Finite Group Databasebeta

← C3×#12226 #10630833

Subgroups — C3×#12226 #10630833

Order 768.

510 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2455C211yesmaximal, characteristic
2C3×#2062C211yesmaximal, characteristic, complemented
3C3×#1614C211yesmaximal, characteristic, complemented
4C3×#1596C211yesmaximal, characteristic
5C3×#1352C211yesmaximal, characteristic, complemented
6C3×#1343C211yesmaximal, characteristic
7C3×#1326C211yesmaximal, characteristic, complemented
8#12226C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#543C2211yescharacteristic
10C2×C3×#123—21no—
11C2×C3×C4×D4—21no—
12C3×#484—21no—
13C3×#450—21no—
14C3×#399C2211yescharacteristic
15C3×#394—21no—
16C3×#392C2211yescharacteristic
17C3×#389—21no—
18C3×#388—21no—
19C3×#385—21no—
20C3×#385—21no—
21C3×#385C2211yescharacteristic
22C3×#342—21no—
23C3×#342C2211yescharacteristic
24C3×#328—21no—
25C3×#328—21no—
26C3×#323C2211yescharacteristic
27C3×#323C2211yescharacteristic
28#2455C2×C311yescharacteristic
29#2062C2×C311yescharacteristic, complemented
30#1614C2×C311yescharacteristic, complemented
31#1596C2×C311yescharacteristic
32#1352C2×C311yescharacteristic, complemented
33#1343C2×C311yescharacteristic
34#1326C2×C311yescharacteristic, complemented
35C22×C3×D4—21no—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4—21no—
38C23×C3×C4D411yescharacteristic
39C2×C3×SD16—41no—
40C3×#123—44no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—41no—
44C3×C22⋊Q8—41no—
45C3×C4×D4—44no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—41no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4D411yescharacteristic
60C2×C3×C22⋊C4C2311yescharacteristic
61C2×C3×C42—21no—
62C3×#107—41no—
63C3×#103—41no—
64C3×#102—41no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95—41no—
70C3×#95D411yescharacteristic
71C3×#95D411yescharacteristic
72#543C22×C311yescharacteristic
73C2×#123—21no—
74C2×C4×D4—21no—
75#484—21no—
76#450—21no—
77#399C22×C311yescharacteristic
78#394—21no—
79#392C22×C311yescharacteristic
80#389—21no—
81#388—21no—
82#385—21no—
83#385—21no—
84#385C22×C311yescharacteristic
85#342—21no—
86#342C22×C311yescharacteristic
87#328—21no—
88#328—21no—
89#323C22×C311yescharacteristic
90#323C22×C311yescharacteristic
91C24×C3—21no—
92C24×C3C2×D411yescharacteristic
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C2×C3×D4—42no—
96C2×C3×D4—41no—
97C2×C3×D4—42no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—42no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—42no—
109C22×C3×C4—42no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—42no—
113C22×C3×C4—81no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C3×SD16—82no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—42no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4—42no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—82no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—21no—
132C3×C22⋊C4—21no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—42no—
142C22×D4—21no—
143C22×D4C3×D411yescharacteristic
144C23×C4—21no—
145C23×C4C3×D411yescharacteristic
146C2×SD16—41no—
147#123—44no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C4×D4—44no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C3×D411yescharacteristic
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4C3×D411yescharacteristic
167C2×C22⋊C4C23×C311yescharacteristic
168C2×C42—21no—
169#107—41no—
170#103—41no—
171#102—41no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174#95—21no—
175#95—21no—
176#95—41no—
177#95C3×D411yescharacteristic
178#95C3×D411yescharacteristic
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3#12011yescharacteristic
191C23×C3#13511yescharacteristic
192C23×C3#13711yescharacteristic
193C3×Q8—81no—
194C3×Q8—81no—
195C3×D4—42no—
196C3×D4—44no—
197C3×D4—44no—
198C3×D4—81no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—81no—
223C2×C3×C4—84no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C3×C8—81no—
229C3×C8—81no—
230C24—21no—
231C24C2×C3×D411yescharacteristic
232C2×Q8—21no—
233C2×Q8—21no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—42no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—81no—
253C22×C4C2×C3×D411yescharacteristic
254C22×C4C2×C3×D411yescharacteristic
255SD16—82no—
256C2×C8—41no—
257C2×C8—41no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—82no—
268C22⋊C4—24no—
269C22⋊C4—22no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C42—42no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—42no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#45211yescharacteristic
298C22×C3#45611yescharacteristic
299C22×C3B(4,𝔽2)11yescharacteristic
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—41no—
303C3×C4—42no—
304C3×C4—41no—
305C3×C4—81no—
306C3×C4—82no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4—81no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C3×#12011yescharacteristic
322C23C3×#13511yescharacteristic
323C23C3×#13711yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—42no—
327D4—44no—
328D4—44no—
329D4—81no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—84no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C8—81no—
360C8—81no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—82no—
366C2×C3#160311yescharacteristic
367C2×C3#219611yescharacteristic
368C2×C3#219811yescharacteristic
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—42no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C3×#45211yescharacteristic
386C22C3×#45611yescharacteristic
387C22C3×B(4,𝔽2)11yescharacteristic
388C4—42no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—41no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4—81no—
397C4—81no—
398C3#1222611yescharacteristic, direct product factor, Sylow
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—82no—
404C2C3×#160311yescharacteristic
405C2C3×#219611yescharacteristic
406C2C3×#219811yescharacteristic
subgroupG/H
center, socleC22×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C3
prime radicalC3×#392C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C2×C3×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2062C2semidirect
3. C3×#1614C2semidirect
5. C3×#1352C2semidirect
7. C3×#1326C2semidirect
8. #12226C3direct (N × H)
29. #2062C2×C3semidirect
30. #1614C2×C3semidirect
32. #1352C2×C3semidirect
34. #1326C2×C3semidirect
398. C3#12226direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24552
C3×#20622
C3×#16142
C3×#15962
C3×#13522
C3×#13432
C3×#13262
#122263