Finite Group Databasebeta

← C3×#61877 #10680484

Subgroups — C3×#61877 #10680484

Order 768.

1600 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3556C211yesmaximal, characteristic, complemented
2C3×#3504C211yesmaximal, characteristic
3C3×#3390C214yesmaximal, complemented
4C3×#3388C214yesmaximal, complemented
5C3×#3374C214yesmaximal, complemented
6C3×#3372C214yesmaximal, complemented
7C3×#3242C214yesmaximal, complemented
8C3×#3236C212yesmaximal, complemented
9C3×#3120C214yesmaximal
10C3×#3116C212yesmaximal
11C3×#2987C211yesmaximal, characteristic, complemented
12#61877C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#564—21no—
14C3×#564—21no—
15C3×#557—22no—
16C3×#557C2214yes—
17C3×#557C2214yes—
18C3×#557C2214yes—
19C3×#554—24no—
20C3×#554C2214yes—
21C3×#544—22no—
22C3×#544C2212yes—
23C3×#541—24no—
24C3×#541C2214yes—
25C3×#533—21no—
26C3×#533C2212yes—
27C3×#533C2211yescharacteristic
28C2×C3×#125—21no—
29C2×C3×#125C2211yescharacteristic
30C3×#520C2212yes—
31C3×C4×#41C2211yescharacteristic
32C3×#501C2214yes—
33C3×#501C2214yes—
34C3×#499C2214yes—
35C3×#491C2214yes—
36C3×#491C2214yes—
37C3×#487C2214yes—
38C3×#485C2214yes—
39C3×#484C2214yes—
40C3×#483C2214yes—
41C3×#479C2214yes—
42C3×#479C2214yes—
43C3×#479C2214yes—
44C3×#479C2214yes—
45C3×#479C2214yes—
46C3×#478C2214yes—
47C3×#477C2212yes—
48C3×#477C2214yes—
49C3×#476C2212yes—
50C3×#475C2214yes—
51C3×#462C2212yes—
52C3×#462C2214yes—
53C3×#461C2212yes—
54C3×#461C2214yes—
55C3×#452C2214yes—
56C3×#446C2214yes—
57C3×#446C2214yes—
58C3×#441C2214yes—
59C3×#441C2214yes—
60C3×#440C2214yes—
61C3×#440C2214yes—
62C3×C4×Q16C2214yes—
63C3×C4×SD16C2214yes—
64C3×#433C2214yes—
65C3×#432C2212yes—
66#3556C2×C311yescharacteristic, complemented
67#3504C2×C311yescharacteristic
68#3390C2×C314yescomplemented
69#3388C2×C314yescomplemented
70#3374C2×C314yescomplemented
71#3372C2×C314yescomplemented
72#3242C2×C314yescomplemented
73#3236C2×C312yescomplemented
74#3120C2×C314yes—
75#3116C2×C312yes—
76#2987C2×C311yescharacteristic, complemented
77C2×C3×#41C2311yescharacteristic
78C2×C3×Q16C2314yes—
79C2×C3×SD16C2314yes—
80C3×C4⋊Q8—24no—
81C3×C4⋊Q8—24no—
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8D412yes—
84C3×C4⋊Q8C2312yes—
85C3×C4⋊Q8C2314yes—
86C3×C4⋊Q8C2314yes—
87C3×C4⋊Q8C2312yes—
88C3×#126—24no—
89C3×#126—24no—
90C3×#125—24no—
91C3×#125—24no—
92C3×#125—24no—
93C3×#125—24no—
94C3×#125—24no—
95C3×#125—24no—
96C3×#125—22no—
97C3×#125—24no—
98C3×#125—24no—
99C3×#125—24no—
100C3×#125—22no—
101C3×#125—21no—
102C3×#125—21no—
103C3×#125D411yescharacteristic
104C3×#125D411yescharacteristic
105C3×#125C2314yes—
106C3×#125C2314yes—
107C3×#125C2314yes—
108C3×C22⋊Q8—24no—
109C3×C22⋊Q8—22no—
110C3×C22⋊Q8—24no—
