
20C22×C3×C16C2212yescomplemented
21C3×#467C2211yescharacteristic
22C3×#467C2211yescharacteristic, complemented
23C3×#466—21no—
24C3×#466—21no—
25C3×#466C2212yes—
26C3×#466C2212yes—
27C3×#466C2211yescharacteristic
28C3×#466C2211yescharacteristic, complemented
29C3×#465—21no—
30C3×#465—21no—
31C3×#427C2211yescharacteristic, complemented
32C2×C3×#107C2211yescharacteristic
33C3×#416—21no—
34C2×C3×#102—21no—
35C3×#367C2212yescomplemented
36C3×#366C2212yescomplemented
37C3×#358C2212yescomplemented
38C3×#357C2212yescomplemented
39C3×#357C2212yescomplemented
40C3×#357C2212yescomplemented
41C3×C22⋊C16C2211yescharacteristic
42C3×C22⋊C16C2211yescharacteristic
43C3×C22⋊C16C2212yescomplemented
44#3050C2×C311yescharacteristic, complemented
45C2×#466C2×C311yescharacteristic, complemented
46#2236C2×C311yescharacteristic, complemented
47#2234C2×C312yescomplemented
48#2233C2×C311yescharacteristic, complemented
49#2217C2×C312yescomplemented
50#2215C2×C312yescomplemented
51#2187C2×C311yescharacteristic, complemented
52#2186C2×C312yescomplemented
53#2184C2×C311yescharacteristic, complemented
54C2×C22⋊C16C2×C311yescharacteristic, complemented
55C2×C3×#41—21no—
56C22×C3×D4—21no—
57C22×C3×D4—41no—
58C23×C3×C4D411yescharacteristic
59C3×#135—42no—
60C2×C3×Q16C2311yescharacteristic
61C2×C3×SD16—21no—
62C2×C3×D8—21no—
63C2×C3×D8—42no—
64C2×C3×D8C2311yescharacteristic
65C2×C3×D8C2312yes—
66C22×C3×C8—42no—
67C22×C3×C8D411yescharacteristic
68C22×C3×C8D411yescharacteristic
69C22×C3×C8D411yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#123—41no—
74C3×C22⋊Q8—21no—
75C3×C22⋊Q8—21no—
76C3×#121—21no—
77C3×#121—21no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—41no—
81C3×#121—41no—
82C3×#120—41no—
83C3×C4×D4—41no—
84C3×C4×D4—41no—
85C3×#117—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C22⋊C4—41no—
88C3×SD32—42no—
89C3×D16—42no—
90C2×C3×C16—42no—
91C2×C3×C16C2312yes—
92C2×C3×C16C2312yes—
93C3×#107—21no—
94C3×#107C2311yescharacteristic
95C3×#107C2312yes—
96C3×#107C2311yescharacteristic
97C3×#106—21no—
98C3×#103—21no—
99C3×#103—21no—
100C3×#102—22no—
101C3×#102—22no—
102C3×#102—21no—
103C3×#102—21no—
104C2×#135C22×C311yescharacteristic, complemented
105C22×D8C22×C311yescharacteristic
106C23×C8C22×C311yescharacteristic
107#525—21no—
108C2×#121—21no—
109C2×SD32C22×C312yescomplemented
110C2×D16C22×C312yescomplemented
111C22×C16C22×C312yescomplemented
112#467C22×C311yescharacteristic
113#467C22×C311yescharacteristic, complemented
114#466—21no—
115#466—21no—
116#466C22×C311yescharacteristic
117#466C22×C312yes—
118#466C22×C312yes—
119#466C22×C311yescharacteristic, complemented
120#465—21no—
121#465—21no—
122#427C22×C311yescharacteristic, complemented
123C2×#107C22×C311yescharacteristic
124#416—21no—
125C2×#102—21no—
126#367C22×C312yescomplemented
127#366C22×C312yescomplemented
128#358C22×C312yescomplemented
129#357C22×C312yescomplemented
130#357C22×C312yescomplemented
131#357C22×C312yescomplemented
132C22⋊C16C22×C311yescharacteristic
133C22⋊C16C22×C311yescharacteristic
134C22⋊C16C22×C312yescomplemented
135C24×C3—41no—
136C24×C3D811yescharacteristic
137C3×#41—42no—
138C3×#41—42no—
139C2×C3×Q8—21no—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—41no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—41no—
