Finite Group Databasebeta

← C3×#25726 #10644333

Subgroups — C3×#25726 #10644333

Order 768.

1560 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#497C212yesmaximal, complemented
3C2×C3×#451C212yesmaximal, complemented
4C2×C3×#448C212yesmaximal
5C3×#2850C211yesmaximal, characteristic
6C3×#2016C212yesmaximal, complemented
7C3×#2013C212yesmaximal, complemented
8C3×#1960C212yesmaximal, complemented
9C3×#1749C211yesmaximal, characteristic, complemented
10#25726C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2212yes—
12C22×C3×Q16C2214yes—
13C22×C3×SD16C2214yes—
14C2×C3×C4⋊Q8—28no—
15C2×C3×C4⋊Q8—22no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×#124C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C22×C3×C4⋊C4—22no—
22C22×C3×C42C2211yescharacteristic
23C3×#497—28no—
24C3×#451—28no—
25C3×#448—28no—
26C2×C3×#103C2214yes—
27C2×C3×#103C2214yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2212yes—
30C3×#391C2211yescharacteristic
31C3×#390C2211yescharacteristic
32C3×#388—22no—
33C3×#386—22no—
34C3×#382C2211yescharacteristic
35C3×#336C2212yes—
36C22×C4⋊Q8C2×C311yescharacteristic, complemented
37C2×#497C2×C312yescomplemented
38C2×#451C2×C312yescomplemented
39C2×#448C2×C312yes—
40#2850C2×C311yescharacteristic
41#2016C2×C312yescomplemented
42#2013C2×C312yescomplemented
43#1960C2×C312yescomplemented
44#1749C2×C311yescharacteristic, complemented
45C22×C3×Q8—28no—
46C22×C3×Q8—22no—
47C22×C3×Q8—24no—
48C22×C3×Q8—24no—
49C22×C3×Q8—44no—
50C22×C3×Q8C2311yescharacteristic
51C22×C3×Q8C2314yes—
52C22×C3×D4—21no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—22no—
55C23×C3×C4D412yes—
56C23×C3×C4C2311yescharacteristic
57C2×C3×Q16—28no—
58C2×C3×Q16—28no—
59C2×C3×Q16—44no—
60C2×C3×SD16—28no—
61C2×C3×SD16—28no—
62C2×C3×SD16—44no—
63C22×C3×C8C2314yes—
64C3×C4⋊Q8—216no—
65C3×C4⋊Q8—28no—
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8D418yescomplemented
68C3×#124—24no—
69C3×C22⋊Q8—24no—
70C3×#121—24no—
71C2×C3×C4⋊C4—216no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—24no—
82C2×C3×C42D412yes—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C3×#103—28no—
86C3×#103—28no—
87C3×C22⋊C8D418yes—
88C3×#97—24no—
89C3×#95—22no—
90C3×#95—22no—
91C23×Q8C22×C312yes—
92C22×Q16C22×C314yes—
93C22×SD16C22×C314yes—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C22×C312yes—
97C2×C4⋊Q8C22×C312yes—
98C2×#124C22×C311yescharacteristic
99C2×C22⋊Q8C22×C311yescharacteristic
100C2×#121C22×C311yescharacteristic
101C22×C4⋊C4—22no—
102C22×C42C22×C311yescharacteristic
103#497—28no—
104#451—28no—
105#448—28no—
106C2×#103C22×C314yes—
107C2×#103C22×C314yes—
108C2×C22⋊C8C22×C312yes—
109C2×#97C22×C312yes—
110#391C22×C311yescharacteristic
111#390C22×C311yescharacteristic
112#388—22no—
113#386—22no—
114#382C22×C311yescharacteristic
115#336C22×C312yes—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C2×C3×Q8—28no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—24no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—24no—
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—416no—
128C2×C3×Q8—416no—
129C2×C3×Q8—44no—
130C2×C3×Q8—44no—
131C2×C3×Q8C2×D418yes—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—42no—
136C2×C3×D4—44no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—28no—
143C22×C3×C4—24no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—44no—
148C22×C3×C4—44no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D414yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2411yescharacteristic
154C3×Q16—416no—
155C3×SD16—416no—
156C2×C3×C8—24no—
157C2×C3×C8—24no—
158C2×C3×C8C2×D418yes—
159C3×C4⋊C4—216no—
160C3×C4⋊C4—216no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—42no—
164C3×C22⋊C4—42no—
165C3×C22⋊C4—44no—
166C3×C22⋊C4—42no—
167C3×C22⋊C4—44no—
168C3×C42—24no—
169C3×C42—22no—
170C3×C42C2×D414yes—
171C22×Q8—28no—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—44no—
176C22×Q8C23×C311yescharacteristic
177C22×Q8C23×C314yes—
178C22×D4—21no—
179C22×D4C23×C311yescharacteristic
180C23×C4—22no—
181C23×C4C3×D412yes—
182C23×C4C23×C311yescharacteristic
183C2×Q16—28no—
184C2×Q16—28no—
185C2×Q16—44no—
186C2×SD16—28no—
187C2×SD16—28no—
188C2×SD16—44no—
189C22×C8C23×C314yes—