111C3×C22⋊Q8—24no—
112C3×C22⋊Q8—24no—
113C3×C22⋊Q8D414yes—
114C3×C22⋊Q8C2314yes—
115C3×C4×Q8—24no—
116C3×C4×Q8—24no—
117C3×C4×Q8—24no—
118C3×C4×Q8—22no—
119C3×C4×Q8C2314yes—
120C3×C4×Q8C2314yes—
121C3×C4×Q8C2314yes—
122C3×C4×Q8C2314yes—
123C3×C4×Q8C2312yes—
124C3×C4×Q8C2314yes—
125C3×C4×D4—22no—
126C3×C4×D4—24no—
127C3×C4×D4C2314yes—
128C3×C4×D4C2312yes—
129C3×#117—22no—
130C3×#117—22no—
131C3×#117—21no—
132C3×#117C2311yescharacteristic
133C3×#117C2312yes—
134C2×C3×C4⋊C4—22no—
135C2×C3×C4⋊C4—22no—
136C2×C3×C4⋊C4—24no—
137C2×C3×C4⋊C4C2312yes—
138C2×C3×C42—21no—
139C2×C3×C42C2311yescharacteristic
140C3×#107C2314yes—
141C3×#107C2314yes—
142C3×#107C2314yes—
143C3×#106C2314yes—
144C3×#106C2314yes—
145C3×#106C2314yes—
146C3×C4⋊C8C2312yes—
147C3×C4⋊C8C2312yes—
148C3×C4⋊C8C2314yes—
149C3×C4⋊C8C2314yes—
150C3×#103C2314yes—
151C3×#103C2314yes—
152C3×#103C2314yes—
153C3×#103C2314yes—
154C3×#103C2314yes—
155C3×#103C2314yes—
156C3×#103C2314yes—
157C3×#102C2314yes—
158C3×C22⋊C8C2314yes—
159C3×#97C2314yes—
160C3×#97C2314yes—
161C3×C4×C8C2314yes—
162#564—21no—
163#564—21no—
164#557—22no—
165#557C22×C314yes—
166#557C22×C314yes—
167#557C22×C314yes—
168#554—24no—
169#554C22×C314yes—
170#544—22no—
171#544C22×C312yes—
172#541—24no—
173#541C22×C314yes—
174#533—21no—
175#533C22×C312yes—
176#533C22×C311yescharacteristic
177C2×#125—21no—
178C2×#125C22×C311yescharacteristic
179#520C22×C312yes—
180C4×#41C22×C311yescharacteristic
181#501C22×C314yes—
182#501C22×C314yes—
183#499C22×C314yes—
184#491C22×C314yes—
185#491C22×C314yes—
186#487C22×C314yes—
187#485C22×C314yes—
188#484C22×C314yes—
189#483C22×C314yes—
190#479C22×C314yes—
191#479C22×C314yes—
192#479C22×C314yes—
193#479C22×C314yes—
194#479C22×C314yes—
195#478C22×C314yes—
196#477C22×C312yes—
197#477C22×C314yes—
198#476C22×C312yes—
199#475C22×C314yes—
200#462C22×C312yes—
201#462C22×C314yes—
202#461C22×C312yes—
203#461C22×C314yes—
204#452C22×C314yes—
205#446C22×C314yes—
206#446C22×C314yes—
207#441C22×C314yes—
208#441C22×C314yes—
209#440C22×C314yes—
210#440C22×C314yes—
211C4×Q16C22×C314yes—
212C4×SD16C22×C314yes—
213#433C22×C314yes—
214#432C22×C312yes—
215C3×#41—22no—
216C3×#41—22no—
217C2×C3×Q8—22no—
218C2×C3×Q8—24no—
219C2×C3×Q8C2×D414yes—
220C2×C3×Q8C2414yes—
221C2×C3×Q8C2411yescharacteristic
222C2×C3×D4—21no—
223C2×C3×D4C2411yescharacteristic
224C22×C3×C4—21no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—24no—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C3×Q16—28no—
231C3×SD16—28no—
232C2×C3×C8C2414yes—
233C2×C3×C8C2414yes—
234C3×C4⋊C4—24no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4C2×D412yes—
254C3×C4⋊C4C2×D414yes—
255C3×C4⋊C4C2×D412yes—
256C3×C4⋊C4C2×D414yes—
257C3×C4⋊C4C2×D414yes—
