Finite Group Databasebeta

← C3×#29705 #10648312

Subgroups — C3×#29705 #10648312

Order 768.

1018 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2C3×#3079C212yesmaximal, complemented
3C3×#2563C211yesmaximal, characteristic, complemented
4C3×#2061C212yesmaximal, complemented
5C3×#2040C212yesmaximal, complemented
6C3×#2037C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1800C211yesmaximal, characteristic, complemented
9#29705C311yesmaximal, 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—
20C3×C4×#41C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C3×#477—22no—
23C3×#476—22no—
24C3×#475—22no—
25C3×#474—22no—
26C3×#428C2214yes—
27C3×#423C2212yes—
28C2×C3×#104C2214yes—
29C3×#417C2214yes—
30C2×C3×#100C2212yes—
31C3×#393—22no—
32C3×#390C2212yes—
33C3×#382C2212yes—
34C3×#380—22no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3460C2×C311yescharacteristic
39#3079C2×C312yescomplemented
40#2563C2×C311yescharacteristic, complemented
41#2061C2×C312yescomplemented
42#2040C2×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×#121—22no—
66C3×#121—24no—
67C3×#121—22no—
68C3×#121—22no—
69C3×#121—22no—
70C3×#121—22no—
71C3×#120—24no—
72C3×C4×Q8D412yes—
73C3×C4×Q8D412yes—
74C3×C4×D4—24no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4D412yes—
80C3×C4×D4D412yes—
81C3×#117—22no—
82C3×#117—22no—
83C3×#117C2312yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—24no—
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42—22no—
91C2×C3×C42C2312yes—
92C2×C3×C42C2311yescharacteristic
93C3×#107—24no—
94C3×#106—24no—
95C3×C4⋊C8—22no—
96C3×C4⋊C8—22no—
97C3×#104—44no—
98C3×#103—24no—
99C3×#102—24no—
100C3×#100—42no—
101C3×#95—21no—
102C3×#95—24no—
103#544C22×C311yescharacteristic
104#542—22no—
105#540—22no—
106#538—22no—
107#537C22×C311yescharacteristic
108#535C22×C312yes—
109#532—22no—
110C2×C4⋊Q8C22×C311yescharacteristic
111#529—22no—
112#525—22no—
113C4×#41C22×C312yes—
114C2×C4×D4C22×C311yescharacteristic
115#477—22no—
116#476—22no—
117#475—22no—
118#474—22no—
119#428C22×C314yes—
120#423C22×C312yes—
121C2×#104C22×C314yes—
122#417C22×C314yes—
123C2×#100C22×C312yes—
124#393—22no—
125#390C22×C312yes—
126#382C22×C312yes—
127#380—22no—
128C4×C4⋊C4C22×C311yescharacteristic
129#343C22×C312yes—
130#339C22×C314yes—
131C24×C3C2×D412yes—
132C3×#41—22no—
133C3×#41—22no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—84no—
137C2×C3×Q8—22no—
138C2×C3×Q8—42no—
139C2×C3×Q8—82no—
140C2×C3×Q8C2×D412yes—
141C2×C3×D4—22no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—22no—
149C2×C3×D4C2×D412yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—44no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C3×M16—24no—
163C3×M16—24no—
164C2×C3×C8—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4C2×D412yes—
176C3×C4⋊C4C2×D412yes—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—84no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192C2×#41—22no—
193C2×#41C23×C312yes—
194C22×Q8—21no—
195C22×D4C23×C311yescharacteristic
196C23×C4C3×D412yes—
197C2×M16C23×C314yes—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201#127—22no—
202#126—24no—
203#125—22no—
204#124—24no—
205#124—22no—
206#123—24no—
207#123—24no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211#121—22no—
212#121—24no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—22no—
217#120—24no—
218C4×Q8C3×D412yes—
219C4×Q8C3×D412yes—
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4C3×D412yes—
226C4×D4C3×D412yes—
227#117—22no—
228#117—22no—
229#117C23×C312yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C23×C312yes—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C22⋊C4—24no—
235C2×C22⋊C4C3×D412yes—
236C2×C42—22no—
237C2×C42C23×C312yes—
238C2×C42C23×C311yescharacteristic
