Finite Group Databasebeta

← C3×#25621 #10644228

Subgroups — C3×#25621 #10644228

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic
2C3×#3152C211yesmaximal, characteristic, complemented
3C3×#3139C211yesmaximal, characteristic, complemented
4C3×#2547C211yesmaximal, characteristic, complemented
5C3×#2004C214yesmaximal, complemented
6C3×#1986C211yesmaximal, characteristic, complemented
7C3×#1980C211yesmaximal, characteristic, complemented
8C3×#1882C214yesmaximal, complemented
9C3×C4×#100C211yesmaximal, characteristic, complemented
10#25621C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137C2214yes—
12C3×#559—22no—
13C3×#556—21no—
14C3×#555—21no—
15C3×#553—22no—
16C3×#550—21no—
17C3×#547—21no—
18C3×#538—22no—
19C3×#535C2212yes—
20C3×#533C2211yescharacteristic
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×#124C2212yes—
24C3×#525—22no—
25C3×#520C2212yes—
26C2×C3×C4×D4C2211yescharacteristic
27C3×#497—22no—
28C3×#494—21no—
29C3×#492—21no—
30C3×#491—22no—
31C3×#487—21no—
32C3×#487—21no—
33C2×C3×#104C2214yes—
34C3×#419C2214yes—
35C2×C3×#100C2212yes—
36C3×C4×M16C2211yescharacteristic
37C3×C4×M16C2211yescharacteristic
38C3×#397—22no—
39C3×#391C2212yes—
40C3×#380—22no—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#343C2212yes—
43C3×#339C2214yes—
44#3506C2×C311yescharacteristic
45#3152C2×C311yescharacteristic, complemented
46#3139C2×C311yescharacteristic, complemented
47#2547C2×C311yescharacteristic, complemented
48#2004C2×C314yescomplemented
49#1986C2×C311yescharacteristic, complemented
50#1980C2×C311yescharacteristic, complemented
51#1882C2×C314yescomplemented
52C4×#100C2×C311yescharacteristic, complemented
53C2×C3×#41—22no—
54C2×C3×#41C2312yes—
55C22×C3×Q8C2312yes—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C3×#137—28no—
59C3×#137—44no—
60C2×C3×Q16—24no—
61C2×C3×SD16—24no—
62C2×C3×M16C2312yes—
63C2×C3×M16C2312yes—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×C4⋊Q8D411yescharacteristic
71C3×#127—22no—
72C3×#127D411yescharacteristic
73C3×#126—24no—
74C3×#125D412yes—
75C3×#124—24no—
76C3×#124—22no—
77C3×#124—44no—
78C3×#124D412yes—
79C3×#123—24no—
80C3×#123—24no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—21no—
84C3×#121—22no—
85C3×#121—24no—
86C3×#121—21no—
87C3×#121—21no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#120—24no—
91C3×C4×Q8—22no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×D4—24no—
95C3×C4×D4—22no—
96C3×C4×D4—24no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×#117—22no—
104C3×#117—22no—
105C3×#117C2312yes—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—24no—
112C2×C3×C22⋊C4D412yes—
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115C3×#104—44no—
116C3×#103—24no—
117C3×#103—24no—
118C3×#100—24no—
119C3×#97—21no—
120C3×#97—21no—
121C3×C4×C8—21no—
122C3×C4×C8—21no—
123C3×#95—21no—
124C3×#95—24no—
125C2×#137C22×C314yes—
126#559—22no—
127#556—21no—
128#555—21no—
129#553—22no—
130#550—21no—
131#547—21no—
132#538—22no—
133#535C22×C312yes—
134#533C22×C311yescharacteristic
135#532C22×C311yescharacteristic
136C2×C4⋊Q8C22×C311yescharacteristic
137C2×#124C22×C312yes—
138#525—22no—
139#520C22×C312yes—
140C2×C4×D4C22×C311yescharacteristic
141#497—22no—
142#494—21no—
143#492—21no—
144#491—22no—
145#487—21no—
146#487—21no—
147C2×#104C22×C314yes—
148#419C22×C314yes—
149C2×#100C22×C312yes—
150C4×M16C22×C311yescharacteristic
151C4×M16C22×C311yescharacteristic
152#397—22no—
153#391C22×C312yes—
154#380—22no—
155C4×C4⋊C4C22×C311yescharacteristic
156#343C22×C312yes—
157#339C22×C314yes—
158C24×C3C2×D412yes—
159C3×#41—24no—
160C3×#41—24no—
161C3×#41—84no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—24no—
167C2×C3×Q8—24no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4C2×D412yes—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—44no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C3×Q16—48no—
191C3×SD16—48no—
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16#4114yescomplemented
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—41no—
209C3×C4⋊C4—41no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4—41no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—48no—
223C3×C42—24no—
224C3×C42—22no—
225C3×C42—24no—
226C3×C42C2×D411yescharacteristic
227C3×C42C2×D411yescharacteristic
228C3×C42C2×D411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C2×#41—22no—
231C2×#41C23×C312yes—
232C22×Q8C23×C312yes—
233C22×D4C23×C311yescharacteristic
234C23×C4C3×D412yes—
235#137—28no—
236#137—44no—
237C2×Q16—24no—
238C2×SD16—24no—
239C2×M16C23×C312yes—
240C2×M16C23×C312yes—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8C3×D411yescharacteristic
