Finite Group Databasebeta

← C3×#27727 #10646334

Subgroups — C3×#27727 #10646334

Order 768.

766 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2552C211yesmaximal, characteristic, complemented
2C3×#2542C211yesmaximal, characteristic
3C3×#2099C212yesmaximal, complemented
4C3×#2071C212yesmaximal
5C3×#1990C211yesmaximal, characteristic, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1934C212yesmaximal
8C3×#1834C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic
10C3×C8×C4⋊C4C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic
12#27727C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125—21no—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#102C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23C2×C3×C4×C8C2212yes—
24C3×#399—22no—
25C3×#398—22no—
26C3×#390C2212yes—
27C3×#389—21no—
28C3×#389C2212yes—
29C3×#384C2211yescharacteristic
30C3×#384C2212yes—
31C3×#384C2212yes—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C4⋊C4C2211yescharacteristic
34C3×#340C2212yes—
35C3×#340C2212yes—
36C3×#340C2211yescharacteristic
37C3×#340C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2212yes—
42#2552C2×C311yescharacteristic, complemented
43#2542C2×C311yescharacteristic
44#2099C2×C312yescomplemented
45#2071C2×C312yes—
46#1990C2×C311yescharacteristic, complemented
47#1988C2×C311yescharacteristic, complemented
48#1934C2×C312yes—
49#1834C2×C312yescomplemented
50#1802C2×C311yescharacteristic
51C8×C4⋊C4C2×C311yescharacteristic, complemented
52C4×#102C2×C311yescharacteristic
53C22×C3×Q8—21no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×C3×C8C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×#125—44no—
62C3×C22⋊Q8—44no—
63C3×C22⋊Q8—44no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×C4⋊C8—22no—
88C3×#102—22no—
89C3×#102—24no—
90C3×#102—22no—
91C3×C4×C8—24no—
92C3×C4×C8—22no—
93C3×#95—21no—
94C3×#95—22no—
95C3×#95D412yes—
96C2×C4⋊Q8C22×C311yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C4×D4C22×C311yescharacteristic
101C2×C4⋊C8C22×C311yescharacteristic
102C2×#102C22×C312yes—
103C2×#102C22×C311yescharacteristic
104C2×#102C22×C311yescharacteristic
105C2×C4×C8C22×C312yes—
106C2×C4×C8C22×C311yescharacteristic
107#399—22no—
108#398—22no—
109#390C22×C312yes—
110#389—21no—
111#389C22×C312yes—
112#384C22×C311yescharacteristic
113#384C22×C312yes—
114#384C22×C312yes—
115C4×C4⋊C4C22×C311yescharacteristic
116C4×C4⋊C4C22×C311yescharacteristic
117#340C22×C312yes—
118#340C22×C312yes—
119#340C22×C312yes—
120#340C22×C311yescharacteristic
121#340C22×C312yes—
122#340C22×C311yescharacteristic
123#340C22×C312yes—
124#336C22×C312yes—
125C24×C3—21no—
126C2×C3×Q8—42no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4#4112yes—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2411yescharacteristic
148C2×C3×C8—21no—
149C2×C3×C8—21no—
150C2×C3×C8—22no—
151C2×C3×C8—21no—
152C2×C3×C8—22no—
153C2×C3×C8—21no—
154C2×C3×C8—22no—
155C2×C3×C8—21no—
156C2×C3×C8#4112yes—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—44no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—44no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4—82no—
174C3×C4⋊C4—81no—
175C3×C4⋊C4D814yes—
176C3×C4⋊C4SD1614yes—
177C3×C22⋊C4—42no—
178C3×C22⋊C4—44no—
179C3×C22⋊C4—44no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—42no—
183C3×C42#4112yes—
184C3×C42#4112yes—
185C22×Q8—21no—
186C22×D4C23×C311yescharacteristic
187C23×C4—21no—
188C22×C8C23×C311yescharacteristic
189C22×C8C23×C312yes—
190C22×C8C23×C311yescharacteristic
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193#125—44no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C4×D4—22no—
197C4×D4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C3×D411yescharacteristic
207C2×C4⋊C4C3×D411yescharacteristic
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C4⋊C4C23×C312yes—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—21no—
217C2×C42C23×C312yes—
218C2×C42C23×C311yescharacteristic
219C4⋊C8—22no—
220#102—24no—
221#102—22no—
222#102—22no—
223C4×C8—24no—
224C4×C8—22no—
225#95—21no—
