Finite Group Databasebeta

← C3×#32974 #10651581

Subgroups — C3×#32974 #10651581

Order 768.

1186 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic
2C3×#3080C212yesmaximal, complemented
3C3×#2565C211yesmaximal, characteristic, complemented
4C3×#1860C212yesmaximal, complemented
5C3×#1859C212yesmaximal, complemented
6C3×#1857C211yesmaximal, characteristic, complemented
7C3×#1857C211yesmaximal, characteristic, complemented
8C3×#1806C212yesmaximal, complemented
9C3×#1801C211yesmaximal, characteristic, complemented
10C3×#1793C211yesmaximal, characteristic
11C3×#1793C211yesmaximal, characteristic
12#32974C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C3×#520—22no—
16C3×#520—22no—
17C3×#520C2212yes—
18C3×#519—22no—
19C3×#519C2211yescharacteristic
20C3×C4×#41—22no—
21C3×C4×#41—22no—
22C3×C4×#41C2212yes—
23C2×C3×C4×Q8—22no—
24C2×C3×C4×Q8C2211yescharacteristic
25C3×#479—22no—
26C3×#479—22no—
27C3×#478—22no—
28C3×#478—22no—
29C3×#428C2212yes—
30C3×#428C2212yes—
31C3×#423C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#417C2212yes—
35C3×#417C2212yes—
36C2×C3×#101C2211yescharacteristic
37C2×C3×#101C2211yescharacteristic
38C3×#398—22no—
39C3×#391C2211yescharacteristic
40C3×#391C2211yescharacteristic
41C3×#387—22no—
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#343C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48C4×2−1+4C2×C311yescharacteristic
49#3080C2×C312yescomplemented
50#2565C2×C311yescharacteristic, complemented
51#1860C2×C312yescomplemented
52#1859C2×C312yescomplemented
53#1857C2×C311yescharacteristic, complemented
54#1857C2×C311yescharacteristic, complemented
55#1806C2×C312yescomplemented
56#1801C2×C311yescharacteristic, complemented
57#1793C2×C311yescharacteristic
58#1793C2×C311yescharacteristic
59C3×2−1+4—24no—
60C3×2−1+4—22no—
61C3×2−1+4Q814yes—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41C2312yes—
65C22×C3×Q8—22no—
66C22×C3×Q8C2311yescharacteristic
67C2×C3×M16C2312yes—
68C2×C3×M16C2312yes—
69C3×#125—22no—
70C3×#125—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—24no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8D412yes—
80C3×C4×Q8D412yes—
81C3×C4×D4—22no—
82C3×C4×D4—24no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—24no—
87C3×C4×D4—22no—
88C3×C4×D4D412yes—
89C3×C4×D4D412yes—
90C3×#117—24no—
91C3×#117—24no—
92C3×#117—22no—
93C3×#117—22no—
94C3×#117—22no—
95C3×#117C2312yes—
96C2×C3×C4⋊C4—24no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D412yes—
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C42—24no—
107C2×C3×C42—22no—
108C2×C3×C42D412yes—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#107—22no—
113C3×#107—22no—
114C3×#106—22no—
115C3×#106—22no—
116C3×C4⋊C8—22no—
117C3×C4⋊C8—22no—
118C3×#104—24no—
119C3×#104—24no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#101—22no—
125C3×#101—22no—
126C3×#95—21no—
127C3×#95—22no—
128C3×#95—22no—
129C3×#95—22no—
130C2×2−1+4C22×C311yescharacteristic
131C2×#125C22×C311yescharacteristic
132#520—22no—
133#520—22no—
134#520C22×C312yes—
135#519—22no—
136#519C22×C311yescharacteristic
137C4×#41—22no—
138C4×#41—22no—
139C4×#41C22×C312yes—
140C2×C4×Q8—22no—
141C2×C4×Q8C22×C311yescharacteristic
142#479—22no—
143#479—22no—
144#478—22no—
145#478—22no—
146#428C22×C312yes—
147#428C22×C312yes—
148#423C22×C312yes—
149C2×#104C22×C312yes—
150C2×#104C22×C312yes—
151#417C22×C312yes—
152#417C22×C312yes—
153C2×#101C22×C311yescharacteristic
154C2×#101C22×C311yescharacteristic
155#398—22no—
156#391C22×C311yescharacteristic
157#391C22×C311yescharacteristic
158#387—22no—
159#384C22×C311yescharacteristic
160#382C22×C311yescharacteristic
161C4×C4⋊C4C22×C311yescharacteristic
162#343C22×C312yes—
163#339C22×C312yes—
164#339C22×C312yes—
165C3×#41—28no—
166C3×#41—28no—
167C3×#41—28no—
168C3×#41—28no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41C2×Q814yes—
172C3×#41#4114yes—
173C2×C3×Q8—22no—
174C2×C3×Q8—28no—
175C2×C3×Q8—28no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8C2×Q812yes—
180C2×C3×Q8#4112yes—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4C2×D412yes—
184C22×C3×C4—24no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—22no—
201C3×M16—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—44no—
215C3×C4⋊C4—41no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4C2×D412yes—
219C3×C4⋊C4C2×D412yes—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C42—24no—
225C3×C42—24no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—22no—
233C3×C42C2×D412yes—
234C3×C42C2×D412yes—
2352−1+4—24no—
2362−1+4—22no—
2372−1+4C3×Q814yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41C23×C312yes—