246C4⋊Q8C3×D411yescharacteristic
247C4⋊Q8C3×D411yescharacteristic
248#127—22no—
249#127C3×D411yescharacteristic
250#126—24no—
251#125C3×D412yes—
252#124—24no—
253#124—22no—
254#124—44no—
255#124C3×D412yes—
256#123—24no—
257#123—24no—
258C22⋊Q8—22no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261#121—22no—
262#121—24no—
263#121—21no—
264#121—21no—
265#121—22no—
266#121—22no—
267#120—24no—
268C4×Q8—22no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×D4—24no—
272C4×D4—22no—
273C4×D4—24no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280#117—22no—
281#117—22no—
282#117C23×C312yes—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—24no—
289C2×C22⋊C4C3×D412yes—
290C2×C42C23×C312yes—
291C2×C42C23×C311yescharacteristic
292#104—44no—
293#103—24no—
294#103—24no—
295#100—24no—
296#97—21no—
297#97—21no—
298C4×C8—21no—
299C4×C8—21no—
300#95—21no—
301#95—24no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3C22×D411yescharacteristic
309C3×Q8—24no—
310C3×Q8—42no—
311C3×Q8—44no—
312C3×Q8—44no—
313C3×Q8—81no—
314C3×D4—24no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—81no—
321C3×D4—84no—
322C3×D4—84no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—41no—
329C2×C3×C4—44no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—82no—
339C2×C3×C4—84no—
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C2×#4112yes—
347C2×C3×C4H(5,𝔽2)12yes—
348C3×C8—24no—
349C3×C8—42no—
350C24C2×C3×D412yes—
351#41—24no—
352#41—24no—
353#41—84no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8C2×C3×D412yes—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4C2×C3×D412yes—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—44no—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D412yes—
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C24×C311yescharacteristic
382Q16—48no—
383SD16—48no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16C3×#4114yescomplemented
388C2×C8—22no—
389C2×C8—22no—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4C2×C3×D412yes—
405C4⋊C4C2×C3×D412yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—48no—
415C42—24no—
416C42—22no—
417C42—24no—
418C42C2×C3×D411yescharacteristic
419C42C2×C3×D411yescharacteristic
420C42C2×C3×D411yescharacteristic
421C42C2×C3×D411yescharacteristic
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—44no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—82no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#53211yescharacteristic
434C22×C3#53511yescharacteristic
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—82no—
443C3×C4—82no—
444C3×C4#45512yes—
445C3×C4#54612yes—
446C3×C4#55012yes—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—42no—
452C23—42no—
453C23C22×C3×D411yescharacteristic
454Q8—24no—
455Q8—42no—
456Q8—44no—
457Q8—44no—
458Q8—81no—
459D4—24no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—81no—
466D4—84no—
467D4—84no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4C3×#12011yescharacteristic
486C2×C4C3×#12011yescharacteristic
487C2×C4C3×#12011yescharacteristic
488C2×C4C3×#12011yescharacteristic
489C2×C4C22×C3×D411yescharacteristic
490C2×C4C22×C3×D411yescharacteristic
491C2×C4C2×C3×#4112yes—
492C2×C4C3×H(5,𝔽2)12yes—
493C8—24no—
494C8—42no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—82no—
498C2×C3—82no—
499C2×C3#243111yescharacteristic
500C2×C3C2×#45511yescharacteristic
501C2×C3#302211yescharacteristic
502C22—21no—
503C22—21no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—82no—
512C22C2×C3×#12011yescharacteristic
513C22C3×#53211yescharacteristic
514C22C3×#53511yescharacteristic
515C4—21no—
516C4—21no—
517C4—21no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—82no—
524C4C3×#45512yes—
525C4C3×#54612yes—
526C4C3×#55012yes—
527C3#2562111yescharacteristic, direct product factor, Sylow
528C2—21no—
529C2—21no—
530C2—82no—
531C2—82no—
532C2C3×#243111yescharacteristic
533C2C2×C3×#45511yescharacteristic
534C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3506C2
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×#3152C2semidirect
3. C3×#3139C2semidirect
4. C3×#2547C2semidirect
5. C3×#2004C2semidirect
6. C3×#1986C2semidirect
7. C3×#1980C2semidirect
8. C3×#1882C2semidirect
9. C3×C4×#100C2semidirect
10. #25621C3direct (N × H)
45. #3152C2×C3semidirect
46. #3139C2×C3semidirect
47. #2547C2×C3semidirect
48. #2004C2×C3semidirect
49. #1986C2×C3semidirect
50. #1980C2×C3semidirect
51. #1882C2×C3semidirect
52. C4×#100C2×C3semidirect
195. C3×M16#41semidirect
387. M16C3×#41semidirect
527. C3#25621direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#31522
C3×#31392
C3×#25472
C3×#20042
C3×#19862
C3×#19802
C3×#18822
C3×C4×#1002
#256213