226#95—22no—
227#95C3×D412yes—
228C23×C3—22no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3C22×D411yescharacteristic
236C3×Q8—84no—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—41no—
240C3×D4—41no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—22no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—84no—
263C2×C3×C4#12412yes—
264C2×C3×C4#12412yes—
265C2×C3×C4C2×D814yes—
266C2×C3×C4C2×D812yes—
267C2×C3×C4C2×SD1614yes—
268C2×C3×C4C2×SD1612yes—
269C2×C3×C4C2×#4111yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C42−1+411yescharacteristic
274C2×C3×C42−1+411yescharacteristic
275C3×C8—22no—
276C3×C8—42no—
277C3×C8—41no—
278C24—21no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4C3×#4112yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D412yes—
298C22×C4C2×C3×D412yes—
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C24×C311yescharacteristic
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8C3×#4112yes—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—82no—
327C4⋊C4—81no—
328C4⋊C4C3×D814yes—
329C4⋊C4C3×SD1614yes—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C42—22no—
334C42—22no—
335C42—42no—
336C42C3×#4112yes—
337C42C3×#4112yes—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3C2×#12411yescharacteristic
346C22×C3#54711yescharacteristic
347C22×C3#54811yescharacteristic
348C22×C3C22×D811yescharacteristic
349C22×C3C22×SD1611yescharacteristic
350C22×C3#57711yescharacteristic
351C22×C3#57811yescharacteristic
352C3×C4—24no—
353C3×C4—24no—
354C3×C4—22no—
355C3×C4—81no—
356C3×C4—81no—
357C3×C4—82no—
358C3×C4—81no—
359C3×C4—82no—
360C3×C4#48614yes—
361C3×C4#52911yescharacteristic
362C3×C4#54111yescharacteristic
363C3×C4#55211yescharacteristic
364C3×C4#55211yescharacteristic
365C23—22no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23C22×C3×D411yescharacteristic
373Q8—84no—
374D4—22no—
375D4—22no—
376D4—41no—
377D4—41no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—84no—
400C2×C4C3×#12412yes—
401C2×C4C3×#12412yes—
402C2×C4C2×C3×D814yes—
403C2×C4C2×C3×D812yes—
404C2×C4C2×C3×SD1614yes—
405C2×C4C2×C3×SD1612yes—
406C2×C4C2×C3×#4111yescharacteristic
407C2×C4C2×C3×#4111yescharacteristic
408C2×C4C2×C3×#4111yescharacteristic
409C2×C4C2×C3×#4111yescharacteristic
410C2×C4C3×2−1+411yescharacteristic
411C2×C4C3×2−1+411yescharacteristic
412C8—22no—
413C8—42no—
414C8—41no—
415C2×C3—82no—
416C2×C3#248911yescharacteristic
417C2×C3C2×#48611yescharacteristic
418C2×C3#313711yescharacteristic
419C2×C3#333411yescharacteristic
420C2×C3#333511yescharacteristic
421C2×C3#335811yescharacteristic
422C2×C3#335911yescharacteristic
423C22—42no—
424C22—42no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22C2×C3×#12411yescharacteristic
431C22C3×#54711yescharacteristic
432C22C3×#54811yescharacteristic
433C22C22×C3×D811yescharacteristic
434C22C22×C3×SD1611yescharacteristic
435C22C3×#57711yescharacteristic
436C22C3×#57811yescharacteristic
437C4—22no—
438C4—24no—
439C4—24no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—82no—
445C4C3×#48614yes—
446C4C3×#52911yescharacteristic
447C4C3×#54111yescharacteristic
448C4C3×#55211yescharacteristic
449C4C3×#55211yescharacteristic
450C3#2772711yescharacteristic, direct product factor, Sylow
451C2—82no—
452C2C3×#248911yescharacteristic
453C2C2×C3×#48611yescharacteristic
454C2C3×#313711yescharacteristic
455C2C3×#333411yescharacteristic
456C2C3×#333511yescharacteristic
457C2C3×#335811yescharacteristic
458C2C3×#335911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2552C2semidirect
3. C3×#2099C2semidirect
5. C3×#1990C2semidirect
6. C3×#1988C2semidirect
8. C3×#1834C2semidirect
10. C3×C8×C4⋊C4C2semidirect
12. #27727C3direct (N × H)
42. #2552C2×C3semidirect
44. #2099C2×C3semidirect
46. #1990C2×C3semidirect
47. #1988C2×C3semidirect
49. #1834C2×C3semidirect
51. C8×C4⋊C4C2×C3semidirect
450. C3#27727direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25522
C3×#25422
C3×#20992
C3×#20712
C3×#19902
C3×#19882
C3×#19342
C3×#18342
C3×#18022
C3×C8×C4⋊C42
C3×C4×#1022
#277273