Finite Group Databasebeta

← C3×#10257 #10628864

Subgroups — C3×#10257 #10628864

Order 768.

528 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3246C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×#3037C211yesmaximal, characteristic
4C3×#1984C211yesmaximal, characteristic, complemented
5C3×#1493C211yesmaximal, characteristic, complemented
6C3×#1492C211yesmaximal, characteristic
7C3×#1292C211yesmaximal, characteristic, complemented
8#10257C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—21no—
10C3×#533C2211yescharacteristic
11C3×#532C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C3×#479—22no—
14C3×#477—21no—
15C3×#476—21no—
16C3×#475—21no—
17C3×#475—21no—
18C3×#465—22no—
19C3×#464—22no—
20C3×#462—21no—
21C3×#461—21no—
22C3×#460—21no—
23C3×#460—21no—
24C3×#428—21no—
25C3×#423C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#419—21no—
29C3×#417—21no—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#336—22no—
32C3×Q8⋊C8—22no—
33C3×D4⋊C8—22no—
34#3246C2×C311yescharacteristic
35#3078C2×C311yescharacteristic, complemented
36#3037C2×C311yescharacteristic
37#1984C2×C311yescharacteristic, complemented
38#1493C2×C311yescharacteristic, complemented
39#1492C2×C311yescharacteristic
40#1292C2×C311yescharacteristic, complemented
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C3×#137—42no—
44C3×#137—42no—
45C2×C3×Q16—22no—
46C2×C3×SD16—22no—
47C2×C3×SD16—22no—
48C2×C3×M16—21no—
49C2×C3×M16—21no—
50C22×C3×C8—21no—
51C22×C3×C8D412yes—
52C3×C4⋊Q8D411yescharacteristic
53C3×C4⋊Q8D411yescharacteristic
54C3×C4⋊Q8D411yescharacteristic
55C3×#127D411yescharacteristic
56C3×#125—21no—
57C3×#124—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×C4×Q8—21no—
63C3×C4×Q8—21no—
64C3×C4×Q8—21no—
65C3×C4×Q8—21no—
66C3×C4×Q8—21no—
67C3×C4×Q8—21no—
68C3×C4×D4—21no—
69C3×C4×D4—21no—
70C3×#117—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75C3×#107—22no—
76C3×#106—22no—
77C3×#106—22no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C4⋊C8—21no—
81C3×C4⋊C8—21no—
82C3×C4⋊C8D412yes—
83C3×C4⋊C8D412yes—
84C3×#103—22no—
85C3×#103—22no—
86C3×#103—22no—
87C3×#103—22no—
88C3×#102—22no—
89C3×#102—22no—
90C3×C4×C8—21no—
91C3×C4×C8—21no—
92C2×#137—21no—
93#533C22×C311yescharacteristic
94#532C22×C311yescharacteristic
95C2×C4×Q8C22×C311yescharacteristic
96#479—22no—
97#477—21no—
98#476—21no—
99#475—21no—
100#475—21no—
101#465—22no—
102#464—22no—
103#462—21no—
104#461—21no—
105#460—21no—
106#460—21no—
107#428—21no—
108#423C22×C311yescharacteristic
109#423C22×C311yescharacteristic
110C2×C4⋊C8C22×C311yescharacteristic
111#419—21no—
112#417—21no—
113C2×C4×C8C22×C311yescharacteristic
114#336—22no—
115Q8⋊C8—22no—
116D4⋊C8—22no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8—21no—
128C2×C3×Q8—21no—
129C2×C3×Q8SD1612yescomplemented
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—41no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C3×Q16—44no—
141C3×SD16—44no—
142C3×SD16—82no—
143C3×M16—41no—
144C3×M16—41no—
145C3×M16—41no—
146C3×M16—41no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—41no—
152C2×C3×C8—41no—
153C2×C3×C8—42no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C42—41no—
170C3×C42—41no—
171C3×C42—41no—
172C3×C42C2×D411yescharacteristic
173C3×C42C2×D411yescharacteristic
174C3×C42C2×D411yescharacteristic
175C3×C42C2×D411yescharacteristic
176C2×#41—21no—
177C22×Q8C3×D411yescharacteristic
178#137—42no—
179#137—42no—
180C2×Q16—22no—
181C2×SD16—22no—
182C2×SD16—22no—
183C2×M16—21no—
184C2×M16—21no—
185C22×C8—21no—
186C22×C8C3×D412yes—
187C4⋊Q8C3×D411yescharacteristic
188C4⋊Q8C3×D411yescharacteristic
189C4⋊Q8C3×D411yescharacteristic
190#127C3×D411yescharacteristic
191#125—21no—
192#124—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195#121—21no—
