Finite Group Databasebeta

← C3×#25619 #10644226

Subgroups — C3×#25619 #10644226

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3512C211yesmaximal, characteristic
2C3×#3152C211yesmaximal, characteristic, complemented
3C3×#3039C211yesmaximal, characteristic, complemented
4C3×#2642C211yesmaximal, characteristic, complemented
5C3×#2007C214yesmaximal, complemented
6C3×#1998C211yesmaximal, characteristic, complemented
7C3×#1963C211yesmaximal, characteristic, complemented
8C3×#1882C214yesmaximal, complemented
9C3×#1759C211yesmaximal, characteristic, complemented
10#25619C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137C2214yes—
12C2×C3×#137C2212yes—
13C3×#553—22no—
14C3×#553—24no—
15C3×#552—22no—
16C3×#541—21no—
17C3×#540—21no—
18C3×#537C2212yes—
19C3×#532C2211yescharacteristic
20C2×C3×C4⋊Q8C2211yescharacteristic
21C3×#529—22no—
22C2×C3×#125C2211yescharacteristic
23C3×#520C2212yes—
24C3×C4×#41C2211yescharacteristic
25C3×#497—22no—
26C3×#494—21no—
27C3×#492—21no—
28C3×#462—22no—
29C3×#461—22no—
30C3×#423C2211yescharacteristic
31C2×C3×#104C2212yes—
32C2×C3×#104C2214yes—
33C3×#419C2212yes—
34C3×C4×M16C2211yescharacteristic
35C3×#400—24no—
36C3×#391C2214yes—
37C3×#376C2211yescharacteristic
38C3×#343C2214yes—
39C3×#339C2212yes—
40#3512C2×C311yescharacteristic
41#3152C2×C311yescharacteristic, complemented
42#3039C2×C311yescharacteristic, complemented
43#2642C2×C311yescharacteristic, complemented
44#2007C2×C314yescomplemented
45#1998C2×C311yescharacteristic, complemented
46#1963C2×C311yescharacteristic, complemented
47#1882C2×C314yescomplemented
48#1759C2×C311yescharacteristic, complemented
49C2×C3×#41C2311yescharacteristic
50C2×C3×#41C2312yes—
51C22×C3×Q8C2312yes—
52C3×#137—28no—
53C3×#137—44no—
54C3×#137—44no—
55C2×C3×Q16—22no—
56C2×C3×Q16—24no—
57C2×C3×SD16—22no—
58C2×C3×SD16—24no—
59C2×C3×M16C2312yes—
60C2×C3×M16C2312yes—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8D411yescharacteristic
65C3×#127D411yescharacteristic
66C3×#126—24no—
67C3×#126—24no—
68C3×#125—24no—
69C3×#125—21no—
70C3×#125—21no—
71C3×#125—22no—
72C3×#124D412yes—
73C3×#123—24no—
74C3×#123—24no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×#121—24no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#121—24no—
82C3×C4×Q8—22no—
83C3×C4×Q8D412yes—
84C3×C4×D4—24no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—24no—
88C3×C4×D4—22no—
89C3×C4×D4D412yes—
90C3×#117—24no—
91C3×#117—22no—
92C3×#117—21no—
93C3×#117C2311yescharacteristic
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4D414yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C42—21no—
99C2×C3×C42C2312yes—
100C2×C3×C42C2311yescharacteristic
101C3×C4⋊C8—22no—
102C3×#104—24no—
103C3×#104—44no—
104C3×#103—24no—
105C3×#97—21no—
106C3×C4×C8—21no—
107C3×#95—22no—
108C3×#95—28no—
109C2×#137C22×C314yes—
110C2×#137C22×C312yes—
111#553—22no—
112#553—24no—
113#552—22no—
114#541—21no—
115#540—21no—
116#537C22×C312yes—
117#532C22×C311yescharacteristic
118C2×C4⋊Q8C22×C311yescharacteristic
119#529—22no—
120C2×#125C22×C311yescharacteristic
121#520C22×C312yes—
122C4×#41C22×C311yescharacteristic
123#497—22no—
124#494—21no—
125#492—21no—
126#462—22no—
127#461—22no—
128#423C22×C311yescharacteristic
129C2×#104C22×C312yes—
130C2×#104C22×C314yes—
131#419C22×C312yes—
132C4×M16C22×C311yescharacteristic
133#400—24no—
134#391C22×C314yes—
135#376C22×C311yescharacteristic
136#343C22×C314yes—
137#339C22×C312yes—
138C3×#41—22no—
139C3×#41—24no—
140C3×#41—24no—
141C3×#41—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—24no—
145C2×C3×Q8—24no—
146C2×C3×Q8C2×D412yes—
147C2×C3×Q8C2×D411yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4C2×D412yes—
153C2×C3×D4C2×D411yescharacteristic
154C22×C3×C4—24no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2411yescharacteristic
163C3×Q16—48no—
164C3×Q16—44no—
165C3×SD16—48no—
166C3×SD16—44no—
167C3×M16—22no—
168C3×M16—24no—
169C3×M16#4114yes—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×D412yes—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—22no—
190C3×C42—24no—
191C3×C42—22no—
192C3×C42—21no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42C2×D411yescharacteristic
196C3×C42C2×D411yescharacteristic
197C3×C42C2×D412yes—
198C2×#41C23×C311yescharacteristic
199C2×#41C23×C312yes—
200C22×Q8C23×C312yes—
201#137—28no—
202#137—44no—
203#137—44no—
204C2×Q16—22no—
205C2×Q16—24no—
206C2×SD16—22no—
207C2×SD16—24no—
