Finite Group Databasebeta

← C3×#32971 #10651578

Subgroups — C3×#32971 #10651578

Order 768.

1294 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#3079C212yesmaximal, complemented
3C3×#2563C211yesmaximal, characteristic, complemented
4C3×#1860C214yesmaximal, complemented
5C3×#1856C212yesmaximal, complemented
6C3×#1806C212yesmaximal, complemented
7C3×#1800C211yesmaximal, characteristic, complemented
8C3×#1792C212yesmaximal
9#32971C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic, complemented
12C3×#520—22no—
13C3×#520C2212yes—
14C3×#518—22no—
15C3×#518—22no—
16C3×#518C2211yescharacteristic
17C3×C4×#41—22no—
18C3×C4×#41C2212yes—
19C2×C3×C4×D4—22no—
20C2×C3×C4×D4—22no—
21C2×C3×C4×D4C2211yescharacteristic
22C3×#477—22no—
23C3×#476—22no—
24C3×#475—22no—
25C3×#474—22no—
26C3×#428C2214yescomplemented
27C3×#423C2212yescomplemented
28C2×C3×#104C2214yes—
29C3×#417C2214yes—
30C2×C3×#100C2212yes—
31C3×#393—22no—
32C3×#390C2212yes—
33C3×#382C2212yescomplemented
34C3×#380—22no—
35C3×C4×C4⋊C4C2211yescharacteristic, complemented
36C3×#343C2212yescomplemented
37C3×#339C2214yescomplemented
38C4×H(5,𝔽2)C2×C311yescharacteristic
39#3079C2×C312yescomplemented
40#2563C2×C311yescharacteristic, complemented
41#1860C2×C314yescomplemented
42#1856C2×C312yescomplemented
43#1806C2×C312yescomplemented
44#1800C2×C311yescharacteristic, complemented
45#1792C2×C312yes—
46C3×H(5,𝔽2)—24no—
47C3×H(5,𝔽2)—22no—
48C3×H(5,𝔽2)Q814yescomplemented
49C2×C3×#41—22no—
50C2×C3×#41C2312yes—
51C22×C3×Q8—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4D412yes—
57C2×C3×M16C2314yes—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4×Q8—22no—
61C3×C4×Q8D412yes—
62C3×C4×Q8D412yes—
63C3×C4×D4—24no—
64C3×C4×D4—24no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C3×C4×D4—24no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4D412yes—
76C3×C4×D4D412yes—
77C3×#117—24no—
78C3×#117—22no—
79C3×#117—22no—
80C3×#117C2312yes—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42—22no—
93C2×C3×C42—22no—
94C2×C3×C42C2312yes—
95C2×C3×C42C2311yescharacteristic
96C3×#107—24no—
97C3×#106—24no—
98C3×C4⋊C8—22no—
99C3×C4⋊C8—22no—
100C3×#104—28no—
101C3×#103—24no—
102C3×#102—24no—
103C3×#100—24no—
104C3×#95—21no—
105C3×#95—24no—
106C2×H(5,𝔽2)C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic, complemented
108#520—22no—
109#520C22×C312yes—
110#518—22no—
111#518—22no—
112#518C22×C311yescharacteristic
113C4×#41—22no—
114C4×#41C22×C312yes—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4C22×C311yescharacteristic
118#477—22no—
119#476—22no—
120#475—22no—
121#474—22no—
122#428C22×C314yescomplemented
123#423C22×C312yescomplemented
124C2×#104C22×C314yes—
125#417C22×C314yes—
126C2×#100C22×C312yes—
127#393—22no—
128#390C22×C312yes—
129#382C22×C312yescomplemented
130#380—22no—
131C4×C4⋊C4C22×C311yescharacteristic, complemented
132#343C22×C312yescomplemented
133#339C22×C314yescomplemented
134C24×C3—22no—
135C24×C3C2×D412yes—
136C3×#41—28no—
137C3×#41—28no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41C2×Q814yes—
141C3×#41#4114yes—
142C2×C3×Q8—42no—
143C2×C3×Q8—82no—
144C2×C3×Q8C2×D412yes—
145C2×C3×D4—24no—
146C2×C3×D4—28no—
147C2×C3×D4—28no—
148C2×C3×D4—28no—
149C2×C3×D4—22no—
150C2×C3×D4—28no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4C2×D412yes—
154C2×C3×D4C2×Q812yes—
155C2×C3×D4#4112yes—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—44no—
168C22×C3×C4—84no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C3×M16—24no—
174C3×M16—24no—
175C2×C3×C8—24no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—41no—
183C3×C4⋊C4—41no—
184C3×C4⋊C4C2×D412yes—
185C3×C4⋊C4C2×D412yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—84no—
193C3×C42—24no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—24no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200H(5,𝔽2)—24no—
201H(5,𝔽2)—22no—
202H(5,𝔽2)C3×Q814yescomplemented
203C2×#41—22no—
204C2×#41C23×C312yes—
205C22×Q8—21no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4C23×C311yescharacteristic
209C23×C4—22no—
210C23×C4C3×D412yes—
211C2×M16C23×C314yes—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4×Q8—22no—
215C4×Q8C3×D412yes—
216C4×Q8C3×D412yes—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4C3×D412yes—
230C4×D4C3×D412yes—
231#117—24no—
232#117—22no—
233#117—22no—
234#117C23×C312yes—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C23×C312yes—
240C2×C4⋊C4C23×C311yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4C3×D412yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C23×C312yes—
249C2×C42C23×C311yescharacteristic