150C2×C3×D4—84no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4D811yescharacteristic
156C22×C3×C4D811yescharacteristic
157C22×C3×C4D811yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C3×Q16—22no—
162C3×SD16—42no—
163C3×D8—22no—
164C3×D8—22no—
165C3×D8—22no—
166C3×D8—42no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—41no—
171C2×C3×C8—41no—
172C2×C3×C8—41no—
173C2×C3×C8C2×D411yescharacteristic
174C2×C3×C8C2×D411yescharacteristic
175C2×C3×C8C2×D411yescharacteristic
176C2×C3×C8C2411yescharacteristic
177C3×C4⋊C4—22no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—41no—
182C3×C4⋊C4—41no—
183C3×C22⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—41no—
190C3×C42—41no—
191C3×C16—22no—
192C3×C16—22no—
193C2×#41—21no—
194C22×D4—21no—
195C22×D4—41no—
196C23×C4C3×D411yescharacteristic
197#135—42no—
198C2×Q16C23×C311yescharacteristic
199C2×SD16—21no—
200C2×D8—21no—
201C2×D8—42no—
202C2×D8C23×C312yes—
203C2×D8C23×C311yescharacteristic
204C22×C8—42no—
205C22×C8C3×D411yescharacteristic
206C22×C8C3×D411yescharacteristic
207C22×C8C3×D411yescharacteristic
208C22×C8C23×C311yescharacteristic
209C22×C8C23×C311yescharacteristic
210C22×C8C23×C311yescharacteristic
211#123—41no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214#121—22no—
215#121—22no—
216#121—21no—
217#121—21no—
218#121—41no—
219#121—41no—
220#120—41no—
221C4×D4—41no—
222C4×D4—41no—
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C22⋊C4—41no—
226SD32—42no—
227D16—42no—
228C2×C16—42no—
229C2×C16C23×C312yes—
230C2×C16C23×C312yes—
231#107—21no—
232#107C23×C311yescharacteristic
233#107C23×C311yescharacteristic
234#107C23×C312yes—
235#106—21no—
236#103—21no—
237#103—21no—
238#102—22no—
239#102—22no—
240#102—21no—
241#102—21no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—82no—
248C23×C3—82no—
249C23×C3—82no—
250C23×C3C2×D811yescharacteristic
251C23×C3C2×D811yescharacteristic
252C23×C3C2×D811yescharacteristic
253C3×Q8—41no—
254C3×Q8—41no—
255C3×D4—42no—
256C3×D4—41no—
257C3×D4—42no—
258C3×D4—41no—
259C3×D4—82no—
260C3×D4—84no—
261C3×D4—82no—
262C3×D4—82no—
263C3×D4—82no—
264C3×D4—84no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—82no—
279C2×C3×C4—82no—
280C2×C3×C4C2×D811yescharacteristic
281C2×C3×C4C2×D811yescharacteristic
282C2×C3×C4C2×D811yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C3×C8—21no—
285C3×C8—21no—
286C3×C8—21no—
287C3×C8H(5,𝔽2)12yes—
288C24—41no—
289C24C3×D811yescharacteristic
290#41—42no—
291#41—42no—
292C2×Q8—21no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—84no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4C3×D811yescharacteristic
309C22×C4C3×D811yescharacteristic
310C22×C4C3×D811yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314Q16—22no—
315SD16—42no—
316D8—22no—
317D8—22no—
318D8—22no—
319D8—42no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8C2×C3×D411yescharacteristic
327C2×C8C2×C3×D411yescharacteristic
328C2×C8C2×C3×D411yescharacteristic
329C2×C8C24×C311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—41no—
344C16—22no—
345C16—22no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—44no—
