Finite Group Databasebeta

← C3×#27728 #10646335

Subgroups — C3×#27728 #10646335

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2571C211yesmaximal, characteristic, complemented
2C3×#2563C211yesmaximal, characteristic
3C3×#2101C212yesmaximal, complemented
4C3×#2073C212yesmaximal
5C3×#1990C211yesmaximal, characteristic, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1934C212yesmaximal
8C3×#1835C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic
10C3×C8×C4⋊C4C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic
12#27728C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×#102C2211yescharacteristic
17C2×C3×#102C2211yescharacteristic
18C2×C3×#102C2212yes—
19C2×C3×C4×C8C2211yescharacteristic
20C2×C3×C4×C8C2212yes—
21C3×#400—22no—
22C3×#393—22no—
23C3×#390C2212yes—
24C3×#389—21no—
25C3×#389C2212yes—
26C3×#384C2211yescharacteristic
27C3×#382—21no—
28C3×#382C2212yes—
29C3×#382C2212yes—
30C3×#380—21no—
31C3×#380—21no—
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#2571C2×C311yescharacteristic, complemented
43#2563C2×C311yescharacteristic
44#2101C2×C312yescomplemented
45#2073C2×C312yes—
46#1990C2×C311yescharacteristic, complemented
47#1987C2×C311yescharacteristic, complemented
48#1934C2×C312yes—
49#1835C2×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×C4×D4—22no—
62C3×C4×D4—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42—21no—
78C2×C3×C42C2312yes—
79C2×C3×C42C2311yescharacteristic
80C3×C4⋊C8—22no—
81C3×#102—22no—
82C3×#102—24no—
83C3×#102—22no—
84C3×C4×C8—24no—
85C3×C4×C8—22no—
86C3×#95—21no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95D412yes—
93C2×C4⋊Q8C22×C311yescharacteristic
94C2×C4×D4C22×C311yescharacteristic
95C2×C4⋊C8C22×C311yescharacteristic
96C2×#102C22×C312yes—
97C2×#102C22×C311yescharacteristic
98C2×#102C22×C311yescharacteristic
99C2×C4×C8C22×C312yes—
100C2×C4×C8C22×C311yescharacteristic
101#400—22no—
102#393—22no—
103#390C22×C312yes—
104#389—21no—
105#389C22×C312yes—
106#384C22×C311yescharacteristic
107#382—21no—
108#382C22×C312yes—
109#382C22×C312yes—
110#380—21no—
111#380—21no—
112C4×C4⋊C4C22×C311yescharacteristic
113C4×C4⋊C4C22×C311yescharacteristic
114#340C22×C312yes—
115#340C22×C312yes—
116#340C22×C312yes—
117#340C22×C311yescharacteristic
118#340C22×C312yes—
119#340C22×C311yescharacteristic
120#340C22×C312yes—
121#336C22×C312yes—
122C24×C3—21no—
123C2×C3×Q8—42no—
124C2×C3×Q8—81no—
125C2×C3×Q8—81no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4#4112yes—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—22no—
148C2×C3×C8—21no—
149C2×C3×C8—22no—
150C2×C3×C8—21no—
151C2×C3×C8—22no—
152C2×C3×C8—21no—
153C2×C3×C8#4112yes—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—81no—
169C3×C4⋊C4—82no—
170C3×C22⋊C4—42no—
171C3×C22⋊C4—82no—
172C3×C22⋊C4—82no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—42no—
176C3×C42#4112yes—
177C3×C42#4112yes—
178C22×Q8—21no—
179C22×D4C23×C311yescharacteristic
180C23×C4—21no—
181C22×C8C23×C311yescharacteristic
182C22×C8C23×C312yes—
183C22×C8C23×C311yescharacteristic
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4×D4—22no—
187C4×D4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C3×D411yescharacteristic
193C2×C4⋊C4C3×D411yescharacteristic
194C2×C4⋊C4C23×C311yescharacteristic
195C2×C4⋊C4C23×C312yes—
196C2×C4⋊C4C23×C311yescharacteristic
197C2×C4⋊C4C23×C311yescharacteristic
198C2×C4⋊C4C23×C312yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42—21no—
203C2×C42C23×C312yes—
204C2×C42C23×C311yescharacteristic
205C4⋊C8—22no—
206#102—24no—
207#102—22no—
208#102—22no—
209C4×C8—24no—
210C4×C8—22no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95C3×D412yes—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3C22×D411yescharacteristic
