Finite Group Databasebeta

← C3×#10054 #10628661

Subgroups — C3×#10054 #10628661

Order 768.

838 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#492C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic, complemented
3C3×#2541C211yesmaximal, characteristic
4C3×#1987C211yesmaximal, characteristic, complemented
5C2×C3×#331C211yesmaximal, characteristic, complemented
6C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10054C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—22no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#127—21no—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#121—22no—
14C2×C3×C4×D4C2211yescharacteristic
15C3×#492—22no—
16C3×#492—22no—
17C3×#476—22no—
18C3×#476—22no—
19C2×C3×#106—21no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#102—21no—
23C2×C3×C4×C8C2211yescharacteristic
24C3×#392—22no—
25C3×#390—21no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×#340—22no—
28C3×#331—22no—
29C3×D4⋊C8—22no—
30C2×#492C2×C311yescharacteristic, complemented
31C2×#476C2×C311yescharacteristic, complemented
32#2541C2×C311yescharacteristic
33#1987C2×C311yescharacteristic, complemented
34C2×#331C2×C311yescharacteristic, complemented
35C2×D4⋊C8C2×C311yescharacteristic, complemented
36#1279C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×D4—24no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C22×C3×D4D412yes—
43C23×C3×C4—21no—
44C2×C3×SD16—28no—
45C2×C3×SD16—48no—
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C22×C3×C8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8D412yescomplemented
53C3×#127—21no—
54C3×#127—21no—
55C3×#127—21no—
56C3×#127—21no—
57C3×#127—42no—
58C3×#127—42no—
59C3×#121—44no—
60C3×#121—44no—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D412yes—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42—21no—
72C2×C3×C42C2311yescharacteristic
73C3×#106—24no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8D412yescomplemented
78C3×#102—24no—
79C3×C4×C8—21no—
80C3×C4×C8D412yescomplemented
81C3×#95—21no—
82C22×SD16—22no—
83C2×C4⋊Q8C22×C311yescharacteristic
84C2×#127—21no—
85C2×#127C22×C311yescharacteristic
86C2×#121—22no—
87C2×C4×D4C22×C311yescharacteristic
88#492—22no—
89#492—22no—
90#476—22no—
91#476—22no—
92C2×#106—21no—
93C2×C4⋊C8C22×C311yescharacteristic
94C2×C4⋊C8C22×C311yescharacteristic
95C2×#102—21no—
96C2×C4×C8C22×C311yescharacteristic
97#392—22no—
98#390—21no—
99C4×C4⋊C4C22×C311yescharacteristic
100#340—22no—
101#331—22no—
102D4⋊C8—22no—
103C24×C3—21no—
104C24×C3—22no—
105C2×C3×Q8—22no—
106C2×C3×Q8—44no—
107C2×C3×Q8—41no—
108C2×C3×D4—21no—
109C2×C3×D4—22no—
110C2×C3×D4—21no—
111C2×C3×D4—22no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—44no—
116C2×C3×D4—48no—
117C2×C3×D4—48no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—42no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—84no—
129C2×C3×D4SD1612yescomplemented
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C3×SD16—416no—
141C2×C3×C8—24no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—42no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4SD1614yescomplemented
156C3×C22⋊C4—48no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C42—41no—
160C3×C42—41no—
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42#4111yescharacteristic
165C22×Q8—21no—
166C22×D4—24no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C3×D411yescharacteristic
170C22×D4C3×D412yes—
171C23×C4—21no—
172C2×SD16—28no—
173C2×SD16—48no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C3×D412yescomplemented
181#127—21no—
182#127—21no—
183#127—21no—
184#127—21no—
185#127—42no—
186#127—42no—
187#121—44no—
188#121—44no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C3×D411yescharacteristic
196C2×C4⋊C4C3×D412yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42C23×C311yescharacteristic
201#106—24no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8C3×D412yescomplemented
206#102—24no—
207C4×C8—21no—
208C4×C8C3×D412yescomplemented
209#95—21no—
210C23×C3—22no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—44no—
220C23×C3#12011yescharacteristic
221C3×Q8—44no—
222C3×Q8—82no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—44no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—82no—
