Finite Group Databasebeta

← C3×#29703 #10648310

Subgroups — C3×#29703 #10648310

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2C3×#3079C212yesmaximal, complemented
3C3×#2542C211yesmaximal, characteristic, complemented
4C3×#2060C212yesmaximal, complemented
5C3×#2039C212yesmaximal, complemented
6C3×#2037C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9#29703C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#544C2211yescharacteristic
11C3×#542—22no—
12C3×#540—22no—
13C3×#538—22no—
14C3×#537C2211yescharacteristic
15C3×#535C2212yes—
16C3×#532—22no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C3×#529—22no—
19C3×#525—22no—
20C2×C3×C22⋊Q8—22no—
21C3×C4×#41C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C3×#477—22no—
24C3×#476—22no—
25C3×#475—22no—
26C3×#474—22no—
27C3×#428C2214yes—
28C3×#423C2212yes—
29C2×C3×#104C2214yes—
30C3×#417C2214yes—
31C2×C3×#100C2212yes—
32C3×#399—22no—
33C3×#390C2212yes—
34C3×#384C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3460C2×C311yescharacteristic
39#3079C2×C312yescomplemented
40#2542C2×C311yescharacteristic, complemented
41#2060C2×C312yescomplemented
42#2039C2×C312yescomplemented
43#2037C2×C314yescomplemented
44#1806C2×C312yescomplemented
45#1800C2×C311yescharacteristic, complemented
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C22×C3×Q8—21no—
49C22×C3×D4C2311yescharacteristic
50C23×C3×C4D412yes—
51C2×C3×M16C2314yes—
52C3×C4⋊Q8—22no—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—22no—
55C3×#127—22no—
56C3×#126—24no—
57C3×#125—22no—
58C3×#124—24no—
59C3×#124—22no—
60C3×#123—24no—
61C3×#123—24no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—48no—
66C3×#121—22no—
67C3×#121—24no—
68C3×#121—22no—
69C3×#121—22no—
70C3×#121—22no—
71C3×#121—22no—
72C3×#120—24no—
73C3×C4×Q8D412yes—
74C3×C4×Q8D412yes—
75C3×C4×D4—24no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4D412yes—
81C3×C4×D4D412yes—
82C3×#117—22no—
83C3×#117—22no—
84C3×#117C2312yes—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42—22no—
93C2×C3×C42C2312yes—
94C2×C3×C42C2311yescharacteristic
95C3×#107—24no—
96C3×#106—24no—
97C3×C4⋊C8—22no—
98C3×C4⋊C8—22no—
99C3×#104—44no—
100C3×#103—24no—
101C3×#102—24no—
102C3×#100—42no—
103C3×#95—21no—
104#544C22×C311yescharacteristic
105#542—22no—
106#540—22no—
107#538—22no—
108#537C22×C311yescharacteristic
109#535C22×C312yes—
110#532—22no—
111C2×C4⋊Q8C22×C311yescharacteristic
112#529—22no—
113#525—22no—
114C2×C22⋊Q8—22no—
115C4×#41C22×C312yes—
116C2×C4×D4C22×C311yescharacteristic
117#477—22no—
118#476—22no—
119#475—22no—
120#474—22no—
121#428C22×C314yes—
122#423C22×C312yes—
123C2×#104C22×C314yes—
124#417C22×C314yes—
125C2×#100C22×C312yes—
126#399—22no—
127#390C22×C312yes—
128#384C22×C312yes—
129C4×C4⋊C4C22×C311yescharacteristic
130#343C22×C312yes—
131#339C22×C314yes—
132C24×C3C2×D412yes—
133C3×#41—22no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—84no—
138C2×C3×Q8—22no—
139C2×C3×Q8—44no—
140C2×C3×Q8—41no—
141C2×C3×Q8—41no—
142C2×C3×Q8C2×D412yes—
143C2×C3×D4—22no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4C2×D412yes—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—44no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C3×M16—24no—
165C3×M16—24no—
166C2×C3×C8—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—48no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4C2×D412yes—
178C3×C4⋊C4C2×D412yes—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—48no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194C2×#41—22no—
195C2×#41C23×C312yes—
196C22×Q8—21no—
197C22×D4C23×C311yescharacteristic
198C23×C4C3×D412yes—
199C2×M16C23×C314yes—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203#127—22no—
204#126—24no—
205#125—22no—
206#124—24no—
207#124—22no—
208#123—24no—
209#123—24no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—48no—
214#121—22no—
215#121—24no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#120—24no—
221C4×Q8C3×D412yes—
222C4×Q8C3×D412yes—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4C3×D412yes—
229C4×D4C3×D412yes—
230#117—22no—