250#107—24no—
251#106—24no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254#104—28no—
255#103—24no—
256#102—24no—
257#100—24no—
258#95—21no—
259#95—24no—
260C23×C3—28no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—82no—
267C23×C3—82no—
268C23×C3C22⋊Q814yes—
269C23×C3C22×D411yescharacteristic
270C3×Q8—22no—
271C3×Q8—84no—
272C3×Q8C22⋊Q814yes—
273C3×D4—216no—
274C3×D4—24no—
275C3×D4—24no—
276C3×D4—24no—
277C3×D4—22no—
278C3×D4—24no—
279C3×D4C22⋊Q814yes—
280C2×C3×C4—28no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—44no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—44no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—82no—
298C2×C3×C4—82no—
299C2×C3×C4—82no—
300C2×C3×C4—84no—
301C2×C3×C4—82no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12011yescharacteristic
304C2×C3×C4#12011yescharacteristic
305C2×C3×C4C22×D412yes—
306C2×C3×C4C22×Q811yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C4C2×#4111yescharacteristic
309C3×C8—44no—
310C24—22no—
311C24C2×C3×D412yes—
312#41—28no—
313#41—28no—
314#41—22no—
315#41—22no—
316#41C2×C3×Q814yes—
317#41C3×#4114yes—
318C2×Q8—42no—
319C2×Q8—82no—
320C2×Q8C2×C3×D412yes—
321C2×D4—24no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—22no—
326C2×D4—28no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4C2×C3×D412yes—
330C2×D4C2×C3×Q812yes—
331C2×D4C3×#4112yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—44no—
344C22×C4—84no—
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C2×C3×D412yes—
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C24×C311yescharacteristic
349M16—24no—
350M16—24no—
351C2×C8—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4C2×C3×D412yes—
361C4⋊C4C2×C3×D412yes—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—24no—
368C22⋊C4—84no—
369C42—24no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—24no—
374C42C2×C3×D412yes—
375C42C2×C3×D412yes—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—44no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—82no—
386C22×C3—88no—
387C22×C3C2×#12011yescharacteristic
388C22×C3C2×C22⋊Q811yescharacteristic
389C22×C3#52511yescharacteristic
390C3×C4—24no—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—82no—
397C3×C4—82no—
398C3×C4#45512yes—
399C3×C4C2×C22⋊Q812yes—
400C3×C4#52512yes—
401C23—28no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—82no—
408C23—82no—
409C23C3×C22⋊Q814yes—
410C23C22×C3×D411yescharacteristic
411Q8—22no—
412Q8—84no—
413Q8C3×C22⋊Q814yes—
414D4—216no—
415D4—24no—
416D4—24no—
417D4—24no—
418D4—22no—
419D4—24no—
420D4C3×C22⋊Q814yes—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4C3×#12012yes—
444C2×C4C3×#12011yescharacteristic
445C2×C4C3×#12011yescharacteristic
446C2×C4C22×C3×D412yes—
447C2×C4C22×C3×Q811yescharacteristic
448C2×C4C2×C3×#4112yes—
449C2×C4C2×C3×#4111yescharacteristic
450C8—44no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—44no—
454C2×C3—44no—
455C2×C3#284811yescharacteristic
456C2×C3C2×#45511yescharacteristic
457C2×C3#302111yescharacteristic
458C22—21no—
459C22—21no—
460C22—24no—
461C22—24no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—82no—
468C22—88no—
469C22C2×C3×#12011yescharacteristic
470C22C2×C3×C22⋊Q811yescharacteristic
471C22C3×#52511yescharacteristic
472C4—21no—
473C4—22no—
474C4—24no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—82no—
479C4—82no—
480C4C3×#45512yes—
481C4C2×C3×C22⋊Q812yes—
482C4C3×#52512yes—
483C3#3297111yescharacteristic, direct product factor, Sylow
484C2—21no—
485C2—21no—
486C2—44no—
487C2—44no—
488C2C3×#284811yescharacteristic
489C2C2×C3×#45511yescharacteristic
490C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3079C2semidirect
3. C3×#2563C2semidirect
4. C3×#1860C2semidirect
5. C3×#1856C2semidirect
6. C3×#1806C2semidirect
7. C3×#1800C2semidirect
9. #32971C3direct (N × H)
11. C2×C3×C4⋊Q8C22semidirect
26. C3×#428C22semidirect
27. C3×#423C22semidirect
33. C3×#382C22semidirect
35. C3×C4×C4⋊C4C22semidirect
36. C3×#343C22semidirect
37. C3×#339C22semidirect
39. #3079C2×C3semidirect
40. #2563C2×C3semidirect
41. #1860C2×C3semidirect
42. #1856C2×C3semidirect
43. #1806C2×C3semidirect
44. #1800C2×C3semidirect
48. C3×H(5,𝔽2)Q8semidirect
107. C2×C4⋊Q8C22×C3semidirect
122. #428C22×C3semidirect
123. #423C22×C3semidirect
129. #382C22×C3semidirect
131. C4×C4⋊C4C22×C3semidirect
132. #343C22×C3semidirect
133. #339C22×C3semidirect
202. H(5,𝔽2)C3×Q8semidirect
483. C3#32971direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C3×#30792
C3×#25632
C3×#18602
C3×#18562
C3×#18062
C3×#18002
C3×#17922
#329713