Finite Group Databasebeta

← C3×#12242 #10630849

Subgroups — C3×#12242 #10630849

Order 768.

588 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2675C211yesmaximal, characteristic
2C3×#2062C211yesmaximal, characteristic, complemented
3C3×#1596C211yesmaximal, characteristic, complemented
4C3×#1596C211yesmaximal, characteristic, complemented
5C3×#1343C211yesmaximal, characteristic, complemented
6C3×#1343C211yesmaximal, characteristic, complemented
7C3×#1326C211yesmaximal, characteristic, complemented
8#12242C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#543C2211yescharacteristic
10C2×C3×#121—21no—
11C2×C3×#121—21no—
12C2×C3×#121—21no—
13C3×#450—21no—
14C3×#450—21no—
15C3×#396—21no—
16C3×#392—21no—
17C3×#392C2211yescharacteristic
18C3×#392C2211yescharacteristic
19C3×#385—21no—
20C3×#385—21no—
21C3×#385C2211yescharacteristic
22C3×#342—21no—
23C3×#342C2211yescharacteristic
24C3×#328—21no—
25C3×#328—21no—
26C3×#323C2211yescharacteristic
27C3×#323C2211yescharacteristic
28#2675C2×C311yescharacteristic
29#2062C2×C311yescharacteristic, complemented
30#1596C2×C311yescharacteristic, complemented
31#1596C2×C311yescharacteristic, complemented
32#1343C2×C311yescharacteristic, complemented
33#1343C2×C311yescharacteristic, complemented
34#1326C2×C311yescharacteristic, complemented
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4D411yescharacteristic
43C2×C3×SD16—41no—
44C2×C3×SD16—41no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—41no—
48C3×C22⋊Q8—41no—
49C3×#121—44no—
50C3×#121—44no—
51C3×#121—42no—
52C3×#121—42no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—41no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4D411yescharacteristic
65C2×C3×C22⋊C4C2311yescharacteristic
66C3×#102—41no—
67C3×#102—41no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3×#95—41no—
72C3×#95D411yescharacteristic
73C3×#95D411yescharacteristic
74#543C22×C311yescharacteristic
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—21no—
78#450—21no—
79#450—21no—
80#396—21no—
81#392—21no—
82#392C22×C311yescharacteristic
83#392C22×C311yescharacteristic
84#385—21no—
85#385—21no—
86#385C22×C311yescharacteristic
87#342—21no—
88#342C22×C311yescharacteristic
89#328—21no—
90#328—21no—
91#323C22×C311yescharacteristic
92#323C22×C311yescharacteristic
93C24×C3—21no—
94C24×C3—21no—
95C24×C3C2×D411yescharacteristic
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—42no—
101C2×C3×D4—42no—
102C2×C3×D4—42no—
103C2×C3×D4—42no—
104C2×C3×D4—42no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—81no—
112C2×C3×D4—81no—
113C2×C3×D4—81no—
114C2×C3×D4—81no—
115C2×C3×D4—81no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—81no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C3×SD16—82no—
131C3×SD16—82no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—82no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—81no—
153C3×C22⋊C4SD1612yescomplemented
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C3×D411yescharacteristic
159C22×D4C3×D411yescharacteristic
160C23×C4—21no—
161C23×C4C3×D411yescharacteristic
162C2×SD16—41no—
163C2×SD16—41no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—41no—
167C22⋊Q8—41no—
168#121—44no—
169#121—44no—
170#121—42no—
171#121—42no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4C3×D411yescharacteristic
184C2×C22⋊C4C23×C311yescharacteristic
185#102—41no—
186#102—41no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189#95—21no—
190#95—41no—
191#95C3×D411yescharacteristic
192#95C3×D411yescharacteristic
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—22no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3#12011yescharacteristic
212C23×C3C2×SD1611yescharacteristic
213C23×C3AGL(1,ℤ/8)11yescharacteristic
214C3×Q8—81no—
215C3×Q8—81no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—81no—
219C3×D4—82no—
220C3×D4—82no—
221C3×D4—84no—
222C3×D4—84no—
223C3×D4—81no—
224C3×D4—82no—
225C3×D4—82no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—81no—
247C2×C3×C4—84no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C3×C8—81no—
253C3×C8—81no—
254C24—21no—
255C24—21no—
256C24C2×C3×D411yescharacteristic
257C2×Q8—21no—
258C2×Q8—21no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—81no—
273C2×D4—81no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—81no—
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291SD16—82no—
292SD16—82no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—82no—
303C22⋊C4—24no—
304C22⋊C4—21no—
305C22⋊C4—22no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—81no—
314C22⋊C4C3×SD1612yescomplemented
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—42no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3#45011yescharacteristic
339C22×C3#45611yescharacteristic
340C22×C3B(4,𝔽2)11yescharacteristic
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—41no—
344C3×C4—42no—
345C3×C4—41no—
346C3×C4—81no—
347C3×C4—82no—
348C3×C4—81no—
349C3×C4—81no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23C3×#12011yescharacteristic
369C23C2×C3×SD1611yescharacteristic
370C23C3×AGL(1,ℤ/8)11yescharacteristic
371Q8—81no—
372Q8—81no—
373D4—42no—
374D4—42no—
375D4—81no—
376D4—82no—
377D4—82no—
378D4—84no—
379D4—84no—
380D4—81no—
381D4—82no—
382D4—82no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—84no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C8—81no—
410C8—81no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—82no—
416C2×C3—161no—
417C2×C3#160411yescharacteristic
418C2×C3#219611yescharacteristic
419C2×C3#220011yescharacteristic
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—41no—
425C22—41no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22C3×#45011yescharacteristic
444C22C3×#45611yescharacteristic
445C22C3×B(4,𝔽2)11yescharacteristic
446C4—42no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—41no—
451C4—81no—
452C4—82no—
453C4—81no—
454C4—81no—
455C3#1224211yescharacteristic, direct product factor, Sylow
456C2—21no—
457C2—21no—
458C2—41no—
459C2—41no—
460C2—82no—
461C2—161no—
462C2C3×#160411yescharacteristic
463C2C3×#219611yescharacteristic
464C2C3×#220011yescharacteristic
subgroupG/H
center, socleC22×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C3
prime radicalC3×#2675C2
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×#1596C2semidirect
4. C3×#1596C2semidirect
5. C3×#1343C2semidirect
6. C3×#1343C2semidirect
7. C3×#1326C2semidirect
8. #12242C3direct (N × H)
29. #2062C2×C3semidirect
30. #1596C2×C3semidirect
31. #1596C2×C3semidirect
32. #1343C2×C3semidirect
33. #1343C2×C3semidirect
34. #1326C2×C3semidirect
153. C3×C22⋊C4SD16semidirect
314. C22⋊C4C3×SD16semidirect
455. C3#12242direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26752
C3×#20622
C3×#15962
C3×#13432
C3×#13262
#122423