241C22×Q8—22no—
242C22×Q8C23×C311yescharacteristic
243C2×M16C23×C312yes—
244C2×M16C23×C312yes—
245#125—22no—
246#125—22no—
247C4×Q8—22no—
248C4×Q8—24no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8C3×D412yes—
256C4×Q8C3×D412yes—
257C4×D4—22no—
258C4×D4—24no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—24no—
263C4×D4—22no—
264C4×D4C3×D412yes—
265C4×D4C3×D412yes—
266#117—24no—
267#117—24no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117C23×C312yes—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C3×D412yes—
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C42—24no—
283C2×C42—22no—
284C2×C42C3×D412yes—
285C2×C42C23×C311yescharacteristic
286C2×C42C23×C311yescharacteristic
287C2×C42C23×C311yescharacteristic
288#107—22no—
289#107—22no—
290#106—22no—
291#106—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294#104—24no—
295#104—24no—
296#103—22no—
297#103—22no—
298#102—22no—
299#102—22no—
300#101—22no—
301#101—22no—
302#95—21no—
303#95—22no—
304#95—22no—
305#95—22no—
306C23×C3—22no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—28no—
309C3×Q8—24no—
310C3×Q8—22no—
311C3×Q8—24no—
312C3×Q8C22⋊Q814yes—
313C3×D4—28no—
314C3×D4—24no—
315C3×D4—24no—
316C3×D4—22no—
317C3×D4C22⋊Q814yes—
318C2×C3×C4—216no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—44no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—82no—
342C2×C3×C4—84no—
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4#12011yescharacteristic
345C2×C3×C4#12012yes—
346C2×C3×C4C22⋊Q814yes—
347C2×C3×C4C22×D412yes—
348C2×C3×C4C22×Q811yescharacteristic
349C2×C3×C4C2×#4112yes—
350C2×C3×C4C2×#4111yescharacteristic
351C3×C8—42no—
352C3×C8—42no—
353#41—28no—
354#41—28no—
355#41—28no—
356#41—28no—
357#41—22no—
358#41—22no—
359#41C2×C3×Q814yes—
360#41C3×#4114yes—
361C2×Q8—22no—
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8C2×C3×D412yes—
367C2×Q8C2×C3×Q812yes—
368C2×Q8C3×#4112yes—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4C2×C3×D412yes—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×D412yes—
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C24×C311yescharacteristic
386M16—22no—
387M16—22no—
388M16—22no—
389M16—22no—
390C2×C8—22no—
391C2×C8—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4C2×C3×D412yes—
407C4⋊C4C2×C3×D412yes—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C42—24no—
413C42—24no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42C2×C3×D412yes—
422C42C2×C3×D412yes—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—24no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3C2×#12011yescharacteristic
429C22×C3C2×C22⋊Q811yescharacteristic
430C22×C3#52511yescharacteristic
431C3×C4—24no—
432C3×C4—24no—
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4C2×C22⋊Q812yes—
444C3×C4#52512yes—
445C23—22no—
446C23C22×C3×D411yescharacteristic
447Q8—28no—
448Q8—24no—
449Q8—22no—
450Q8—24no—
451Q8C3×C22⋊Q814yes—
452D4—28no—
453D4—24no—
454D4—24no—
455D4—22no—
456D4C3×C22⋊Q814yes—
457C2×C4—216no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4C3×#12011yescharacteristic
483C2×C4C3×#12011yescharacteristic
484C2×C4C3×#12012yes—
485C2×C4C3×C22⋊Q814yes—
486C2×C4C22×C3×D412yes—
487C2×C4C22×C3×Q811yescharacteristic
488C2×C4C2×C3×#4112yes—
489C2×C4C2×C3×#4111yescharacteristic
490C8—42no—
491C8—42no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—44no—
495C2×C3#284811yescharacteristic
496C2×C3#301811yescharacteristic
497C2×C3#302011yescharacteristic
498C22—21no—
499C22—21no—
500C22—24no—
501C22—42no—
502C22—42no—
503C22C2×C3×#12011yescharacteristic
504C22C2×C3×C22⋊Q811yescharacteristic
505C22C3×#52511yescharacteristic
506C4—22no—
507C4—21no—
508C4—21no—
509C4—24no—
510C4—24no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4C2×C3×C22⋊Q812yes—
519C4C3×#52512yes—
520C3#3297411yescharacteristic, direct product factor, Sylow
521C2—21no—
522C2—21no—
523C2—44no—
524C2C3×#284811yescharacteristic
525C2C3×#301811yescharacteristic
526C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3080C2semidirect
3. C3×#2565C2semidirect
4. C3×#1860C2semidirect
5. C3×#1859C2semidirect
6. C3×#1857C2semidirect
7. C3×#1857C2semidirect
8. C3×#1806C2semidirect
9. C3×#1801C2semidirect
12. #32974C3direct (N × H)
49. #3080C2×C3semidirect
50. #2565C2×C3semidirect
51. #1860C2×C3semidirect
52. #1859C2×C3semidirect
53. #1857C2×C3semidirect
54. #1857C2×C3semidirect
55. #1806C2×C3semidirect
56. #1801C2×C3semidirect
520. C3#32974direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C3×#30802
C3×#25652
C3×#18602
C3×#18592
C3×#18572
C3×#18062
C3×#18012
C3×#17932
#329743