231#117—22no—
232#117C23×C312yes—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C23×C312yes—
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C22⋊C4—24no—
239C2×C22⋊C4C3×D412yes—
240C2×C42—22no—
241C2×C42C23×C312yes—
242C2×C42C23×C311yescharacteristic
243#107—24no—
244#106—24no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247#104—44no—
248#103—24no—
249#102—24no—
250#100—42no—
251#95—21no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3C22×D411yescharacteristic
259C3×Q8—22no—
260C3×Q8—22no—
261C3×Q8—82no—
262C3×Q8—82no—
263C3×Q8—82no—
264C3×D4—22no—
265C3×D4—22no—
266C3×D4—42no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—42no—
270C3×D4—84no—
271C3×D4—82no—
272C3×D4—84no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
286C2×C3×C4—42no—
287C2×C3×C4—44no—
288C2×C3×C4—42no—
289C2×C3×C4—82no—
290C2×C3×C4—82no—
291C2×C3×C4#12012yes—
292C2×C3×C4#12011yescharacteristic
293C2×C3×C4#12011yescharacteristic
294C2×C3×C4C22×D412yes—
295C2×C3×C4H(5,𝔽2)11yescharacteristic
296C2×C3×C42−1+411yescharacteristic
297C2×C3×C42−1+412yes—
298C3×C8—44no—
299C24C2×C3×D412yes—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—22no—
304#41—84no—
305C2×Q8—22no—
306C2×Q8—44no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8C2×C3×D412yes—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—22no—
318C2×D4C2×C3×D412yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—44no—
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D412yes—
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C24×C311yescharacteristic
331M16—24no—
332M16—24no—
333C2×C8—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4C2×C3×D412yes—
345C4⋊C4C2×C3×D412yes—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—24no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—48no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42C2×C3×D412yes—
360C42C2×C3×D412yes—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—44no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—82no—
371C22×C3C2×#12011yescharacteristic
372C22×C3#53611yescharacteristic
373C22×C3#53711yescharacteristic
374C3×C4—21no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—82no—
382C3×C4#45312yes—
383C3×C4#53612yes—
384C3×C4#53712yes—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—42no—
390C23—42no—
391C23C22×C3×D411yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—82no—
395Q8—82no—
396Q8—82no—
397D4—22no—
398D4—22no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—82no—
404D4—84no—
405D4—84no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4C3×#12012yes—
425C2×C4C3×#12011yescharacteristic
426C2×C4C3×#12011yescharacteristic
427C2×C4C22×C3×D412yes—
428C2×C4C3×H(5,𝔽2)11yescharacteristic
429C2×C4C3×2−1+411yescharacteristic
430C2×C4C3×2−1+412yes—
431C8—44no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—82no—
435C2×C3—82no—
436C2×C3#261611yescharacteristic
437C2×C3C2×#45311yescharacteristic
438C2×C3#302211yescharacteristic
439C22—21no—
440C22—21no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—82no—
449C22C2×C3×#12011yescharacteristic
450C22C3×#53611yescharacteristic
451C22C3×#53711yescharacteristic
452C4—21no—
453C4—22no—
454C4—44no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—82no—
460C4C3×#45312yes—
461C4C3×#53612yes—
462C4C3×#53712yes—
463C3#2970311yescharacteristic, direct product factor, Sylow
464C2—21no—
465C2—21no—
466C2—82no—
467C2—82no—
468C2C3×#261611yescharacteristic
469C2C2×C3×#45311yescharacteristic
470C2C3×#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×#3079C2semidirect
3. C3×#2542C2semidirect
4. C3×#2060C2semidirect
5. C3×#2039C2semidirect
6. C3×#2037C2semidirect
7. C3×#1806C2semidirect
8. C3×#1800C2semidirect
9. #29703C3direct (N × H)
39. #3079C2×C3semidirect
40. #2542C2×C3semidirect
41. #2060C2×C3semidirect
42. #2039C2×C3semidirect
43. #2037C2×C3semidirect
44. #1806C2×C3semidirect
45. #1800C2×C3semidirect
463. C3#29703direct (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×#30792
C3×#25422
C3×#20602
C3×#20392
C3×#20372
C3×#18062
C3×#18002
#297033