258C3×C4⋊C4C2414yes—
259C3×C4⋊C4C2414yes—
260C3×C4⋊C4C2412yes—
261C3×C22⋊C4—24no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—22no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4C2×D414yes—
266C3×C42—21no—
267C3×C42—22no—
268C3×C42—24no—
269C3×C42—24no—
270C3×C42—24no—
271C3×C42—24no—
272C3×C42C2×D411yescharacteristic
273C3×C42C2×D411yescharacteristic
274C3×C42C2414yes—
275C3×C42C2412yes—
276C2×#41C23×C311yescharacteristic
277C2×Q16C23×C314yes—
278C2×SD16C23×C314yes—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8C3×D412yes—
283C4⋊Q8C23×C312yes—
284C4⋊Q8C23×C314yes—
285C4⋊Q8C23×C312yes—
286C4⋊Q8C23×C314yes—
287#126—24no—
288#126—24no—
289#125—22no—
290#125—24no—
291#125—24no—
292#125—24no—
293#125—24no—
294#125—24no—
295#125—24no—
296#125—24no—
297#125—24no—
298#125—24no—
299#125—21no—
300#125—21no—
301#125—22no—
302#125C3×D411yescharacteristic
303#125C3×D411yescharacteristic
304#125C23×C314yes—
305#125C23×C314yes—
306#125C23×C314yes—
307C22⋊Q8—24no—
308C22⋊Q8—22no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—24no—
312C22⋊Q8C3×D414yes—
313C22⋊Q8C23×C314yes—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—22no—
318C4×Q8C23×C312yes—
319C4×Q8C23×C314yes—
320C4×Q8C23×C314yes—
321C4×Q8C23×C314yes—
322C4×Q8C23×C314yes—
323C4×Q8C23×C314yes—
324C4×D4—22no—
325C4×D4—24no—
326C4×D4C23×C312yes—
327C4×D4C23×C314yes—
328#117—22no—
329#117—22no—
330#117—21no—
331#117C23×C311yescharacteristic
332#117C23×C312yes—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4C23×C312yes—
337C2×C42—21no—
338C2×C42C23×C311yescharacteristic
339#107C23×C314yes—
340#107C23×C314yes—
341#107C23×C314yes—
342#106C23×C314yes—
343#106C23×C314yes—
344#106C23×C314yes—
345C4⋊C8C23×C312yes—
346C4⋊C8C23×C312yes—
347C4⋊C8C23×C314yes—
348C4⋊C8C23×C314yes—
349#103C23×C314yes—
350#103C23×C314yes—
351#103C23×C314yes—
352#103C23×C314yes—
353#103C23×C314yes—
354#103C23×C314yes—
355#103C23×C314yes—
356#102C23×C314yes—
357C22⋊C8C23×C314yes—
358#97C23×C314yes—
359#97C23×C314yes—
360C4×C8C23×C314yes—
361C23×C3—21no—
362C23×C3C22×D411yescharacteristic
363C3×Q8—21no—
364C3×Q8—24no—
365C3×Q8—24no—
366C3×Q8—44no—
367C3×Q8—44no—
368C3×Q8—84no—
369C3×Q82−1+412yes—
370C3×D4—24no—
371C3×D4—21no—
372C3×D42−1+412yes—
373C2×C3×C4—24no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—42no—
382C2×C3×C4—82no—
383C2×C3×C4—84no—
384C2×C3×C4—82no—
385C2×C3×C4C22×D412yes—
386C2×C3×C4C22×D414yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D414yes—
390C2×C3×C4C2511yescharacteristic
391C3×C8—24no—
392C3×C8—24no—
393#41—22no—
394#41—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8C2×C3×D414yes—
398C2×Q8C24×C314yes—
399C2×Q8C24×C311yescharacteristic
400C2×D4—21no—