208C2×M16C23×C312yes—
209C2×M16C23×C312yes—
210C4⋊Q8—22no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C3×D411yescharacteristic
214#127C3×D411yescharacteristic
215#126—24no—
216#126—24no—
217#125—24no—
218#125—21no—
219#125—21no—
220#125—22no—
221#124C3×D412yes—
222#123—24no—
223#123—24no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226#121—24no—
227#121—22no—
228#121—22no—
229#121—22no—
230#121—24no—
231C4×Q8—22no—
232C4×Q8C3×D412yes—
233C4×D4—24no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—24no—
237C4×D4—22no—
238C4×D4C3×D412yes—
239#117—24no—
240#117—22no—
241#117—21no—
242#117C23×C311yescharacteristic
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C3×D414yes—
246C2×C4⋊C4C23×C312yes—
247C2×C42—21no—
248C2×C42C23×C312yes—
249C2×C42C23×C311yescharacteristic
250C4⋊C8—22no—
251#104—24no—
252#104—44no—
253#103—24no—
254#97—21no—
255C4×C8—21no—
256#95—22no—
257#95—28no—
258C23×C3—22no—
259C23×C3—21no—
260C23×C3C22×D411yescharacteristic
261C3×Q8—21no—
262C3×Q8—21no—
263C3×Q8—24no—
264C3×Q8—42no—
265C3×Q8—42no—
266C3×Q8—44no—
267C3×D4—21no—
268C3×D4—21no—
269C3×D4—24no—
270C3×D4—42no—
271C3×D4—44no—
272C3×D4—82no—
273C3×D4—84no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—24no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—44no—
287C2×C3×C4—82no—
288C2×C3×C4—84no—
289C2×C3×C4#12012yes—
290C2×C3×C4#12012yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4H(5,𝔽2)12yes—
295C3×C8—24no—
296C3×C8—42no—
297#41—22no—
298#41—24no—
299#41—24no—
300#41—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8C2×C3×D412yes—
306C2×Q8C2×C3×D411yescharacteristic
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—22no—
311C2×D4C2×C3×D412yes—
312C2×D4C2×C3×D411yescharacteristic
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4C2×C3×D414yes—
320C22×C4C2×C3×D412yes—
321C22×C4C24×C311yescharacteristic
322Q16—48no—
323Q16—44no—
324SD16—48no—
325SD16—44no—
326M16—22no—
327M16—24no—
328M16C3×#4114yes—
329C2×C8—22no—
330C2×C8—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4C2×C3×D412yes—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—21no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C42—24no—
350C42—22no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42C2×C3×D411yescharacteristic
355C42C2×C3×D411yescharacteristic
356C42C2×C3×D412yes—
357C22×C3—22no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3C2×#12011yescharacteristic
364C22×C3#55012yes—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—42no—
368C3×C4—41no—
369C3×C4—44no—
370C3×C4—42no—
371C3×C4—82no—
372C3×C4—82no—
373C3×C4#53211yescharacteristic
374C3×C4#53511yescharacteristic
375C3×C4#54612yes—
376C23—22no—
377C23—21no—
378C23C22×C3×D411yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—24no—
382Q8—42no—
383Q8—42no—
384Q8—44no—
385D4—21no—
386D4—21no—
387D4—24no—
388D4—42no—
389D4—44no—
390D4—82no—
391D4—84no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—82no—
406C2×C4—84no—
407C2×C4C3×#12012yes—
408C2×C4C3×#12012yes—
409C2×C4C22×C3×D411yescharacteristic
410C2×C4C22×C3×D411yescharacteristic
411C2×C4C2×C3×#4112yes—
412C2×C4C3×H(5,𝔽2)12yes—
413C8—24no—
414C8—42no—
415C2×C3—22no—
416C2×C3—82no—
417C2×C3—81no—
418C2×C3#243111yescharacteristic
419C2×C3#302212yes—
420C22—22no—
421C22—42no—
422C22—41no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22C2×C3×#12011yescharacteristic
427C22C3×#55012yes—
428C4—22no—
429C4—22no—
430C4—42no—
431C4—41no—
432C4—44no—
433C4—42no—
434C4—82no—
435C4—82no—
436C4C3×#53211yescharacteristic
437C4C3×#53511yescharacteristic
438C4C3×#54612yes—
439C3#2561911yescharacteristic, direct product factor, Sylow
440C2—22no—
441C2—82no—
442C2—81no—
443C2C3×#243111yescharacteristic
444C2C3×#302212yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3512C2
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×#3152C2semidirect
3. C3×#3039C2semidirect
4. C3×#2642C2semidirect
5. C3×#2007C2semidirect
6. C3×#1998C2semidirect
7. C3×#1963C2semidirect
8. C3×#1882C2semidirect
9. C3×#1759C2semidirect
10. #25619C3direct (N × H)
41. #3152C2×C3semidirect
42. #3039C2×C3semidirect
43. #2642C2×C3semidirect
44. #2007C2×C3semidirect
45. #1998C2×C3semidirect
46. #1963C2×C3semidirect
47. #1882C2×C3semidirect
48. #1759C2×C3semidirect
439. C3#25619direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35122
C3×#31522
C3×#30392
C3×#26422
C3×#20072
C3×#19982
C3×#19632
C3×#18822
C3×#17592
#256193