Finite Group Databasebeta

← C3×#16700 #10635307

Subgroups — C3×#16700 #10635307

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#387C211yesmaximal, characteristic
2C2×C3×#382C211yesmaximal, characteristic
3C3×#1445C212yesmaximal
4C3×#1444C211yesmaximal, characteristic
5C3×#1443C212yesmaximal
6C3×#1442C211yesmaximal, characteristic
7C3×#1441C212yesmaximal
8C3×#1441C212yesmaximal
9C3×#1441C211yesmaximal, characteristic
10C3×#1441C211yesmaximal, characteristic
11C3×C4⋊#95C211yesmaximal, characteristic
12#16700C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C3×#387—24no—
21C3×#382—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×#387C2×C311yescharacteristic
39C2×#382C2×C311yescharacteristic
40#1445C2×C312yes—
41#1444C2×C311yescharacteristic
42#1443C2×C312yes—
43#1442C2×C311yescharacteristic
44#1441C2×C312yes—
45#1441C2×C312yes—
46#1441C2×C311yescharacteristic
47#1441C2×C311yescharacteristic
48C4⋊#95C2×C311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C4⋊C4—42no—
74C2×C3×C4⋊C4Q814yes—
75C2×C3×C42—24no—
76C2×C3×C42D414yes—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—22no—
80C3×#95—22no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—24no—
85C3×#95—22no—
86C3×#95—42no—
87C3×#95—42no—
88C3×#95—42no—
89C3×#95—42no—
90C3×#95—42no—
91C3×#95—42no—
92C3×#95—42no—
93C22×C4⋊C4C22×C311yescharacteristic
94C22×C4⋊C4C22×C312yes—
95C22×C4⋊C4C22×C311yescharacteristic
96C22×C4⋊C4C22×C311yescharacteristic
97C22×C4⋊C4C22×C312yes—
98C22×C4⋊C4C22×C311yescharacteristic
99C22×C42C22×C311yescharacteristic
100#387—24no—
101#382—24no—
102C2×#95C22×C311yescharacteristic
103C2×#95C22×C311yescharacteristic
104C2×#95C22×C312yes—
105C2×#95C22×C311yescharacteristic
106C2×#95C22×C312yes—
107C2×#95C22×C311yescharacteristic
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C312yes—
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C24×C3C2411yescharacteristic
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×Q812yes—
179C22×C3×C4C2×Q812yes—
180C22×C3×C4C2×Q812yes—
181C22×C3×C4#4114yes—
182C22×C3×C4#4112yes—
183C22×C3×C4#4114yes—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—48no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4—44no—
188C3×C4⋊C4—48no—
189C3×C4⋊C4—44no—
190C3×C42—28no—
191C23×C4C23×C311yescharacteristic
192C23×C4C23×C312yes—
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C312yes—
195C23×C4C23×C312yes—
196C23×C4C23×C311yescharacteristic
197C23×C4C23×C311yescharacteristic
198C23×C4C23×C311yescharacteristic
199C23×C4C23×C312yes—
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C311yescharacteristic
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C3×Q814yes—
217C2×C42—24no—
218C2×C42C3×D414yes—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—24no—
227#95—22no—
228#95—42no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235C23×C3C22×D411yescharacteristic
236C23×C3C22×Q811yescharacteristic
237C23×C3C2×#4111yescharacteristic
238C23×C3C2×#4112yes—
239C23×C3C2×#4112yes—
240C23×C3H(5,𝔽2)12yes—
241C23×C3H(5,𝔽2)11yescharacteristic
242C23×C3H(5,𝔽2)11yescharacteristic
243C23×C32−1+412yes—
244C23×C32−1+411yescharacteristic
245C23×C32−1+411yescharacteristic
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4C22⋊Q814yes—
310C2×C3×C4#12314yes—
