Finite Group Databasebeta

← C3×#10204 #10628811

Subgroups — C3×#10204 #10628811

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3245C211yesmaximal, characteristic
2C3×#3226C211yesmaximal, characteristic, complemented
3C3×#3152C211yesmaximal, characteristic, complemented
4C3×#1988C211yesmaximal, characteristic, complemented
5C3×#1980C211yesmaximal, characteristic, complemented
6C3×#1502C211yesmaximal, characteristic, complemented
7C3×#1283C211yesmaximal, characteristic, complemented
8#10204C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—22no—
10C3×#532C2211yescharacteristic
11C3×#532C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C3×#500—21no—
14C3×#497—22no—
15C3×#494—21no—
16C3×#492—21no—
17C3×#486—21no—
18C3×#480—22no—
19C3×#474—22no—
20C3×#474—21no—
21C3×#474—21no—
22C3×#466—22no—
23C3×#459—21no—
24C3×#459—21no—
25C3×#431C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#104—22no—
28C3×C4×M16C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×#340—22no—
31C3×#327—22no—
32#3245C2×C311yescharacteristic
33#3226C2×C311yescharacteristic, complemented
34#3152C2×C311yescharacteristic, complemented
35#1988C2×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—
42C3×#137—44no—
43C3×#137—44no—
44C2×C3×Q16—24no—
45C2×C3×SD16—24no—
46C2×C3×D8—22no—
47C2×C3×M16D412yes—
48C22×C3×C8D412yes—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8D412yes—
53C3×C4⋊Q8D411yescharacteristic
54C3×C4⋊Q8D411yescharacteristic
55C3×#127D411yescharacteristic
56C3×#127D411yescharacteristic
57C3×#124—21no—
58C3×#124—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×C4×D4—21no—
66C3×C4×D4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72C3×#107—22no—
73C3×#107—24no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×#104—44no—
78C3×#102—22no—
79C3×#102—22no—
80C3×#102—24no—
81C3×C22⋊C8D412yes—
82C3×#97—21no—
83C3×C4×C8—21no—
84C3×C4×C8—21no—
85C3×#95—21no—
86C2×#137—22no—
87#532C22×C311yescharacteristic
88#532C22×C311yescharacteristic
89C2×C4⋊Q8C22×C311yescharacteristic
90#500—21no—
91#497—22no—
92#494—21no—
93#492—21no—
94#486—21no—
95#480—22no—
96#474—22no—
97#474—21no—
98#474—21no—
99#466—22no—
100#459—21no—
101#459—21no—
102#431C22×C311yescharacteristic
103C2×C4⋊C8C22×C311yescharacteristic
104C2×#104—22no—
105C4×M16C22×C311yescharacteristic
106C4×C4⋊C4C22×C311yescharacteristic
107#340—22no—
108#327—22no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—42no—
112C3×#41—42no—
113C2×C3×Q8—21no—
114C2×C3×Q8—22no—
115C2×C3×Q8—21no—
116C2×C3×Q8—41no—
117C2×C3×Q8—42no—
118C2×C3×Q8—42no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C3×Q16—48no—
135C3×SD16—48no—
136C3×D8—82no—
137C3×M16—22no—
138C3×M16D814yescomplemented
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—42no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C42—42no—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42#4111yescharacteristic
162C2×#41—21no—
163C2×#41—21no—
164C22×Q8—21no—
165#137—44no—
166#137—44no—
167C2×Q16—24no—
168C2×SD16—24no—
169C2×D8—22no—
170C2×M16C3×D412yes—
171C22×C8C3×D412yes—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8C3×D412yes—
176C4⋊Q8C3×D411yescharacteristic
177C4⋊Q8C3×D411yescharacteristic
178#127C3×D411yescharacteristic
179#127C3×D411yescharacteristic
180#124—21no—
181#124—21no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C42—21no—
194C2×C42C23×C311yescharacteristic
195#107—22no—
196#107—24no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200#104—44no—
201#102—22no—
202#102—22no—
203#102—24no—
204C22⋊C8C3×D412yes—
205#97—21no—
206C4×C8—21no—
207C4×C8—21no—
208#95—21no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3#12711yescharacteristic
212C3×Q8—44no—
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—82no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—41no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—42no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C3×D4—81no—
227C3×D4—81no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4—82no—
240C2×C3×C4—82no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4C2×D812yes—
248C2×C3×C4AGL(1,ℤ/8)12yes—
249C3×C8—24no—
250C3×C8—42no—
251C3×C8—42no—
252#41—42no—
253#41—42no—
254#41—42no—
255#41—42no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4C2×C3×D411yescharacteristic
275C22×C4C2×C3×D411yescharacteristic
276C22×C4C2×C3×D411yescharacteristic
277Q16—48no—
278SD16—48no—
279D8—82no—
280M16—22no—
281M16C3×D814yescomplemented
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—42no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—42no—
301C42C2×C3×D411yescharacteristic
302C42C2×C3×D411yescharacteristic
303C42C2×C3×D411yescharacteristic
304C42C3×#4111yescharacteristic
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#39211yescharacteristic
314C22×C3#49311yescharacteristic
315C22×C3#49611yescharacteristic
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—82no—
320C3×C4—81no—
321C3×C4—81no—
322C3×C4—82no—
323C3×C4#44712yes—
324C3×C4#45511yescharacteristic
325C3×C4#45611yescharacteristic
326C3×C4#45912yes—
327C23—41no—
328C23—41no—
329C23C3×#12711yescharacteristic
330Q8—44no—
331Q8—42no—
332Q8—42no—
333Q8—82no—
334D4—42no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—42no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4C3×#12011yescharacteristic
360C2×C4C3×#12011yescharacteristic
361C2×C4C3×#12011yescharacteristic
362C2×C4C3×#12111yescharacteristic
363C2×C4C3×#12111yescharacteristic
364C2×C4C3×#12111yescharacteristic
365C2×C4C2×C3×D812yes—
366C2×C4C3×AGL(1,ℤ/8)12yes—
367C8—24no—
368C8—42no—
369C8—42no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—161no—
373C2×C3—161no—
374C2×C3#200711yescharacteristic
375C2×C3#200911yescharacteristic
376C2×C3#201111yescharacteristic
377C22—21no—
378C22—21no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C3×#39211yescharacteristic
386C22C3×#49311yescharacteristic
387C22C3×#49611yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—81no—
393C4—81no—
394C4—82no—
395C4C3×#44712yes—
396C4C3×#45511yescharacteristic
397C4C3×#45611yescharacteristic
398C4C3×#45912yes—
399C3#1020411yescharacteristic, direct product factor, Sylow
400C2—21no—
401C2—21no—
402C2—161no—
403C2—161no—
404C2C3×#200711yescharacteristic
405C2C3×#200911yescharacteristic
406C2C3×#201111yescharacteristic
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×#3226C2semidirect
3. C3×#3152C2semidirect
4. C3×#1988C2semidirect
5. C3×#1980C2semidirect
6. C3×#1502C2semidirect
7. C3×#1283C2semidirect
8. #10204C3direct (N × H)
33. #3226C2×C3semidirect
34. #3152C2×C3semidirect
35. #1988C2×C3semidirect
36. #1980C2×C3semidirect
37. #1502C2×C3semidirect
38. #1283C2×C3semidirect
138. C3×M16D8semidirect
281. M16C3×D8semidirect
399. C3#10204direct (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×#32262
C3×#31522
C3×#19882
C3×#19802
C3×#15022
C3×#12832
#102043