Finite Group Databasebeta

← C3×#33037 #10651644

Subgroups — C3×#33037 #10651644

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#475C211yesmaximal, characteristic, complemented
2C2×C3×#464C211yesmaximal, characteristic
3C3×#2640C211yesmaximal, characteristic
4C3×#2276C211yesmaximal, characteristic, complemented
5C3×#1987C211yesmaximal, characteristic, complemented
6C3×#1983C211yesmaximal, characteristic
7C3×#1927C212yesmaximal, complemented
8C3×#1926C212yesmaximal
9C3×#1880C212yesmaximal
10C3×#1878C212yesmaximal, complemented
11C3×#1832C211yesmaximal, characteristic, complemented
12#33037C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C4×Q8C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C3×#475—24no—
20C3×#464—24no—
21C2×C3×#106C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2211yescharacteristic
29C3×#397—22no—
30C3×#393—21no—
31C3×#393—21no—
32C3×#391C2212yes—
33C3×#390C2212yes—
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4—22no—
36C3×C4×C4⋊C4C2212yes—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#376—21no—
39C3×#376C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#475C2×C311yescharacteristic, complemented
45C2×#464C2×C311yescharacteristic
46#2640C2×C311yescharacteristic
47#2276C2×C311yescharacteristic, complemented
48#1987C2×C311yescharacteristic, complemented
49#1983C2×C311yescharacteristic
50#1927C2×C312yescomplemented
51#1926C2×C312yes—
52#1880C2×C312yes—
53#1878C2×C312yescomplemented
54#1832C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C2×C3×Q16—22no—
60C2×C3×Q16—24no—
61C2×C3×SD16—22no—
62C2×C3×SD16—24no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×#124—22no—
69C3×#124—22no—
70C3×C4×Q8—24no—
71C3×C4×Q8—24no—
72C3×C4×Q8—22no—
73C3×C4×Q8D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—22no—
85C2×C3×C42—21no—
86C2×C3×C42—22no—
87C2×C3×C42—21no—
88C2×C3×C42C2312yes—
89C2×C3×C42C2311yescharacteristic
90C3×#106—24no—
91C3×C4⋊C8—22no—
92C3×C4⋊C8—22no—
93C3×#103—24no—
94C3×#103—22no—
95C3×#103—24no—
96C3×#102—22no—
97C3×#102—24no—
98C3×#95—22no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×Q16C22×C311yescharacteristic
106C22×SD16C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×#124C22×C311yescharacteristic
109C2×C4×Q8C22×C312yes—
110C2×C4×Q8C22×C311yescharacteristic
111#475—24no—
112#464—24no—
113C2×#106C22×C312yes—
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C311yescharacteristic
116C2×#103C22×C312yes—
117C2×#103C22×C312yes—
118C2×#103C22×C311yescharacteristic
119C2×#102C22×C312yes—
120C2×#102C22×C311yescharacteristic
121#397—22no—
122#393—21no—
123#393—21no—
124#391C22×C312yes—
125#390C22×C312yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C22×C312yes—
129C4×C4⋊C4C22×C311yescharacteristic
130#376—21no—
131#376C22×C311yescharacteristic
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135#336C22×C312yes—
136C24×C3—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—42no—
144C2×C3×Q8—82no—
145C2×C3×Q8C2×D412yes—
146C2×C3×Q8#4114yes—
147C2×C3×D4—21no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C3×Q16—44no—
162C3×SD16—44no—
163C2×C3×C8—21no—
164C2×C3×C8—22no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4C2×D412yes—
178C3×C4⋊C4#4114yes—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—82no—
182C3×C22⋊C4—82no—
183C3×C42—28no—
184C3×C42—24no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—81no—
188C3×C42—82no—
189C3×C42C2×D412yes—
190C3×C42#4112yes—
191C22×Q8—21no—
192C22×Q8C23×C311yescharacteristic
193C22×Q8C23×C311yescharacteristic
194C22×D4C23×C311yescharacteristic
195C2×Q16—22no—
196C2×Q16—24no—
197C2×SD16—22no—
198C2×SD16—24no—
199C22×C8C23×C311yescharacteristic
200C22×C8C23×C311yescharacteristic
201C22×C8C23×C312yes—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204#124—22no—