226C3×Q8—84no—
227C3×D4—22no—
228C3×D4—22no—
229C3×D4—41no—
230C3×D4—41no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—84no—
255C2×C3×C4#12412yes—
256C2×C3×C4#12412yes—
257C2×C3×C4#13512yes—
258C2×C3×C4#13512yes—
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C42−1+411yescharacteristic
264C2×C3×C42−1+411yescharacteristic
265C3×C8—22no—
266C3×C8—42no—
267C3×C8—41no—
268C24—21no—
269C2×Q8—42no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C3×#4112yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C2×C3×D412yes—
287C22×C4C2×C3×D412yes—
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C24×C311yescharacteristic
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8C3×#4112yes—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C4⋊C4—82no—
316C22⋊C4—42no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—22no—
320C42—22no—
321C42—42no—
322C42C3×#4112yes—
323C42C3×#4112yes—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3C2×#12411yescharacteristic
332C22×C3#54711yescharacteristic
333C22×C3#54811yescharacteristic
334C22×C3C2×#13511yescharacteristic
335C22×C3C2×#13511yescharacteristic
336C22×C3#57611yescharacteristic
337C22×C3#57711yescharacteristic
338C3×C4—22no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—81no—
342C3×C4—81no—
343C3×C4—82no—
344C3×C4—81no—
345C3×C4—82no—
346C3×C4#48814yes—
347C3×C4#52911yescharacteristic
348C3×C4#54111yescharacteristic
349C3×C4#55211yescharacteristic
350C3×C4#55211yescharacteristic
351C23—22no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C22×C3×D411yescharacteristic
359Q8—84no—
360D4—22no—
361D4—22no—
362D4—41no—
363D4—41no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—84no—
388C2×C4C3×#12412yes—
389C2×C4C3×#12412yes—
390C2×C4C3×#13512yes—
391C2×C4C3×#13512yes—
392C2×C4C2×C3×#4111yescharacteristic
393C2×C4C2×C3×#4111yescharacteristic
394C2×C4C2×C3×#4111yescharacteristic
395C2×C4C2×C3×#4111yescharacteristic
396C2×C4C3×2−1+411yescharacteristic
397C2×C4C3×2−1+411yescharacteristic
398C8—22no—
399C8—42no—
400C8—41no—
401C2×C3—82no—
402C2×C3#248911yescharacteristic
403C2×C3C2×#48811yescharacteristic
404C2×C3#313611yescharacteristic
405C2×C3#334011yescharacteristic
406C2×C3#334111yescharacteristic
407C2×C3#336011yescharacteristic
408C2×C3#336111yescharacteristic
409C22—42no—
410C22—42no—
411C22—42no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22C2×C3×#12411yescharacteristic
417C22C3×#54711yescharacteristic
418C22C3×#54811yescharacteristic
419C22C2×C3×#13511yescharacteristic
420C22C2×C3×#13511yescharacteristic
421C22C3×#57611yescharacteristic
422C22C3×#57711yescharacteristic
423C4—22no—
424C4—42no—
425C4—42no—
426C4—81no—
427C4—82no—
428C4—81no—
429C4—81no—
430C4—82no—
431C4C3×#48814yes—
432C4C3×#52911yescharacteristic
433C4C3×#54111yescharacteristic
434C4C3×#55211yescharacteristic
435C4C3×#55211yescharacteristic
436C3#2772811yescharacteristic, direct product factor, Sylow
437C2—82no—
438C2C3×#248911yescharacteristic
439C2C2×C3×#48811yescharacteristic
440C2C3×#313611yescharacteristic
441C2C3×#334011yescharacteristic
442C2C3×#334111yescharacteristic
443C2C3×#336011yescharacteristic
444C2C3×#336111yescharacteristic
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×#2571C2semidirect
3. C3×#2101C2semidirect
5. C3×#1990C2semidirect
6. C3×#1987C2semidirect
8. C3×#1835C2semidirect
10. C3×C8×C4⋊C4C2semidirect
12. #27728C3direct (N × H)
42. #2571C2×C3semidirect
44. #2101C2×C3semidirect
46. #1990C2×C3semidirect
47. #1987C2×C3semidirect
49. #1835C2×C3semidirect
51. C8×C4⋊C4C2×C3semidirect
436. C3#27728direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25712
C3×#25632
C3×#21012
C3×#20732
C3×#19902
C3×#19872
C3×#19342
C3×#18352
C3×#18022
C3×C8×C4⋊C42
C3×C4×#1022
#277283