Finite Group Databasebeta

← C3×#10918 #10629525

Subgroups — C3×#10918 #10629525

Order 768.

1246 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#385C211yesmaximal, characteristic
2C3×#2273C211yesmaximal, characteristic
3C2×C3×#342C211yesmaximal, characteristic
4C2×C3×#342C212yesmaximal
5C3×C22⋊#95C212yesmaximal
6#10918C311yesmaximal, characteristic, direct product factor, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C4⋊C4—21no—
9C22×C3×C22⋊C4C2212yes—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2212yes—
12C22×C3×C22⋊C4C2211yescharacteristic
13C3×#385—22no—
14C3×#385C414yescomplemented
15C3×C4×C22⋊C4—21no—
16C3×C4×C22⋊C4—21no—
17C3×C4×C22⋊C4—24no—
18C2×C3×#95—21no—
19C2×C3×#95—24no—
20C3×#342—22no—
21C3×#342C418yescomplemented
22C2×#385C2×C311yescharacteristic
23#2273C2×C311yescharacteristic
24C2×#342C2×C311yescharacteristic
25C2×#342C2×C312yes—
26C22⋊#95C2×C312yes—
27C25×C3C2311yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—21no—
30C23×C3×C4—21no—
31C23×C3×C4—21no—
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—44no—
35C23×C3×C4C2×C412yes—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×C3×C4D414yes—
39C2×C3×C4⋊C4—24no—
40C2×C3×C4⋊C4—44no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4C2×C414yes—
55C2×C3×C22⋊C4C2×C412yes—
56C2×C3×C22⋊C4C2×C414yes—
57C2×C3×C22⋊C4C2×C412yes—
58C2×C3×C22⋊C4C2×C414yes—
59C2×C3×C22⋊C4D412yes—
60C2×C3×C22⋊C4Q814yes—
61C2×C3×C42—22no—
62C2×C3×C42—22no—
63C3×#95—22no—
64C3×#95—22no—
65C3×#95—24no—
66C3×#95—48no—
67C24×C4C22×C311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C22⋊C4C22×C312yes—
70C22×C22⋊C4C22×C311yescharacteristic
71C22×C22⋊C4C22×C312yes—
72C22×C22⋊C4C22×C311yescharacteristic
73#385—22no—
74#385C3×C414yescomplemented
75C4×C22⋊C4—21no—
76C4×C22⋊C4—21no—
77C4×C22⋊C4—24no—
78C2×#95—21no—
79C2×#95—24no—
80#342—22no—
81#342C3×C418yescomplemented
82C24×C3—42no—
83C24×C3—42no—
84C24×C3C22×C411yescharacteristic
85C24×C3C22×C412yes—
86C24×C3C2×D411yescharacteristic
87C24×C3C2×D412yes—
88C24×C3C2×D411yescharacteristic
89C24×C3C2×Q812yes—
90C24×C3#4112yes—
91C24×C3#4112yes—
92C24×C3#4112yes—
93C22×C3×C4—28no—
94C22×C3×C4—24no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—24no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—24no—
106C22×C3×C4—44no—
107C22×C3×C4—44no—
108C22×C3×C4—48no—
109C22×C3×C4—42no—
110C22×C3×C4—42no—
111C22×C3×C4—44no—
112C22×C3×C4—42no—
113C22×C3×C4—41no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—44no—
117C22×C3×C4—42no—
118C22×C3×C4—42no—
119C22×C3×C4—44no—
120C22×C3×C4—44no—
121C22×C3×C4—44no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4C22⋊C412yes—
125C22×C3×C4C22⋊C412yes—
126C22×C3×C4C4⋊C418yescomplemented
127C3×C4⋊C4—48no—
128C3×C22⋊C4—22no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—24no—
131C3×C22⋊C4—24no—
132C3×C22⋊C4—24no—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—42no—
140C3×C22⋊C4C4214yescomplemented
141C3×C42—82no—
142C3×C42—82no—
143C25C23×C311yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4C2×C3×C412yes—
152C23×C4C3×D411yescharacteristic
153C23×C4C3×D411yescharacteristic
154C23×C4C3×D414yes—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—44no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4C2×C3×C414yes—
171C2×C22⋊C4C2×C3×C412yes—
172C2×C22⋊C4C2×C3×C414yes—
173C2×C22⋊C4C2×C3×C414yes—
174C2×C22⋊C4C2×C3×C412yes—
175C2×C22⋊C4C3×D412yes—
176C2×C22⋊C4C3×Q814yes—
177C2×C42—22no—
178C2×C42—22no—
179#95—22no—
180#95—22no—
181#95—24no—
182#95—48no—
183C23×C3—24no—
184C23×C3—22no—
185C23×C3—24no—
186C23×C3—22no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—42no—
190C23×C3—44no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—44no—
194C23×C3—44no—
195C23×C3—44no—
196C23×C3—42no—
197C23×C3C2×C4211yescharacteristic
198C23×C3C2×C22⋊C411yescharacteristic
199C23×C3C2×C4⋊C412yes—
200C23×C3#11711yescharacteristic
201C23×C3#11712yes—
202C23×C3C4×D412yes—
203C23×C3C4×D412yes—
204C23×C3C4×D412yes—
205C23×C3C4×D412yes—
206C23×C3C4×Q812yes—
207C23×C3C4×Q812yes—
208C23×C3#12011yescharacteristic
209C23×C3#12112yes—
210C23×C3C22⋊Q812yes—