311C24C24×C311yescharacteristic
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4C2×C3×D412yes—
369C22×C4C2×C3×D412yes—
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×Q812yes—
372C22×C4C2×C3×Q812yes—
373C22×C4C2×C3×Q812yes—
374C22×C4C3×#4114yes—
375C22×C4C3×#4112yes—
376C22×C4C3×#4114yes—
377C4⋊C4—28no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C42—28no—
384C22×C3C2×C22⋊Q811yescharacteristic
385C22×C3C2×#12311yescharacteristic
386C22×C3#53211yescharacteristic
387C22×C3#53511yescharacteristic
388C22×C3#53611yescharacteristic
389C22×C3#53711yescharacteristic
390C22×C3#53711yescharacteristic
391C22×C3#53812yes—
392C22×C3#54011yescharacteristic
393C22×C3#54112yes—
394C22×C3#54111yescharacteristic
395C22×C3#54212yes—
396C22×C3#54212yes—
397C22×C3#54411yescharacteristic
398C22×C3#54411yescharacteristic
399C22×C3#55112yes—
400C22×C3#55212yes—
401C22×C3#55212yes—
402C22×C3#55312yes—
403C22×C3#55412yes—
404C22×C3#55712yes—
405C22×C3#56212yes—
406C22×C3#56312yes—
407C3×C4—24no—
408C3×C4—24no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—44no—
413C3×C4—82no—
414C3×C4—82no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—82no—
418C23C22×C3×D411yescharacteristic
419C23C22×C3×Q811yescharacteristic
420C23C2×C3×#4111yescharacteristic
421C23C2×C3×#4112yes—
422C23C2×C3×#4112yes—
423C23C3×H(5,𝔽2)11yescharacteristic
424C23C3×H(5,𝔽2)12yes—
425C23C3×H(5,𝔽2)11yescharacteristic
426C23C3×2−1+412yes—
427C23C3×2−1+411yescharacteristic
428C23C3×2−1+411yescharacteristic
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4C3×C22⋊Q814yes—
493C2×C4C3×#12314yes—
494C2×C3#248711yescharacteristic
495C2×C3#249811yescharacteristic
496C2×C3#256511yescharacteristic
497C2×C3#262511yescharacteristic
498C2×C3#262911yescharacteristic
499C2×C3#266211yescharacteristic
500C2×C3#275512yes—
501C2×C3#276112yes—
502C2×C3#276612yes—
503C2×C3#277412yes—
504C2×C3#282011yescharacteristic
505C22C2×C3×C22⋊Q811yescharacteristic
506C22C2×C3×#12311yescharacteristic
507C22C3×#53211yescharacteristic
508C22C3×#53511yescharacteristic
509C22C3×#53611yescharacteristic
510C22C3×#53711yescharacteristic
511C22C3×#53711yescharacteristic
512C22C3×#53812yes—
513C22C3×#54011yescharacteristic
514C22C3×#54112yes—
515C22C3×#54111yescharacteristic
516C22C3×#54212yes—
517C22C3×#54212yes—
518C22C3×#54411yescharacteristic
519C22C3×#54411yescharacteristic
520C22C3×#55112yes—
521C22C3×#55212yes—
522C22C3×#55212yes—
523C22C3×#55312yes—
524C22C3×#55412yes—
525C22C3×#55712yes—
526C22C3×#56212yes—
527C22C3×#56312yes—
528C4—24no—
529C4—24no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—44no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4—82no—
538C4—82no—
539C3#1670011yescharacteristic, direct product factor, Sylow
540C2C3×#248711yescharacteristic
541C2C3×#249811yescharacteristic
542C2C3×#256511yescharacteristic
543C2C3×#262511yescharacteristic
544C2C3×#262911yescharacteristic
545C2C3×#266211yescharacteristic
546C2C3×#275512yes—
547C2C3×#276112yes—
548C2C3×#276612yes—
549C2C3×#277412yes—
550C2C3×#282011yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
12. #16700C3direct (N × H)
539. C3#16700direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3872
C2×C3×#3822
C3×#14452
C3×#14442
C3×#14432
C3×#14422
C3×#14412
C3×C4⋊#952
#167003