Finite Group Databasebeta

← C3×#10201 #10628808

Subgroups — C3×#10201 #10628808

Order 768.

528 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3246C211yesmaximal, characteristic
2C3×#3227C211yesmaximal, characteristic
3C3×#3165C211yesmaximal, characteristic, complemented
4C3×#1981C211yesmaximal, characteristic, complemented
5C3×#1980C211yesmaximal, characteristic
6C3×#1503C211yesmaximal, characteristic, complemented
7C3×#1283C211yesmaximal, characteristic, complemented
8#10201C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#498—21no—
16C3×#486—21no—
17C3×#481—22no—
18C3×#476—22no—
19C3×#476—21no—
20C3×#475—21no—
21C3×#465—22no—
22C3×#461—21no—
23C3×#460—21no—
24C3×#431C2211yescharacteristic
25C3×#428—22no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#104—22no—
28C2×C3×#103—22no—
29C3×C4×M16C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×#328—22no—
32#3246C2×C311yescharacteristic
33#3227C2×C311yescharacteristic
34#3165C2×C311yescharacteristic, complemented
35#1981C2×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⋊Q8D411yescharacteristic
49C3×C4⋊Q8D412yes—
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—24no—
70C3×#106—22no—
71C3×#106—24no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×#104—44no—
76C3×#103—22no—
77C3×#103—44no—
78C3×#102—22no—
79C3×#102—24no—
80C3×C22⋊C8D412yes—
81C3×#97—21no—
82C3×C4×C8—21no—
83C3×C4×C8—21no—
84C3×#95—21no—
85#533C22×C311yescharacteristic
86#532C22×C311yescharacteristic
87C2×C4⋊Q8C22×C311yescharacteristic
88#501—22no—
89#500—21no—
90#498—21no—
91#498—21no—
92#486—21no—
93#481—22no—
94#476—22no—
95#476—21no—
96#475—21no—
97#465—22no—
98#461—21no—
99#460—21no—
100#431C22×C311yescharacteristic
101#428—22no—
102C2×C4⋊C8C22×C311yescharacteristic
103C2×#104—22no—
104C2×#103—22no—
105C4×M16C22×C311yescharacteristic
106C4×C4⋊C4C22×C311yescharacteristic
107#328—22no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—22no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×Q8—41no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C3×SD16—82no—
128C3×M16—22no—
129C3×M16SD1614yescomplemented
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—42no—
134C3×C4⋊C4—22no—
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—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42—41no—
152C3×C42—42no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157C2×#41—21no—
158C22×Q8—21no—
159C2×SD16—22no—
160C2×M16C3×D412yes—
161C22×C8C3×D412yes—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C3×D412yes—
166C4⋊Q8C3×D411yescharacteristic
167C4⋊Q8C3×D411yescharacteristic
168C4⋊Q8C3×D411yescharacteristic
169#127C3×D411yescharacteristic
170#125—21no—
171#124—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174#121—21no—
175#121—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—21no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C42—21no—
185C2×C42C23×C311yescharacteristic
186#107—24no—
187#106—24no—
188#106—22no—
189#106—24no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193#104—44no—
194#103—22no—
195#103—44no—
196#102—22no—
197#102—24no—
198C22⋊C8C3×D412yes—
199#97—21no—
200C4×C8—21no—
201C4×C8—21no—
202#95—21no—
203C23×C3—41no—
204C23×C3#12711yescharacteristic
205C3×Q8—44no—
206C3×Q8—41no—
207C3×Q8—41no—
208C3×Q8—42no—
209C3×Q8—81no—
210C3×Q8—81no—
211C3×Q8—81no—
212C3×D4—42no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—42no—
216C3×D4—81no—
217C3×D4—81no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—81no—
227C2×C3×C4—82no—
228C2×C3×C4—81no—
229C2×C3×C4—82no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4C2×SD1612yes—
237C2×C3×C4#13712yes—
238C3×C8—24no—
239C3×C8—42no—
240C3×C8—42no—
241#41—42no—
242#41—42no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4C2×C3×D411yescharacteristic
258C22×C4C2×C3×D411yescharacteristic
259C22×C4C2×C3×D411yescharacteristic
260SD16—82no—
261M16—22no—
262M16C3×SD1614yescomplemented
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—42no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—41no—
285C42—42no—
286C42C2×C3×D411yescharacteristic
287C42C2×C3×D411yescharacteristic
288C42C2×C3×D411yescharacteristic
289C42C3×#4111yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3#39211yescharacteristic
296C22×C3#49211yescharacteristic
297C22×C3#49711yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—82no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4#44812yes—
307C3×C4#45311yescharacteristic
308C3×C4#45411yescharacteristic
309C3×C4#46112yes—
310C23—41no—
311C23C3×#12711yescharacteristic
312Q8—44no—
313Q8—41no—
314Q8—41no—
315Q8—42no—
316Q8—81no—
317Q8—81no—
318Q8—81no—
319D4—42no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4C3×#12011yescharacteristic
338C2×C4C3×#12011yescharacteristic
339C2×C4C3×#12011yescharacteristic
340C2×C4C3×#12111yescharacteristic
341C2×C4C3×#12111yescharacteristic
342C2×C4C3×#12111yescharacteristic
343C2×C4C2×C3×SD1612yes—
344C2×C4C3×#13712yes—
345C8—24no—
346C8—42no—
347C8—42no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—161no—
351C2×C3#200611yescharacteristic
352C2×C3#200911yescharacteristic
353C2×C3#201311yescharacteristic
354C22—21no—
355C22—21no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22C3×#39211yescharacteristic
360C22C3×#49211yescharacteristic
361C22C3×#49711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4C3×#44812yes—
371C4C3×#45311yescharacteristic
372C4C3×#45411yescharacteristic
373C4C3×#46112yes—
374C3#1020111yescharacteristic, direct product factor, Sylow
375C2—21no—
376C2—21no—
377C2—161no—
378C2C3×#200611yescharacteristic
379C2C3×#200911yescharacteristic
380C2C3×#201311yescharacteristic
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×#3165C2semidirect
4. C3×#1981C2semidirect
6. C3×#1503C2semidirect
7. C3×#1283C2semidirect
8. #10201C3direct (N × H)
34. #3165C2×C3semidirect
35. #1981C2×C3semidirect
37. #1503C2×C3semidirect
38. #1283C2×C3semidirect
129. C3×M16SD16semidirect
262. M16C3×SD16semidirect
374. C3#10201direct (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×#32272
C3×#31652
C3×#19812
C3×#19802
C3×#15032
C3×#12832
#102013