Finite Group Databasebeta

← C3×#29590 #10648197

Subgroups — C3×#29590 #10648197

Order 768.

644 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2647C211yesmaximal, characteristic, complemented
2C3×#2319C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4C3×#2092C211yesmaximal, characteristic
5C3×#2084C211yesmaximal, characteristic, complemented
6C3×#2071C211yesmaximal, characteristic
7C3×#1944C211yesmaximal, characteristic, complemented
8C3×#1941C211yesmaximal, characteristic, complemented
9C3×#1936C211yesmaximal, characteristic
10C3×#1934C211yesmaximal, characteristic
11C3×#1919C211yesmaximal, characteristic, complemented
12C3×#1918C211yesmaximal, characteristic, complemented
13C3×#1864C211yesmaximal, characteristic
14C3×#1864C211yesmaximal, characteristic
15C3×#1746C211yesmaximal, characteristic, complemented
16#29590C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#107C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×#400—21no—
29C3×#400—21no—
30C3×#398—21no—
31C3×#398—21no—
32C3×#390C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#389C2211yescharacteristic
40C3×#388—21no—
41C3×#388—21no—
42C3×#384C2211yescharacteristic
43C3×#382C2211yescharacteristic
44C3×#381—21no—
45C3×#380—21no—
46C3×C4×C4⋊C4C2211yescharacteristic
47C3×#376C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2647C2×C311yescharacteristic, complemented
61#2319C2×C311yescharacteristic
62#2101C2×C311yescharacteristic, complemented
63#2092C2×C311yescharacteristic
64#2084C2×C311yescharacteristic, complemented
65#2071C2×C311yescharacteristic
66#1944C2×C311yescharacteristic, complemented
67#1941C2×C311yescharacteristic, complemented
68#1936C2×C311yescharacteristic
69#1934C2×C311yescharacteristic
70#1919C2×C311yescharacteristic, complemented
71#1918C2×C311yescharacteristic, complemented
72#1864C2×C311yescharacteristic
73#1864C2×C311yescharacteristic
74#1746C2×C311yescharacteristic, complemented
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C3×#107—22no—
108C3×#106—22no—
109C3×#106—22no—
110C3×#106—22no—
111C3×#102—22no—
112C3×#102—22no—
113C3×#102—22no—
114C3×#102—22no—
115C3×#97—22no—
116C3×C4×C8—22no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130C2×C4×D4C22×C311yescharacteristic
131C2×#107C22×C311yescharacteristic
132C2×#106C22×C311yescharacteristic
133C2×#106C22×C311yescharacteristic
134C2×#106C22×C311yescharacteristic
135C2×#102C22×C311yescharacteristic
136C2×#102C22×C311yescharacteristic
137C2×#102C22×C311yescharacteristic
138C2×#102C22×C311yescharacteristic
139C2×#97C22×C311yescharacteristic
140C2×C4×C8C22×C311yescharacteristic
141#400—21no—
142#400—21no—
143#398—21no—
144#398—21no—
145#390C22×C311yescharacteristic
146#390C22×C311yescharacteristic
147#389C22×C311yescharacteristic
148#389C22×C311yescharacteristic
149#389C22×C311yescharacteristic
150#389C22×C311yescharacteristic
151#389C22×C311yescharacteristic
152#389C22×C311yescharacteristic
153#388—21no—
154#388—21no—
155#384C22×C311yescharacteristic
156#382C22×C311yescharacteristic
157#381—21no—
158#380—21no—
159C4×C4⋊C4C22×C311yescharacteristic
160#376C22×C311yescharacteristic
161#340C22×C311yescharacteristic
162#340C22×C311yescharacteristic
163#340C22×C311yescharacteristic
164#340C22×C311yescharacteristic
165#340C22×C311yescharacteristic
166#340C22×C311yescharacteristic
167#340C22×C311yescharacteristic
168#340C22×C311yescharacteristic
169#336C22×C311yescharacteristic
170#336C22×C311yescharacteristic
171#336C22×C311yescharacteristic
172#336C22×C311yescharacteristic
173C24×C3—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4#4112yes—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—81no—
189C22×C3×C4—81no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8#4112yes—
208C2×C3×C8#4112yes—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—81no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4—81no—
225C3×C4⋊C4—81no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4—81no—
229C3×C4⋊C4—81no—
230C3×C4⋊C4#4112yes—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—81no—
233C3×C22⋊C4—81no—
234C3×C22⋊C4—81no—
235C3×C22⋊C4—81no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—81no—
243C22×D4C23×C311yescharacteristic
244C23×C4—21no—
245C22×C8C23×C311yescharacteristic
246C22×C8C23×C311yescharacteristic
247C22×C8C23×C311yescharacteristic
248C22×C8C23×C311yescharacteristic
