Finite Group Databasebeta

← C3×#29680 #10648287

Subgroups — C3×#29680 #10648287

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2C3×#3039C212yesmaximal, complemented
3C3×#2547C211yesmaximal, characteristic, complemented
4C3×#2028C212yesmaximal, complemented
5C3×#2007C212yesmaximal, complemented
6C3×#2004C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9#29680C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×#137C2214yes—
11C3×#544C2211yescharacteristic
12C3×#542—22no—
13C3×#540—22no—
14C3×#538—22no—
15C3×#537C2211yescharacteristic
16C3×#535C2212yes—
17C3×#532—22no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C3×#529—22no—
20C2×C3×#124C2212yes—
21C3×#525—22no—
22C3×C4×#41C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C3×#462—22no—
25C3×#462—22no—
26C3×#461—22no—
27C3×#461—22no—
28C3×#423C2212yes—
29C2×C3×#104C2214yes—
30C3×#419C2214yes—
31C2×C3×#100C2212yes—
32C3×#397—22no—
33C3×#391C2212yes—
34C3×#380—22no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3460C2×C311yescharacteristic
39#3039C2×C312yescomplemented
40#2547C2×C311yescharacteristic, complemented
41#2028C2×C312yescomplemented
42#2007C2×C312yescomplemented
43#2004C2×C314yescomplemented
44#1806C2×C312yescomplemented
45#1800C2×C311yescharacteristic, complemented
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C22×C3×Q8C2312yes—
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4D412yes—
51C3×#137—44no—
52C3×#137—44no—
53C2×C3×Q16—24no—
54C2×C3×SD16—24no—
55C2×C3×M16C2314yes—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—22no—
59C3×#127—22no—
60C3×#126—24no—
61C3×#125—22no—
62C3×#124—24no—
63C3×#124—22no—
64C3×#124—44no—
65C3×#123—24no—
66C3×#123—24no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×#121—22no—
71C3×#121—24no—
72C3×#121—22no—
73C3×#121—22no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#120—24no—
77C3×C4×Q8D412yes—
78C3×C4×Q8D412yes—
79C3×C4×D4—24no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4D412yes—
85C3×C4×D4D412yes—
86C3×#117—22no—
87C3×#117—22no—
88C3×#117C2312yes—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—24no—
94C2×C3×C22⋊C4D412yes—
95C2×C3×C42—22no—
96C2×C3×C42C2312yes—
97C2×C3×C42C2311yescharacteristic
98C3×C4⋊C8—22no—
99C3×C4⋊C8—22no—
100C3×#104—44no—
101C3×#103—24no—
102C3×#103—24no—
103C3×#100—42no—
104C3×#95—21no—
105C3×#95—24no—
106C2×#137C22×C314yes—
107#544C22×C311yescharacteristic
108#542—22no—
109#540—22no—
110#538—22no—
111#537C22×C311yescharacteristic
112#535C22×C312yes—
113#532—22no—
114C2×C4⋊Q8C22×C311yescharacteristic
115#529—22no—
116C2×#124C22×C312yes—
117#525—22no—
118C4×#41C22×C312yes—
119C2×C4×D4C22×C311yescharacteristic
120#462—22no—
121#462—22no—
122#461—22no—
123#461—22no—
124#423C22×C312yes—
125C2×#104C22×C314yes—
126#419C22×C314yes—
127C2×#100C22×C312yes—
128#397—22no—
129#391C22×C312yes—
130#380—22no—
131C4×C4⋊C4C22×C311yescharacteristic
132#343C22×C312yes—
133#339C22×C314yes—
134C24×C3C2×D412yes—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—84no—
140C2×C3×Q8—22no—
141C2×C3×Q8—24no—
142C2×C3×Q8—24no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8C2×D412yes—
146C2×C3×D4—22no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C2×C3×D4C2×D412yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—44no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C3×Q16—48no—
168C3×SD16—48no—
169C3×M16—24no—
170C3×M16—24no—
171C2×C3×C8—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—41no—
182C3×C4⋊C4C2×D412yes—
183C3×C4⋊C4C2×D412yes—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—48no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—24no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198C2×#41—22no—
199C2×#41C23×C312yes—
200C22×Q8C23×C312yes—
201C22×D4C23×C311yescharacteristic
202C23×C4C3×D412yes—
203#137—44no—
204#137—44no—
205C2×Q16—24no—
206C2×SD16—24no—
207C2×M16C23×C314yes—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211#127—22no—
212#126—24no—
213#125—22no—
214#124—24no—
215#124—22no—
216#124—44no—
217#123—24no—
218#123—24no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222#121—22no—
223#121—24no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—22no—
228#120—24no—