205#124—22no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—22no—
209C4×Q8C3×D414yes—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C4⋊C4C23×C312yes—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42C23×C312yes—
225C2×C42C23×C311yescharacteristic
226#106—24no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229#103—24no—
230#103—22no—
231#103—24no—
232#102—22no—
233#102—24no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95C3×D411yescharacteristic
239#95C3×D411yescharacteristic
240#95C3×D412yes—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3C22×D411yescharacteristic
247C3×Q8—24no—
248C3×Q8—22no—
249C3×Q8—41no—
250C3×Q8—42no—
251C3×Q8—41no—
252C3×Q8—84no—
253C3×Q8#12414yes—
254C3×D4—41no—
255C3×D4—42no—
256C3×D4—41no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4#12112yes—
280C2×C3×C4#12112yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4112yes—
285C3×C8—41no—
286C3×C8—41no—
287C3×C8—42no—
288C24—21no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—42no—
296C2×Q8—82no—
297C2×Q8C2×C3×D412yes—
298C2×Q8C3×#4114yes—
299C2×D4—21no—
300C2×D4—22no—
301C2×D4—21no—
302C2×D4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4C2×C3×D412yes—
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C24×C311yescharacteristic
313Q16—44no—
314SD16—44no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4C2×C3×D412yes—
330C4⋊C4C3×#4114yes—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C42—28no—
336C42—24no—
337C42—22no—
338C42—22no—
339C42—81no—
340C42—82no—
341C42C2×C3×D412yes—
342C42C3×#4112yes—
343C22×C3—81no—
344C22×C3—82no—
345C22×C3—82no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3C2×#12111yescharacteristic
349C22×C3#52512yes—
350C22×C3#57712yes—
351C22×C3#57711yescharacteristic
352C22×C3#57811yescharacteristic
353C3×C4—24no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—42no—
357C3×C4—42no—
358C3×C4—44no—
359C3×C4—42no—
360C3×C4—81no—
361C3×C4—82no—
362C3×C4C2×#12411yescharacteristic
363C3×C4#52912yes—
364C3×C4#53211yescharacteristic
365C23—42no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23C22×C3×D411yescharacteristic
371Q8—24no—
372Q8—22no—
373Q8—41no—
374Q8—42no—
375Q8—41no—
376Q8—84no—
377Q8C3×#12414yes—
378D4—41no—
379D4—42no—
380D4—41no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4C3×#12112yes—
404C2×C4C3×#12112yes—
405C2×C4C22×C3×D411yescharacteristic
406C2×C4C2×C3×#4112yes—
407C2×C4C2×C3×#4111yescharacteristic
408C2×C4C2×C3×#4112yes—
409C8—41no—
410C8—41no—
411C8—42no—
412C2×C3—161no—
413C2×C3#285011yescharacteristic
414C2×C3#304311yescharacteristic
415C2×C3#304611yescharacteristic
416C2×C3#312112yes—
417C2×C3#312512yes—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—82no—
422C22—82no—
423C22C2×C3×#12111yescharacteristic
424C22C3×#52512yes—
425C22C3×#57712yes—
426C22C3×#57711yescharacteristic
427C22C3×#57811yescharacteristic
428C4—22no—
429C4—22no—
430C4—24no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—44no—
435C4—82no—
436C4—81no—
437C4C2×C3×#12411yescharacteristic
438C4C3×#52912yes—
439C4C3×#53211yescharacteristic
440C3#3303711yescharacteristic, direct product factor, Sylow
441C2—161no—
442C2C3×#285011yescharacteristic
443C2C3×#304311yescharacteristic
444C2C3×#304611yescharacteristic
445C2C3×#312112yes—
446C2C3×#312512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#475C2semidirect
4. C3×#2276C2semidirect
5. C3×#1987C2semidirect
7. C3×#1927C2semidirect
10. C3×#1878C2semidirect
11. C3×#1832C2semidirect
12. #33037C3direct (N × H)
44. C2×#475C2×C3semidirect
47. #2276C2×C3semidirect
48. #1987C2×C3semidirect
50. #1927C2×C3semidirect
53. #1878C2×C3semidirect
54. #1832C2×C3semidirect
440. C3#33037direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4752
C2×C3×#4642
C3×#26402
C3×#22762
C3×#19872
C3×#19832
C3×#19272
C3×#19262
C3×#18802
C3×#18782
C3×#18322
#330373