211C23×C3C22⋊Q812yes—
212C23×C3#12312yes—
213C23×C3#12311yescharacteristic
214C23×C3#12411yescharacteristic
215C23×C3#12512yes—
216C23×C3#12612yes—
217C23×C3C4⋊Q811yescharacteristic
218C2×C3×C4—24no—
219C2×C3×C4—28no—
220C2×C3×C4—24no—
221C2×C3×C4—28no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—42no—
225C2×C3×C4—48no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—44no—
229C2×C3×C4—48no—
230C2×C3×C4—44no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4—42no—
234C2×C3×C4—44no—
235C2×C3×C4—44no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—48no—
240C2×C3×C4—48no—
241C2×C3×C4—44no—
242C2×C3×C4—84no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—84no—
246C2×C3×C4—82no—
247C2×C3×C4—82no—
248C24—42no—
249C24—42no—
250C24C22×C3×C411yescharacteristic
251C24C22×C3×C412yes—
252C24C2×C3×D411yescharacteristic
253C24C2×C3×D412yes—
254C24C2×C3×D411yescharacteristic
255C24C2×C3×Q812yes—
256C24C3×#4112yes—
257C24C3×#4112yes—
258C24C3×#4112yes—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—24no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—48no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C3×C22⋊C412yes—
291C22×C4C3×C22⋊C412yes—
292C22×C4C3×C4⋊C418yescomplemented
293C4⋊C4—48no—
294C22⋊C4—22no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4C3×C4214yescomplemented
307C42—82no—
308C42—82no—
309C22×C3—24no—
310C22×C3—24no—
311C22×C3—24no—
312C22×C3—24no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3C4×C22⋊C411yescharacteristic
334C22×C3C4×C4⋊C412yes—
335C22×C3#38512yes—
336C22×C3#38611yescharacteristic
337C22×C3#38712yes—
338C22×C3#38712yes—
339C22×C3#38812yes—
340C22×C3#38912yes—
341C22×C3#39111yescharacteristic
342C22×C3B(4,𝔽2)12yes—
343C22×C3#45812yes—
344C3×C4—44no—
345C3×C4—48no—
346C3×C4—44no—
347C3×C4—42no—
348C3×C4—42no—
349C3×C4—84no—
350C23—24no—
351C23—22no—
352C23—24no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—44no—
358C23—41no—
359C23—41no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—42no—
364C23C2×C3×C4211yescharacteristic
365C23C2×C3×C22⋊C411yescharacteristic
366C23C2×C3×C4⋊C412yes—
367C23C3×#11711yescharacteristic
368C23C3×#11712yes—
369C23C3×C4×D412yes—
370C23C3×C4×D412yes—
371C23C3×C4×D412yes—
372C23C3×C4×D412yes—
373C23C3×C4×Q812yes—
374C23C3×C4×Q812yes—
375C23C3×#12011yescharacteristic
376C23C3×#12112yes—
377C23C3×C22⋊Q812yes—
378C23C3×C22⋊Q812yes—
379C23C3×#12312yes—
380C23C3×#12311yescharacteristic
381C23C3×#12411yescharacteristic
382C23C3×#12512yes—
383C23C3×#12612yes—
384C23C3×C4⋊Q811yescharacteristic
385C2×C4—24no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—84no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—84no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C3—21no—
416C2×C3—22no—
417C2×C3—22no—
418C2×C3—22no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3#143611yescharacteristic
423C2×C3#186714yes—
424C2×C3#188911yescharacteristic
425C2×C3#189011yescharacteristic
426C22—24no—
427C22—24no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22C3×C4×C22⋊C411yescharacteristic
451C22C3×C4×C4⋊C412yes—
452C22C3×#38512yes—
453C22C3×#38611yescharacteristic
454C22C3×#38712yes—
455C22C3×#38712yes—
456C22C3×#38812yes—
457C22C3×#38912yes—
458C22C3×#39111yescharacteristic
459C22C3×B(4,𝔽2)12yes—
460C22C3×#45812yes—
461C4—44no—
462C4—48no—
463C4—44no—
464C4—42no—
465C4—42no—
466C4—84no—
467C3#1091811yescharacteristic, direct product factor, Sylow
468C2—21no—
469C2—22no—
470C2—22no—
471C2—22no—
472C2—22no—
473C2—22no—
474C2—21no—
475C2C3×#143611yescharacteristic
476C2C3×#186714yes—
477C2C3×#188911yescharacteristic
478C2C3×#189011yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutatorC23C2×C3×C42
Frattini, cosocleC25C23×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
6. #10918C3direct (N × H)
14. C3×#385C4semidirect
21. C3×#342C4semidirect
74. #385C3×C4semidirect
81. #342C3×C4semidirect
126. C22×C3×C4C4⋊C4semidirect
140. C3×C22⋊C4C42semidirect
292. C22×C4C3×C4⋊C4semidirect
306. C22⋊C4C3×C42semidirect
467. C3#10918direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3852
C3×#22732
C2×C3×#3422
C3×C22⋊#952
#109183