229C4×Q8C3×D412yes—
230C4×Q8C3×D412yes—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4C3×D412yes—
237C4×D4C3×D412yes—
238#117—22no—
239#117—22no—
240#117C23×C312yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C23×C311yescharacteristic
245C2×C22⋊C4—24no—
246C2×C22⋊C4C3×D412yes—
247C2×C42—22no—
248C2×C42C23×C312yes—
249C2×C42C23×C311yescharacteristic
250C4⋊C8—22no—
251C4⋊C8—22no—
252#104—44no—
253#103—24no—
254#103—24no—
255#100—42no—
256#95—21no—
257#95—24no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3C22×D411yescharacteristic
265C3×Q8—22no—
266C3×Q8—22no—
267C3×Q8—44no—
268C3×Q8—44no—
269C3×Q8—82no—
270C3×D4—22no—
271C3×D4—22no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—84no—
277C3×D4—82no—
278C3×D4—84no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—44no—
288C2×C3×C4—44no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—82no—
297C2×C3×C4—82no—
298C2×C3×C4#12012yes—
299C2×C3×C4#12011yescharacteristic
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4C22×D412yes—
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×C3×C4H(5,𝔽2)12yes—
304C2×C3×C42−1+411yescharacteristic
305C3×C8—44no—
306C24C2×C3×D412yes—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—84no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8C2×C3×D412yes—
318C2×D4—22no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—22no—
325C2×D4—21no—
326C2×D4C2×C3×D412yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—44no—
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C24×C311yescharacteristic
339Q16—48no—
340SD16—48no—
341M16—24no—
342M16—24no—
343C2×C8—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4C2×C3×D412yes—
355C4⋊C4C2×C3×D412yes—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—22no—
360C22⋊C4—24no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—48no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—24no—
368C42C2×C3×D412yes—
369C42C2×C3×D412yes—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—44no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—82no—
380C22×C3C2×#12011yescharacteristic
381C22×C3#53511yescharacteristic
382C22×C3#53711yescharacteristic
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—42no—
386C3×C4—42no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4—82no—
390C3×C4—82no—
391C3×C4#45512yes—
392C3×C4#53512yes—
393C3×C4#53712yes—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23C22×C3×D411yescharacteristic
401Q8—22no—
402Q8—22no—
403Q8—44no—
404Q8—44no—
405Q8—82no—
406D4—22no—
407D4—22no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—82no—
413D4—84no—
414D4—84no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4C3×#12012yes—
435C2×C4C3×#12011yescharacteristic
436C2×C4C3×#12011yescharacteristic
437C2×C4C22×C3×D412yes—
438C2×C4C3×H(5,𝔽2)11yescharacteristic
439C2×C4C3×H(5,𝔽2)12yes—
440C2×C4C3×2−1+411yescharacteristic
441C8—44no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—82no—
445C2×C3—82no—
446C2×C3#261511yescharacteristic
447C2×C3C2×#45511yescharacteristic
448C2×C3#302211yescharacteristic
449C22—21no—
450C22—21no—
451C22—44no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22C2×C3×#12011yescharacteristic
460C22C3×#53511yescharacteristic
461C22C3×#53711yescharacteristic
462C4—21no—
463C4—22no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—82no—
469C4—82no—
470C4C3×#45512yes—
471C4C3×#53512yes—
472C4C3×#53712yes—
473C3#2968011yescharacteristic, direct product factor, Sylow
474C2—21no—
475C2—21no—
476C2—82no—
477C2—82no—
478C2C3×#261511yescharacteristic
479C2C2×C3×#45511yescharacteristic
480C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3460C2
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×#3039C2semidirect
3. C3×#2547C2semidirect
4. C3×#2028C2semidirect
5. C3×#2007C2semidirect
6. C3×#2004C2semidirect
7. C3×#1806C2semidirect
8. C3×#1800C2semidirect
9. #29680C3direct (N × H)
39. #3039C2×C3semidirect
40. #2547C2×C3semidirect
41. #2028C2×C3semidirect
42. #2007C2×C3semidirect
43. #2004C2×C3semidirect
44. #1806C2×C3semidirect
45. #1800C2×C3semidirect
473. C3#29680direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×#30392
C3×#25472
C3×#20282
C3×#20072
C3×#20042
C3×#18062
C3×#18002
#296803