239#107—24no—
240#106—24no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243#104—44no—
244#103—24no—
245#102—24no—
246#100—42no—
247#95—21no—
248#95—24no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3C22×D411yescharacteristic
256C3×Q8—22no—
257C3×Q8—22no—
258C3×Q8—84no—
259C3×Q8—82no—
260C3×D4—22no—
261C3×D4—22no—
262C3×D4—42no—
263C3×D4—42no—
264C3×D4—42no—
265C3×D4—42no—
266C3×D4—84no—
267C3×D4—82no—
268C3×D4—84no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—82no—
286C2×C3×C4—82no—
287C2×C3×C4#12012yes—
288C2×C3×C4#12011yescharacteristic
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4C22×D412yes—
291C2×C3×C4H(5,𝔽2)11yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×C3×C42−1+412yes—
294C3×C8—44no—
295C24C2×C3×D412yes—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—84no—
301C2×Q8—22no—
302C2×Q8—42no—
303C2×Q8—82no—
304C2×Q8C2×C3×D412yes—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4C2×C3×D412yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—44no—
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D412yes—
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C24×C311yescharacteristic
326M16—24no—
327M16—24no—
328C2×C8—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4C2×C3×D412yes—
340C4⋊C4C2×C3×D412yes—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—22no—
345C22⋊C4—24no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—84no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42C2×C3×D412yes—
355C42C2×C3×D412yes—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—44no—
359C22×C3—42no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—82no—
366C22×C3C2×#12011yescharacteristic
367C22×C3#53611yescharacteristic
368C22×C3#53711yescharacteristic
369C3×C4—21no—
370C3×C4—22no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—42no—
375C3×C4—82no—
376C3×C4—82no—
377C3×C4#45512yes—
378C3×C4#53612yes—
379C3×C4#53712yes—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—42no—
385C23—42no—
386C23C22×C3×D411yescharacteristic
387Q8—22no—
388Q8—22no—
389Q8—84no—
390Q8—82no—
391D4—22no—
392D4—22no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—82no—
398D4—84no—
399D4—84no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4C3×#12012yes—
419C2×C4C3×#12011yescharacteristic
420C2×C4C3×#12011yescharacteristic
421C2×C4C22×C3×D412yes—
422C2×C4C3×H(5,𝔽2)11yescharacteristic
423C2×C4C3×2−1+411yescharacteristic
424C2×C4C3×2−1+412yes—
425C8—44no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—82no—
429C2×C3—82no—
430C2×C3#261611yescharacteristic
431C2×C3C2×#45511yescharacteristic
432C2×C3#302111yescharacteristic
433C22—21no—
434C22—21no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—82no—
443C22C2×C3×#12011yescharacteristic
444C22C3×#53611yescharacteristic
445C22C3×#53711yescharacteristic
446C4—21no—
447C4—22no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—82no—
453C4—82no—
454C4C3×#45512yes—
455C4C3×#53612yes—
456C4C3×#53712yes—
457C3#2970511yescharacteristic, direct product factor, Sylow
458C2—21no—
459C2—21no—
460C2—82no—
461C2—82no—
462C2C3×#261611yescharacteristic
463C2C2×C3×#45511yescharacteristic
464C2C3×#302111yescharacteristic
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×#2563C2semidirect
4. C3×#2061C2semidirect
5. C3×#2040C2semidirect
6. C3×#2037C2semidirect
7. C3×#1806C2semidirect
8. C3×#1800C2semidirect
9. #29705C3direct (N × H)
39. #3079C2×C3semidirect
40. #2563C2×C3semidirect
41. #2061C2×C3semidirect
42. #2040C2×C3semidirect
43. #2037C2×C3semidirect
44. #1806C2×C3semidirect
45. #1800C2×C3semidirect
457. C3#29705direct (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×#25632
C3×#20612
C3×#20402
C3×#20372
C3×#18062
C3×#18002
#297053