354C22×C3—81no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—82no—
360C22×C3—164no—
361C22×C3#50812yes—
362C22×C3#50912yes—
363C22×C3C22×D811yescharacteristic
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—81no—
368C3×C4—82no—
369C3×C4—81no—
370C3×C4—81no—
371C3×C4#53411yescharacteristic
372C3×C4#57511yescharacteristic
373C23—22no—
374C23—22no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23C2×C3×D811yescharacteristic
382C23C2×C3×D811yescharacteristic
383C23C2×C3×D811yescharacteristic
384Q8—41no—
385Q8—41no—
386D4—42no—
387D4—41no—
388D4—42no—
389D4—41no—
390D4—82no—
391D4—84no—
392D4—82no—
393D4—82no—
394D4—84no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4C2×C3×D811yescharacteristic
412C2×C4C2×C3×D811yescharacteristic
413C2×C4C2×C3×D811yescharacteristic
414C2×C4C22×C3×D411yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8C3×H(5,𝔽2)12yes—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—41no—
422C2×C3—161no—
423C2×C3—162no—
424C2×C3#318611yescharacteristic
425C2×C3C2×#50811yescharacteristic
426C2×C3C2×#50911yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—44no—
435C22—81no—
436C22—82no—
437C22—82no—
438C22—81no—
439C22—81no—
440C22—82no—
441C22—164no—
442C22C3×#50812yes—
443C22C3×#50912yes—
444C22C22×C3×D811yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—82no—
450C4—81no—
451C4—81no—
452C4C3×#53411yescharacteristic
453C4C3×#57511yescharacteristic
454C3#3375411yescharacteristic, direct product factor, Sylow
455C2—22no—
456C2—22no—
457C2—41no—
458C2—162no—
459C2—161no—
460C2C3×#318611yescharacteristic
461C2C2×C3×#50811yescharacteristic
462C2C2×C3×#50911yescharacteristic
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3050C2semidirect
2. C2×C3×#466C2semidirect
3. C3×#2236C2semidirect
4. C3×#2234C2semidirect
5. C3×#2233C2semidirect
6. C3×#2217C2semidirect
7. C3×#2215C2semidirect
8. C3×#2187C2semidirect
9. C3×#2186C2semidirect
10. C3×#2184C2semidirect
11. C2×C3×C22⋊C16C2semidirect
12. #33754C3direct (N × H)
13. C2×C3×#135C22semidirect
18. C2×C3×SD32C22semidirect
19. C2×C3×D16C22semidirect
20. C22×C3×C16C22semidirect
22. C3×#467C22semidirect
28. C3×#466C22semidirect
31. C3×#427C22semidirect
35. C3×#367C22semidirect
36. C3×#366C22semidirect
37. C3×#358C22semidirect
38. C3×#357C22semidirect
39. C3×#357C22semidirect
40. C3×#357C22semidirect
43. C3×C22⋊C16C22semidirect
44. #3050C2×C3semidirect
45. C2×#466C2×C3semidirect
46. #2236C2×C3semidirect
47. #2234C2×C3semidirect
48. #2233C2×C3semidirect
49. #2217C2×C3semidirect
50. #2215C2×C3semidirect
51. #2187C2×C3semidirect
52. #2186C2×C3semidirect
53. #2184C2×C3semidirect
54. C2×C22⋊C16C2×C3semidirect
104. C2×#135C22×C3semidirect
109. C2×SD32C22×C3semidirect
110. C2×D16C22×C3semidirect
111. C22×C16C22×C3semidirect
113. #467C22×C3semidirect
119. #466C22×C3semidirect
122. #427C22×C3semidirect
126. #367C22×C3semidirect
127. #366C22×C3semidirect
128. #358C22×C3semidirect
129. #357C22×C3semidirect
130. #357C22×C3semidirect
131. #357C22×C3semidirect
134. C22⋊C16C22×C3semidirect
454. C3#33754direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30502
C2×C3×#4662
C3×#22362
C3×#22342
C3×#22332
C3×#22172
C3×#22152
C3×#21872
C3×#21862
C3×#21842
C2×C3×C22⋊C162
#337543