Finite Group Databasebeta

← C3×#29834 #10648441

Subgroups — C3×#29834 #10648441

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#478C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic
3C3×#2641C211yesmaximal, characteristic, complemented
4C3×#2319C211yesmaximal, characteristic, complemented
5C3×#2067C212yesmaximal, complemented
6C3×#2065C212yesmaximal, complemented
7C3×#2054C212yesmaximal, complemented
8C3×#2054C212yesmaximal, complemented
9C3×#1989C211yesmaximal, characteristic, complemented
10C3×#1982C211yesmaximal, characteristic, complemented
11C3×#1843C211yesmaximal, characteristic, complemented
12#29834C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×#478—24no—
18C3×#459—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C3×#394—21no—
31C3×#394—21no—
32C3×#394—22no—
33C3×#390C2212yes—
34C3×#390C2212yes—
35C3×#390C2212yes—
36C3×#389C2212yes—
37C3×#388—22no—
38C3×#381—21no—
39C3×#380—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#478C2×C311yescharacteristic, complemented
45C2×#459C2×C311yescharacteristic
46#2641C2×C311yescharacteristic, complemented
47#2319C2×C311yescharacteristic, complemented
48#2067C2×C312yescomplemented
49#2065C2×C312yescomplemented
50#2054C2×C312yescomplemented
51#2054C2×C312yescomplemented
52#1989C2×C311yescharacteristic, complemented
53#1982C2×C311yescharacteristic, complemented
54#1843C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C2×C3×D8—24no—
61C2×C3×D8—42no—
62C2×C3×D8—42no—
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×#127—22no—
67C3×#127—22no—
68C3×#125—22no—
69C3×#125—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—21no—
87C2×C3×C42C2312yes—
88C2×C3×C42C2311yescharacteristic
89C3×#107—22no—
90C3×#106—22no—
91C3×C4⋊C8—22no—
92C3×C4⋊C8—22no—
93C3×C4⋊C8—42no—
94C3×C4⋊C8—42no—
95C3×#102—22no—
96C3×#102—22no—
97C3×#102—24no—
98C3×#102—42no—
99C3×#102—42no—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×D8C22×C312yes—
106C2×#127C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×C4×D4C22×C311yescharacteristic
109#478—24no—
110#459—24no—
111C2×#107C22×C311yescharacteristic
112C2×#106C22×C311yescharacteristic
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#102C22×C311yescharacteristic
118C2×#102C22×C311yescharacteristic
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×#102C22×C312yes—
122#394—21no—
123#394—21no—
124#394—22no—
125#390C22×C312yes—
126#390C22×C312yes—
127#390C22×C312yes—
128#389C22×C312yes—
129#388—22no—
130#381—21no—
131#380—21no—
132C4×C4⋊C4C22×C311yescharacteristic
133#376C22×C311yescharacteristic
134#340C22×C312yes—
135#340C22×C312yes—
136C24×C3—21no—
137C24×C3—22no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—81no—
150C2×C3×D4—82no—
151C2×C3×D4—81no—
152C2×C3×D4C2×D412yes—
153C22×C3×C4—21no—
154C22×C3×C4—24no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—82no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C3×D8—48no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—82no—
182C3×C4⋊C4C2×D412yes—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—82no—
185C3×C22⋊C4—82no—
186C3×C22⋊C4—82no—
187C3×C22⋊C4—82no—
188C3×C22⋊C4—82no—
189C3×C22⋊C4—82no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42—81no—
193C3×C42C2×D412yes—
194C3×C42#4112yes—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4C23×C312yes—
198C22×D4C23×C311yescharacteristic
199C23×C4—21no—
200C2×D8—24no—
201C2×D8—42no—
202C2×D8—42no—
203C22×C8C23×C312yes—
204C22×C8C23×C311yescharacteristic
205C22×C8C23×C311yescharacteristic
206#127—22no—
207#127—22no—
208#125—22no—
209#125—22no—
210C4×D4—22no—
211C4×D4C3×D414yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C42—21no—
227C2×C42C23×C312yes—
228C2×C42C23×C311yescharacteristic