190C4⋊Q8—216no—
191C4⋊Q8—28no—
192C4⋊Q8—24no—
193C4⋊Q8C3×D418yescomplemented
194#124—24no—
195C22⋊Q8—24no—
196#121—24no—
197C2×C4⋊C4—216no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C23×C311yescharacteristic
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C42—24no—
208C2×C42C3×D412yes—
209C2×C42C23×C311yescharacteristic
210C2×C42C23×C311yescharacteristic
211#103—28no—
212#103—28no—
213C22⋊C8C3×D418yes—
214#97—24no—
215#95—22no—
216#95—22no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3#12712yes—
226C23×C3#13714yes—
227C23×C3C22×D411yescharacteristic
228C3×Q8—28no—
229C3×Q8—28no—
230C3×Q8—416no—
231C3×Q8—44no—
232C3×Q8—44no—
233C3×Q8—44no—
234C3×D4—44no—
235C3×D4—84no—
236C2×C3×C4—216no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—28no—
244C2×C3×C4—28no—
245C2×C3×C4—416no—
246C2×C3×C4—44no—
247C2×C3×C4—44no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4#12014yes—
256C2×C3×C4#12712yes—
257C2×C3×C4#13714yes—
258C2×C3×C4C22×D412yes—
259C2×C3×C4C22×D412yes—
260C2×C3×C4H(5,𝔽2)12yes—
261C3×C8—28no—
262C24—21no—
263C24C2×C3×D411yescharacteristic
264C2×Q8—28no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—416no—
274C2×Q8—416no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8C2×C3×D418yes—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—44no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4C2×C3×D412yes—
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D414yes—
298C22×C4C2×C3×D412yes—
299C22×C4C24×C311yescharacteristic
300Q16—416no—
301SD16—416no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8C2×C3×D418yes—
305C4⋊C4—216no—
306C4⋊C4—216no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C42—24no—
315C42—22no—
316C42C2×C3×D414yes—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—24no—
320C22×C3—24no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3#49714yescomplemented
327C22×C3C2×#12011yescharacteristic
328C22×C3C2×#12711yescharacteristic
329C22×C3#53511yescharacteristic
330C22×C3C2×#13712yes—
331C22×C3C2×#13712yes—
332C3×C4—22no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—48no—
336C3×C4—81no—
337C3×C4—81no—
338C3×C4—81no—
339C3×C4D4214yes—
340C23—22no—
341C23—21no—
342C23—21no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23C3×#12712yes—
349C23C3×#13714yes—
350C23C22×C3×D411yescharacteristic
351Q8—28no—
352Q8—28no—
353Q8—416no—
354Q8—44no—
355Q8—44no—
356Q8—44no—
357D4—44no—
358D4—84no—
359C2×C4—216no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—416no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4C3×#12014yes—
379C2×C4C3×#12712yes—
380C2×C4C3×#13714yes—
381C2×C4C22×C3×D412yes—
382C2×C4C22×C3×D412yes—
383C2×C4C3×H(5,𝔽2)12yes—
384C8—28no—
385C2×C3—24no—
386C2×C3—161no—
387C2×C3#243311yescharacteristic
388C2×C3#300412yes—
389C2×C3C2×#49711yescharacteristic
390C2×C3#315211yescharacteristic
391C2×C3#321812yes—
392C22—22no—
393C22—22no—
394C22—24no—
395C22—24no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—82no—
400C22—82no—
401C22C3×#49714yescomplemented
402C22C2×C3×#12011yescharacteristic
403C22C2×C3×#12711yescharacteristic
404C22C3×#53511yescharacteristic
405C22C2×C3×#13712yes—
406C22C2×C3×#13712yes—
407C4—22no—
408C4—24no—
409C4—24no—
410C4—48no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4C3×D4214yes—
415C3#2572611yescharacteristic, direct product factor, Sylow
416C2—24no—
417C2—161no—
418C2C3×#243311yescharacteristic
419C2C3×#300412yes—
420C2C2×C3×#49711yescharacteristic
421C2C3×#315211yescharacteristic
422C2C3×#321812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
2. C2×C3×#497C2semidirect
3. C2×C3×#451C2semidirect
6. C3×#2016C2semidirect
7. C3×#2013C2semidirect
8. C3×#1960C2semidirect
9. C3×#1749C2semidirect
10. #25726C3direct (N × H)
36. C22×C4⋊Q8C2×C3semidirect
37. C2×#497C2×C3semidirect
38. C2×#451C2×C3semidirect
41. #2016C2×C3semidirect
42. #2013C2×C3semidirect
43. #1960C2×C3semidirect
44. #1749C2×C3semidirect
67. C3×C4⋊Q8D4semidirect
193. C4⋊Q8C3×D4semidirect
326. C22×C3#497semidirect
401. C22C3×#497semidirect
415. C3#25726direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#4972
C2×C3×#4512
C2×C3×#4482
C3×#28502
C3×#20162
C3×#20132
C3×#19602
C3×#17492
#257263