Finite Group Databasebeta

← C3×#10205 #10628812

Subgroups — C3×#10205 #10628812

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3246C211yesmaximal, characteristic
2C3×#3226C211yesmaximal, characteristic
3C3×#3139C211yesmaximal, characteristic, complemented
4C3×#1988C211yesmaximal, characteristic, complemented
5C3×#1980C211yesmaximal, characteristic
6C3×#1503C211yesmaximal, characteristic, complemented
7C3×#1283C211yesmaximal, characteristic, complemented
8#10205C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×#500—21no—
13C3×#491—22no—
14C3×#487—21no—
15C3×#487—21no—
16C3×#486—21no—
17C3×#480—22no—
18C3×#476—21no—
19C3×#475—21no—
20C3×#474—22no—
21C3×#465—22no—
22C3×#461—21no—
23C3×#460—21no—
24C3×#431C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#104—22no—
27C3×#419—22no—
28C3×C4×M16C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×#340—22no—
31C3×#328—22no—
32#3246C2×C311yescharacteristic
33#3226C2×C311yescharacteristic
34#3139C2×C311yescharacteristic, complemented
35#1988C2×C311yescharacteristic, complemented
36#1980C2×C311yescharacteristic
37#1503C2×C311yescharacteristic, complemented
38#1283C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C2×C3×SD16—22no—
42C2×C3×M16D412yes—
43C22×C3×C8D412yes—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×C4⋊Q8D412yes—
49C3×C4⋊Q8D411yescharacteristic
50C3×C4⋊Q8D411yescharacteristic
51C3×#127D411yescharacteristic
52C3×#125—21no—
53C3×#124—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×#117—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C3×#107—24no—
69C3×#106—22no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×#104—44no—
74C3×#103—24no—
75C3×#103—24no—
76C3×#103—22no—
77C3×#102—22no—
78C3×#102—24no—
79C3×C22⋊C8D412yes—
80C3×#97—21no—
81C3×C4×C8—21no—
82C3×C4×C8—21no—
83C3×#95—21no—
84#533C22×C311yescharacteristic
85#532C22×C311yescharacteristic
86C2×C4⋊Q8C22×C311yescharacteristic
87#500—21no—
88#491—22no—
89#487—21no—
90#487—21no—
91#486—21no—
92#480—22no—
93#476—21no—
94#475—21no—
95#474—22no—
96#465—22no—
97#461—21no—
98#460—21no—
99#431C22×C311yescharacteristic
100C2×C4⋊C8C22×C311yescharacteristic
101C2×#104—22no—
102#419—22no—
103C4×M16C22×C311yescharacteristic
104C4×C4⋊C4C22×C311yescharacteristic
105#340—22no—
106#328—22no—
107C3×#41—42no—
108C3×#41—42no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C3×SD16—82no—
127C3×M16—22no—
128C3×M16—22no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—42no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42—41no—
151C3×C42—42no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C2×#41—21no—
157C22×Q8—21no—
158C2×SD16—22no—
159C2×M16C3×D412yes—
160C22×C8C3×D412yes—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C3×D412yes—
165C4⋊Q8C3×D411yescharacteristic
166C4⋊Q8C3×D411yescharacteristic
167C4⋊Q8C3×D411yescharacteristic
168#127C3×D411yescharacteristic
169#125—21no—
170#124—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173#121—21no—
174#121—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×D4—21no—
178C4×D4—21no—
179#117—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C42—21no—
184C2×C42C23×C311yescharacteristic
185#107—24no—
186#106—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190#104—44no—
191#103—24no—
192#103—24no—
193#103—22no—
194#102—22no—
195#102—24no—
196C22⋊C8C3×D412yes—
197#97—21no—
198C4×C8—21no—
199C4×C8—21no—
200#95—21no—
201C23×C3—41no—
202C23×C3#12711yescharacteristic
203C3×Q8—44no—
204C3×Q8—41no—
205C3×Q8—41no—
206C3×Q8—42no—
207C3×Q8—81no—
208C3×Q8—82no—
209C3×D4—42no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—42no—
213C3×D4—81no—
214C3×D4—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—82no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#13512yes—
235C2×C3×C4AGL(1,ℤ/8)12yes—
236C3×C8—42no—
237C3×C8—42no—
238C3×C8—42no—
239#41—42no—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×C3×D411yescharacteristic
256C22×C4C2×C3×D411yescharacteristic
257C22×C4C2×C3×D411yescharacteristic
258SD16—82no—
259M16—22no—
260M16—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—42no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C42—41no—
283C42—42no—
284C42C2×C3×D411yescharacteristic
285C42C2×C3×D411yescharacteristic
286C42C2×C3×D411yescharacteristic
287C42C3×#4111yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3#39211yescharacteristic
294C22×C3#49511yescharacteristic
295C22×C3#49611yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—82no—
300C3×C4—81no—
301C3×C4—81no—
302C3×C4—81no—
303C3×C4—82no—
304C3×C4#44912yes—
305C3×C4#45511yescharacteristic
306C3×C4#45611yescharacteristic
307C3×C4#46312yes—
308C23—41no—
309C23C3×#12711yescharacteristic
310Q8—44no—
311Q8—41no—
312Q8—41no—
313Q8—42no—
314Q8—81no—
315Q8—82no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—42no—
320D4—81no—
321D4—81no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4C3×#12011yescharacteristic
336C2×C4C3×#12011yescharacteristic
337C2×C4C3×#12011yescharacteristic
338C2×C4C3×#12111yescharacteristic
339C2×C4C3×#12111yescharacteristic
340C2×C4C3×#12111yescharacteristic
341C2×C4C3×#13512yes—
342C2×C4C3×AGL(1,ℤ/8)12yes—
343C8—42no—
344C8—42no—
345C8—42no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—161no—
349C2×C3#200711yescharacteristic
350C2×C3#200911yescharacteristic
351C2×C3#201211yescharacteristic
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22C3×#39211yescharacteristic
358C22C3×#49511yescharacteristic
359C22C3×#49611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—82no—
368C4C3×#44912yes—
369C4C3×#45511yescharacteristic
370C4C3×#45611yescharacteristic
371C4C3×#46312yes—
372C3#1020511yescharacteristic, direct product factor, Sylow
373C2—21no—
374C2—21no—
375C2—161no—
376C2C3×#200711yescharacteristic
377C2C3×#200911yescharacteristic
378C2C3×#201211yescharacteristic
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
3. C3×#3139C2semidirect
4. C3×#1988C2semidirect
6. C3×#1503C2semidirect
7. C3×#1283C2semidirect
8. #10205C3direct (N × H)
34. #3139C2×C3semidirect
35. #1988C2×C3semidirect
37. #1503C2×C3semidirect
38. #1283C2×C3semidirect
372. C3#10205direct (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×#32262
C3×#31392
C3×#19882
C3×#19802
C3×#15032
C3×#12832
#102053