229#107—22no—
230#106—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235#102—22no—
236#102—22no—
237#102—24no—
238#102—42no—
239#102—42no—
240#95—21no—
241#95—22no—
242#95C3×D411yescharacteristic
243#95C3×D411yescharacteristic
244#95C3×D412yes—
245C23×C3—22no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3C22×D411yescharacteristic
258C3×D4—22no—
259C3×D4—22no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—42no—
263C3×D4—42no—
264C3×D4—42no—
265C3×D4—84no—
266C3×D4—84no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—84no—
287C2×C3×C4#12112yes—
288C2×C3×C4#12112yes—
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4H(5,𝔽2)12yes—
292C2×C3×C42−1+412yes—
293C3×C8—41no—
294C3×C8—41no—
295C3×C8—42no—
296C24—21no—
297C24—22no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—81no—
310C2×D4—82no—
311C2×D4—81no—
312C2×D4C2×C3×D412yes—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—82no—
319C22×C4C2×C3×D412yes—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C24×C311yescharacteristic
324D8—48no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—82no—
342C4⋊C4C2×C3×D412yes—
343C22⋊C4—24no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C42—22no—
351C42—22no—
352C42—81no—
353C42C2×C3×D412yes—
354C42C3×#4112yes—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3C2×#12111yescharacteristic
368C22×C3#53712yes—
369C22×C3#57612yes—
370C22×C3#57611yescharacteristic
371C22×C3#57711yescharacteristic
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—42no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—82no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4#53511yescharacteristic
381C3×C4#53611yescharacteristic
382C3×C4#54212yes—
383C23—22no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23C22×C3×D411yescharacteristic
396D4—22no—
397D4—22no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—84no—
404D4—84no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—84no—
425C2×C4C3×#12112yes—
426C2×C4C3×#12112yes—
427C2×C4C22×C3×D411yescharacteristic
428C2×C4C2×C3×#4111yescharacteristic
429C2×C4C3×H(5,𝔽2)12yes—
430C2×C4C3×2−1+412yes—
431C8—42no—
432C8—41no—
433C8—41no—
434C2×C3—82no—
435C2×C3—162no—
436C2×C3#262011yescharacteristic
437C2×C3#304111yescharacteristic
438C2×C3#304511yescharacteristic
439C2×C3#325712yes—
440C2×C3#325912yes—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22C2×C3×#12111yescharacteristic
454C22C3×#53712yes—
455C22C3×#57612yes—
456C22C3×#57611yescharacteristic
457C22C3×#57711yescharacteristic
458C4—22no—
459C4—22no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4—81no—
465C4—81no—
466C4C3×#53511yescharacteristic
467C4C3×#53611yescharacteristic
468C4C3×#54212yes—
469C3#2983411yescharacteristic, direct product factor, Sylow
470C2—82no—
471C2—162no—
472C2C3×#262011yescharacteristic
473C2C3×#304111yescharacteristic
474C2C3×#304511yescharacteristic
475C2C3×#325712yes—
476C2C3×#325912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#459C2
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×#478C2semidirect
3. C3×#2641C2semidirect
4. C3×#2319C2semidirect
5. C3×#2067C2semidirect
6. C3×#2065C2semidirect
7. C3×#2054C2semidirect
8. C3×#2054C2semidirect
9. C3×#1989C2semidirect
10. C3×#1982C2semidirect
11. C3×#1843C2semidirect
12. #29834C3direct (N × H)
44. C2×#478C2×C3semidirect
46. #2641C2×C3semidirect
47. #2319C2×C3semidirect
48. #2067C2×C3semidirect
49. #2065C2×C3semidirect
50. #2054C2×C3semidirect
51. #2054C2×C3semidirect
52. #1989C2×C3semidirect
53. #1982C2×C3semidirect
54. #1843C2×C3semidirect
469. C3#29834direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4782
C2×C3×#4592
C3×#26412
C3×#23192
C3×#20672
C3×#20652
C3×#20542
C3×#19892
C3×#19822
C3×#18432
#298343