Finite Group Databasebeta

← C3×#10200 #10628807

Subgroups — C3×#10200 #10628807

Order 768.

550 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3245C211yesmaximal, characteristic
2C3×#3227C211yesmaximal, characteristic, complemented
3C3×#3141C211yesmaximal, characteristic, complemented
4C3×#1981C211yesmaximal, characteristic, complemented
5C3×#1980C211yesmaximal, characteristic, complemented
6C3×#1502C211yesmaximal, characteristic, complemented
7C3×#1283C211yesmaximal, characteristic, complemented
8#10200C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#532C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×#498—21no—
13C3×#489—22no—
14C3×#487—21no—
15C3×#486—21no—
16C3×#486—21no—
17C3×#481—22no—
18C3×#476—22no—
19C3×#474—21no—
20C3×#474—21no—
21C3×#466—22no—
22C3×#459—21no—
23C3×#459—21no—
24C3×#431C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#104—22no—
27C3×#417—22no—
28C2×C3×#103—22no—
29C3×C4×M16C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×#327—22no—
32#3245C2×C311yescharacteristic
33#3227C2×C311yescharacteristic, complemented
34#3141C2×C311yescharacteristic, complemented
35#1981C2×C311yescharacteristic, complemented
36#1980C2×C311yescharacteristic, complemented
37#1502C2×C311yescharacteristic, complemented
38#1283C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C2×C3×#41—21no—
41C22×C3×Q8—21no—
42C2×C3×D8—22no—
43C2×C3×M16D412yes—
44C22×C3×C8D412yes—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×C4⋊Q8D411yescharacteristic
50C3×C4⋊Q8D412yes—
51C3×#127D411yescharacteristic
52C3×#127D411yescharacteristic
53C3×#124—21no—
54C3×#124—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×C4×D4—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—22no—
69C3×#106—24no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×#104—44no—
74C3×#103—24no—
75C3×#103—44no—
76C3×#102—24no—
77C3×#102—22no—
78C3×#102—22no—
79C3×#102—24no—
80C3×C22⋊C8D412yes—
81C3×#97—21no—
82C3×C4×C8—21no—
83C3×C4×C8—21no—
84C3×#95—21no—
85#532C22×C311yescharacteristic
86#532C22×C311yescharacteristic
87C2×C4⋊Q8C22×C311yescharacteristic
88#498—21no—
89#489—22no—
90#487—21no—
91#486—21no—
92#486—21no—
93#481—22no—
94#476—22no—
95#474—21no—
96#474—21no—
97#466—22no—
98#459—21no—
99#459—21no—
100#431C22×C311yescharacteristic
101C2×C4⋊C8C22×C311yescharacteristic
102C2×#104—22no—
103#417—22no—
104C2×#103—22no—
105C4×M16C22×C311yescharacteristic
106C4×C4⋊C4C22×C311yescharacteristic
107#327—22no—
108C3×#41—42no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—42no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—22no—
115C2×C3×Q8—42no—
116C2×C3×Q8—42no—
117C2×C3×Q8—41no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C3×D8—82no—
134C3×M16—22no—
135C3×M16—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—42no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42—42no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159C2×#41—21no—
160C2×#41—21no—
161C22×Q8—21no—
162C2×D8—22no—
163C2×M16C3×D412yes—
164C22×C8C3×D412yes—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C3×D412yes—
169C4⋊Q8C3×D411yescharacteristic
170C4⋊Q8C3×D411yescharacteristic
171#127C3×D411yescharacteristic
172#127C3×D411yescharacteristic
173#124—21no—
174#124—21no—
175#121—21no—
176#121—21no—
177#121—21no—
178#121—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C42—21no—
187C2×C42C23×C311yescharacteristic
188#107—22no—
189#106—24no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193#104—44no—
194#103—24no—
195#103—44no—
196#102—24no—
197#102—22no—
198#102—22no—
199#102—24no—
200C22⋊C8C3×D412yes—
201#97—21no—
202C4×C8—21no—
203C4×C8—21no—
204#95—21no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3#12711yescharacteristic
208C3×Q8—44no—
209C3×Q8—42no—
210C3×Q8—42no—
211C3×Q8—81no—
212C3×Q8—81no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—41no—
216C3×D4—41no—
217C3×D4—41no—
218C3×D4—41no—
219C3×D4—42no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—81no—
233C2×C3×C4—82no—
234C2×C3×C4—81no—
235C2×C3×C4—82no—
236C2×C3×C4—82no—
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#13512yes—
244C2×C3×C4AGL(1,ℤ/8)12yes—
245C3×C8—42no—
246C3×C8—42no—
247C3×C8—42no—
248#41—42no—
249#41—42no—
250#41—42no—
251#41—42no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C2×C3×D411yescharacteristic
271C22×C4C2×C3×D411yescharacteristic
272C22×C4C2×C3×D411yescharacteristic
273D8—82no—
274M16—22no—
275M16—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—42no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—42no—
295C42C2×C3×D411yescharacteristic
296C42C2×C3×D411yescharacteristic
297C42C2×C3×D411yescharacteristic
298C42C3×#4111yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3#39211yescharacteristic
308C22×C3#49511yescharacteristic
309C22×C3#49611yescharacteristic
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—82no—
314C3×C4—82no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4#44912yes—
318C3×C4#45311yescharacteristic
319C3×C4#45411yescharacteristic
320C3×C4#46312yes—
321C23—41no—
322C23—41no—
323C23C3×#12711yescharacteristic
324Q8—44no—
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329D4—42no—
330D4—42no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4C3×#12011yescharacteristic
354C2×C4C3×#12011yescharacteristic
355C2×C4C3×#12011yescharacteristic
356C2×C4C3×#12111yescharacteristic
357C2×C4C3×#12111yescharacteristic
358C2×C4C3×#12111yescharacteristic
359C2×C4C3×#13512yes—
360C2×C4C3×AGL(1,ℤ/8)12yes—
361C8—42no—
362C8—42no—
363C8—42no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—161no—
367C2×C3—161no—
368C2×C3#200611yescharacteristic
369C2×C3#200911yescharacteristic
370C2×C3#201211yescharacteristic
371C22—21no—
372C22—21no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C3×#39211yescharacteristic
380C22C3×#49511yescharacteristic
381C22C3×#49611yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—82no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4C3×#44912yes—
390C4C3×#45311yescharacteristic
391C4C3×#45411yescharacteristic
392C4C3×#46312yes—
393C3#1020011yescharacteristic, direct product factor, Sylow
394C2—21no—
395C2—21no—
396C2—161no—
397C2—161no—
398C2C3×#200611yescharacteristic
399C2C3×#200911yescharacteristic
400C2C3×#201211yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#3245C2
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×#3227C2semidirect
3. C3×#3141C2semidirect
4. C3×#1981C2semidirect
5. C3×#1980C2semidirect
6. C3×#1502C2semidirect
7. C3×#1283C2semidirect
8. #10200C3direct (N × H)
33. #3227C2×C3semidirect
34. #3141C2×C3semidirect
35. #1981C2×C3semidirect
36. #1980C2×C3semidirect
37. #1502C2×C3semidirect
38. #1283C2×C3semidirect
393. C3#10200direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32452
C3×#32272
C3×#31412
C3×#19812
C3×#19802
C3×#15022
C3×#12832
#102003