229C3×D4—816no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—82no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—82no—
245C2×C3×C4—82no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12711yescharacteristic
254C2×C3×C4C2×SD1611yescharacteristic
255C2×C3×C4C2×SD1612yes—
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)12yes—
259C2×C3×C4#13711yescharacteristic
260C3×C8—24no—
261C3×C8—42no—
262C3×C8—81no—
263C24—21no—
264C24—22no—
265C2×Q8—22no—
266C2×Q8—44no—
267C2×Q8—41no—
268C2×D4—21no—
269C2×D4—22no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—44no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—84no—
289C2×D4C3×SD1612yescomplemented
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C2×C3×D411yescharacteristic
300SD16—416no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4C3×SD1614yescomplemented
316C22⋊C4—48no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42—41no—
320C42—41no—
321C42C2×C3×D411yescharacteristic
322C42C2×C3×D411yescharacteristic
323C42C2×C3×D411yescharacteristic
324C42C3×#4111yescharacteristic
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—82no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—84no—
331C22×C3—84no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3#39211yescharacteristic
336C22×C3#45011yescharacteristic
337C22×C3#45012yes—
338C22×C3#45211yescharacteristic
339C22×C3#45312yes—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—44no—
343C3×C4—82no—
344C3×C4—81no—
345C3×C4#44811yescharacteristic
346C3×C4#44911yescharacteristic
347C3×C4#46012yes—
348C3×C4#46511yescharacteristic
349C3×C4#46811yescharacteristic
350C3×C4#49211yescharacteristic
351C3×C4#49611yescharacteristic
352C23—22no—
353C23—41no—
354C23—42no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—42no—
360C23—42no—
361C23—44no—
362C23C3×#12011yescharacteristic
363Q8—44no—
364Q8—82no—
365D4—22no—
366D4—22no—
367D4—44no—
368D4—41no—
369D4—41no—
370D4—82no—
371D4—816no—
372D4—81no—
373D4—81no—
374D4—82no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4C3×#12011yescharacteristic
391C2×C4C3×#12011yescharacteristic
392C2×C4C3×#12111yescharacteristic
393C2×C4C3×#12111yescharacteristic
394C2×C4C3×#12111yescharacteristic
395C2×C4C3×#12711yescharacteristic
396C2×C4C2×C3×SD1611yescharacteristic
397C2×C4C2×C3×SD1612yes—
398C2×C4C3×#13511yescharacteristic
399C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
400C2×C4C3×AGL(1,ℤ/8)12yes—
401C2×C4C3×#13711yescharacteristic
402C8—24no—
403C8—42no—
404C8—81no—
405C2×C3—82no—
406C2×C3—162no—
407C2×C3#162912yes—
408C2×C3#163112yes—
409C2×C3#200011yescharacteristic
410C2×C3#200211yescharacteristic
411C2×C3#201411yescharacteristic
412C22—42no—
413C22—44no—
414C22—81no—
415C22—81no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—84no—
421C22—84no—
422C22C3×#39211yescharacteristic
423C22C3×#45011yescharacteristic
424C22C3×#45012yes—
425C22C3×#45211yescharacteristic
426C22C3×#45312yes—
427C4—21no—
428C4—21no—
429C4—44no—
430C4—82no—
431C4—81no—
432C4C3×#44811yescharacteristic
433C4C3×#44911yescharacteristic
434C4C3×#46012yes—
435C4C3×#46511yescharacteristic
436C4C3×#46811yescharacteristic
437C4C3×#49211yescharacteristic
438C4C3×#49611yescharacteristic
439C3#1005411yescharacteristic, direct product factor, Sylow
440C2—82no—
441C2—162no—
442C2C3×#162912yes—
443C2C3×#163112yes—
444C2C3×#200011yescharacteristic
445C2C3×#200211yescharacteristic
446C2C3×#201411yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#2541C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#492C2semidirect
2. C2×C3×#476C2semidirect
4. C3×#1987C2semidirect
5. C2×C3×#331C2semidirect
6. C2×C3×D4⋊C8C2semidirect
7. C3×#1279C2semidirect
8. #10054C3direct (N × H)
30. C2×#492C2×C3semidirect
31. C2×#476C2×C3semidirect
33. #1987C2×C3semidirect
34. C2×#331C2×C3semidirect
35. C2×D4⋊C8C2×C3semidirect
36. #1279C2×C3semidirect
52. C3×C4⋊Q8D4semidirect
77. C3×C4⋊C8D4semidirect
80. C3×C4×C8D4semidirect
129. C2×C3×D4SD16semidirect
155. C3×C4⋊C4SD16semidirect
180. C4⋊Q8C3×D4semidirect
205. C4⋊C8C3×D4semidirect
208. C4×C8C3×D4semidirect
289. C2×D4C3×SD16semidirect
315. C4⋊C4C3×SD16semidirect
439. C3#10054direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4922
C2×C3×#4762
C3×#25412
C3×#19872
C2×C3×#3312
C2×C3×D4⋊C82
C3×#12792
#100543