401C2×D4C24×C311yescharacteristic
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4C2×C3×D412yes—
407C22×C4C24×C311yescharacteristic
408Q16—28no—
409SD16—28no—
410C2×C8C24×C314yes—
411C2×C8C24×C314yes—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4C2×C3×D412yes—
432C4⋊C4C2×C3×D414yes—
433C4⋊C4C2×C3×D412yes—
434C4⋊C4C2×C3×D414yes—
435C4⋊C4C2×C3×D414yes—
436C4⋊C4C24×C314yes—
437C4⋊C4C24×C314yes—
438C4⋊C4C24×C312yes—
439C22⋊C4—24no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4C2×C3×D414yes—
444C42—21no—
445C42—22no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42C2×C3×D411yescharacteristic
451C42C2×C3×D411yescharacteristic
452C42C24×C314yes—
453C42C24×C312yes—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—22no—
457C22×C3—42no—
458C22×C3C23×D411yescharacteristic
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—24no—
462C3×C4—24no—
463C3×C4—21no—
464C3×C4—22no—
465C3×C4—44no—
466C3×C4—42no—
467C3×C4—44no—
468C3×C4—44no—
469C3×C4C2×2−1+411yescharacteristic
470C3×C4C2×2−1+411yescharacteristic
471C23—21no—
472C23C22×C3×D411yescharacteristic
473Q8—21no—
474Q8—24no—
475Q8—24no—
476Q8—44no—
477Q8—44no—
478Q8—84no—
479Q8C3×2−1+412yes—
480D4—24no—
481D4—21no—
482D4C3×2−1+412yes—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—82no—
493C2×C4—84no—
494C2×C4—82no—
495C2×C4C22×C3×D412yes—
496C2×C4C22×C3×D414yes—
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C22×C3×D412yes—
499C2×C4C22×C3×D414yes—
500C2×C4C25×C311yescharacteristic
501C8—24no—
502C8—24no—
503C2×C3—41no—
504C2×C3—42no—
505C2×C3#348611yescharacteristic
506C2×C3#358612yes—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—42no—
511C22C23×C3×D411yescharacteristic
512C4—21no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—24no—
517C4—24no—
518C4—44no—
519C4—44no—
520C4—42no—
521C4—44no—
522C4C2×C3×2−1+411yescharacteristic
523C4C2×C3×2−1+411yescharacteristic
524C3#6187711yescharacteristic, direct product factor, Sylow
525C2—41no—
526C2—42no—
527C2C3×#348611yescharacteristic
528C2C3×#358612yes—
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3556C2semidirect
3. C3×#3390C2semidirect
4. C3×#3388C2semidirect
5. C3×#3374C2semidirect
6. C3×#3372C2semidirect
7. C3×#3242C2semidirect
8. C3×#3236C2semidirect
11. C3×#2987C2semidirect
12. #61877C3direct (N × H)
66. #3556C2×C3semidirect
68. #3390C2×C3semidirect
69. #3388C2×C3semidirect
70. #3374C2×C3semidirect
71. #3372C2×C3semidirect
72. #3242C2×C3semidirect
73. #3236C2×C3semidirect
76. #2987C2×C3semidirect
524. C3#61877direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35562
C3×#35042
C3×#33902
C3×#33882
C3×#33742
C3×#33722
C3×#32422
C3×#32362
C3×#31202
C3×#31162
C3×#29872
#618773