249C4×D4—22no—
250C4×D4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C4⋊C4C23×C311yescharacteristic
264C2×C4⋊C4C23×C311yescharacteristic
265C2×C4⋊C4C23×C311yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C42—21no—
272C2×C42C23×C311yescharacteristic
273C2×C42C23×C311yescharacteristic
274C2×C42C23×C311yescharacteristic
275#107—22no—
276#106—22no—
277#106—22no—
278#106—22no—
279#102—22no—
280#102—22no—
281#102—22no—
282#102—22no—
283#97—22no—
284C4×C8—22no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95C3×D411yescharacteristic
295#95C3×D411yescharacteristic
296#95C3×D411yescharacteristic
297#95C3×D411yescharacteristic
298C23×C3—22no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3C22×D411yescharacteristic
306C3×D4—22no—
307C3×D4—22no—
308C3×D4—41no—
309C3×D4—41no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—84no—
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C42−1+411yescharacteristic
355C2×C3×C42−1+411yescharacteristic
356C3×C8—22no—
357C3×C8—22no—
358C3×C8—41no—
359C3×C8—41no—
360C24—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4C3×#4112yes—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C24×C311yescharacteristic
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8C3×#4112yes—
395C2×C8C3×#4112yes—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—81no—
410C4⋊C4—81no—
411C4⋊C4—81no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4C3×#4112yes—
418C22⋊C4—24no—
419C22⋊C4—81no—
420C22⋊C4—81no—
421C22⋊C4—81no—
422C22⋊C4—81no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—81no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3#52511yescharacteristic
438C22×C3#53211yescharacteristic
439C22×C3#53611yescharacteristic
440C22×C3#57611yescharacteristic
441C22×C3#57711yescharacteristic
442C22×C3#57711yescharacteristic
443C22×C3#57711yescharacteristic
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—42no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—81no—
452C3×C4—81no—
453C3×C4—81no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4—81no—
457C3×C4#55211yescharacteristic
458C3×C4#55211yescharacteristic
459C3×C4#55211yescharacteristic
460C3×C4#55211yescharacteristic
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23C22×C3×D411yescharacteristic
469D4—22no—
470D4—22no—
471D4—41no—
472D4—41no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—84no—
513C2×C4C2×C3×#4111yescharacteristic
514C2×C4C2×C3×#4111yescharacteristic
515C2×C4C2×C3×#4111yescharacteristic
516C2×C4C2×C3×#4111yescharacteristic
517C2×C4C3×2−1+411yescharacteristic
518C2×C4C3×2−1+411yescharacteristic
519C8—22no—
520C8—22no—
521C8—41no—
522C8—41no—
523C2×C3—82no—
524C2×C3#261111yescharacteristic
525C2×C3#311911yescharacteristic
526C2×C3#312311yescharacteristic
527C2×C3#317311yescharacteristic
528C2×C3#317911yescharacteristic
529C2×C3#324111yescharacteristic
530C2×C3#324411yescharacteristic
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22C3×#52511yescharacteristic
539C22C3×#53211yescharacteristic
540C22C3×#53611yescharacteristic
541C22C3×#57611yescharacteristic
542C22C3×#57711yescharacteristic
543C22C3×#57711yescharacteristic
544C22C3×#57711yescharacteristic
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4C3×#55211yescharacteristic
559C4C3×#55211yescharacteristic
560C4C3×#55211yescharacteristic
561C4C3×#55211yescharacteristic
562C3#2959011yescharacteristic, direct product factor, Sylow
563C2—82no—
564C2C3×#261111yescharacteristic
565C2C3×#311911yescharacteristic
566C2C3×#312311yescharacteristic
567C2C3×#317311yescharacteristic
568C2C3×#317911yescharacteristic
569C2C3×#324111yescharacteristic
570C2C3×#324411yescharacteristic
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. C3×#2647C2semidirect
3. C3×#2101C2semidirect
5. C3×#2084C2semidirect
7. C3×#1944C2semidirect
8. C3×#1941C2semidirect
11. C3×#1919C2semidirect
12. C3×#1918C2semidirect
15. C3×#1746C2semidirect
16. #29590C3direct (N × H)
60. #2647C2×C3semidirect
62. #2101C2×C3semidirect
64. #2084C2×C3semidirect
66. #1944C2×C3semidirect
67. #1941C2×C3semidirect
70. #1919C2×C3semidirect
71. #1918C2×C3semidirect
74. #1746C2×C3semidirect
562. C3#29590direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26472
C3×#23192
C3×#21012
C3×#20922
C3×#20842
C3×#20712
C3×#19442
C3×#19412
C3×#19362
C3×#19342
C3×#19192
C3×#19182
C3×#18642
C3×#17462
#295903