196#121—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×D4—21no—
204C4×D4—21no—
205#117—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C3×D411yescharacteristic
208C2×C42—21no—
209C2×C42C23×C311yescharacteristic
210#107—22no—
211#106—22no—
212#106—22no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8C3×D412yes—
218C4⋊C8C3×D412yes—
219#103—22no—
220#103—22no—
221#103—22no—
222#103—22no—
223#102—22no—
224#102—22no—
225C4×C8—21no—
226C4×C8—21no—
227C23×C3—41no—
228C23×C3#12111yescharacteristic
229C3×Q8—22no—
230C3×Q8—22no—
231C3×Q8—42no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×D4—42no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—81no—
244C3×D4—81no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12711yescharacteristic
264C2×C3×C4C2×SD1611yescharacteristic
265C2×C3×C4#13511yescharacteristic
266C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
267C2×C3×C4#13711yescharacteristic
268C3×C8—42no—
269C3×C8—42no—
270C3×C8—42no—
271C3×C8—42no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8C3×SD1612yescomplemented
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4C2×C3×D411yescharacteristic
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C3×#4111yescharacteristic
295Q16—44no—
296SD16—44no—
297SD16—82no—
298M16—41no—
299M16—41no—
300M16—41no—
301M16—41no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42C2×C3×D411yescharacteristic
328C42C2×C3×D411yescharacteristic
329C42C2×C3×D411yescharacteristic
330C42C2×C3×D411yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#39211yescharacteristic
337C22×C3#46511yescharacteristic
338C22×C3#46811yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—42no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4#44811yescharacteristic
348C3×C4#44911yescharacteristic
349C3×C4#45011yescharacteristic
350C3×C4#45211yescharacteristic
351C3×C4#45511yescharacteristic
352C3×C4#45611yescharacteristic
353C23—41no—
354C23C3×#12111yescharacteristic
355Q8—22no—
356Q8—22no—
357Q8—42no—
358Q8—41no—
359Q8—41no—
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—41no—
364D4—42no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4C3×#12011yescharacteristic
385C2×C4C3×#12011yescharacteristic
386C2×C4C3×#12011yescharacteristic
387C2×C4C3×#12111yescharacteristic
388C2×C4C3×#12111yescharacteristic
389C2×C4C3×#12711yescharacteristic
390C2×C4C2×C3×SD1611yescharacteristic
391C2×C4C3×#13511yescharacteristic
392C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
393C2×C4C3×#13711yescharacteristic
394C8—42no—
395C8—42no—
396C8—42no—
397C8—42no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—161no—
401C2×C3#200011yescharacteristic
402C2×C3#200711yescharacteristic
403C2×C3#201711yescharacteristic
404C22—21no—
405C22—21no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22C3×#39211yescharacteristic
410C22C3×#46511yescharacteristic
411C22C3×#46811yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4C3×#44811yescharacteristic
421C4C3×#44911yescharacteristic
422C4C3×#45011yescharacteristic
423C4C3×#45211yescharacteristic
424C4C3×#45511yescharacteristic
425C4C3×#45611yescharacteristic
426C3#1025711yescharacteristic, direct product factor, Sylow
427C2—21no—
428C2—21no—
429C2—161no—
430C2C3×#200011yescharacteristic
431C2C3×#200711yescharacteristic
432C2C3×#201711yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3078C2semidirect
4. C3×#1984C2semidirect
5. C3×#1493C2semidirect
7. C3×#1292C2semidirect
8. #10257C3direct (N × H)
35. #3078C2×C3semidirect
37. #1984C2×C3semidirect
38. #1493C2×C3semidirect
40. #1292C2×C3semidirect
129. C2×C3×Q8SD16semidirect
284. C2×Q8C3×SD16semidirect
426. C3#10257direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32462
C3×#30782
C3×#30372
C3×#19842
C3×#14932
C3×#14922
C3×#12922
#102573