Finite Group Databasebeta

← C3×#10066 #10628673

Subgroups — C3×#10066 #10628673

Order 768.

634 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#500C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic, complemented
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×D4⋊C8C211yesmaximal, characteristic
7C3×#1279C211yesmaximal, characteristic, complemented
8#10066C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#124—21no—
13C2×C3×C4×D4C2211yescharacteristic
14C3×#500—22no—
15C3×#500—22no—
16C3×#461—22no—
17C3×#461—22no—
18C2×C3×#107—22no—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#103—21no—
22C2×C3×C4×C8C2211yescharacteristic
23C3×#397—22no—
24C3×#391—21no—
25C3×#380—22no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×#340—22no—
28C3×#332—22no—
29C3×D4⋊C8—22no—
30C2×#500C2×C311yescharacteristic, complemented
31C2×#461C2×C311yescharacteristic
32#2547C2×C311yescharacteristic
33#1988C2×C311yescharacteristic, complemented
34C2×#332C2×C311yescharacteristic, complemented
35C2×D4⋊C8C2×C311yescharacteristic
36#1279C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C2×C3×SD16—24no—
42C2×C3×SD16—42no—
43C2×C3×SD16—42no—
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8D412yes—
55C3×#124—42no—
56C3×#124—42no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D412yes—
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C3×#107—28no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×#103—24no—
77C3×C4×C8—21no—
78C3×C4×C8D412yes—
79C3×#95—24no—
80C3×#95—21no—
81C22×SD16—21no—
82C2×C4⋊Q8C22×C311yescharacteristic
83C2×C4⋊Q8C22×C311yescharacteristic
84C2×#124—21no—
85C2×C4×D4C22×C311yescharacteristic
86#500—22no—
87#500—22no—
88#461—22no—
89#461—22no—
90C2×#107—22no—
91C2×C4⋊C8C22×C311yescharacteristic
92C2×C4⋊C8C22×C311yescharacteristic
93C2×#103—21no—
94C2×C4×C8C22×C311yescharacteristic
95#397—22no—
96#391—21no—
97#380—22no—
98C4×C4⋊C4C22×C311yescharacteristic
99#340—22no—
100#332—22no—
101D4⋊C8—22no—
102C24×C3—21no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4D812yes—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C3×SD16—48no—
133C2×C3×C8—24no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8—42no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—48no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161C22×Q8—21no—
162C22×Q8—21no—
163C22×D4C3×D411yescharacteristic
164C23×C4—21no—
165C2×SD16—24no—
166C2×SD16—42no—
167C2×SD16—42no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C3×D412yes—
179#124—42no—
180#124—42no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C3×D411yescharacteristic
189C2×C4⋊C4C3×D412yes—
190C2×C4⋊C4C3×D412yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42C23×C311yescharacteristic
195#107—28no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8C3×D412yes—
200#103—24no—
201C4×C8—21no—
202C4×C8C3×D412yes—
203#95—24no—
204#95—21no—
205C23×C3—22no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3#12011yescharacteristic
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8—81no—
214C3×Q8—81no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×Q8—81no—
218C3×Q8—81no—
219C3×D4—22no—
220C3×D4—22no—
221C3×D4—41no—
222C3×D4—41no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4—82no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4#13512yes—
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4#13712yes—
250C3×C8—24no—
251C3×C8—42no—
252C3×C8—81no—
253C24—21no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4C3×D812yes—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C2×C3×D411yescharacteristic
282C22×C4C2×C3×D411yescharacteristic
283SD16—48no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—48no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C42—41no—
307C42—41no—
308C42C2×C3×D411yescharacteristic
309C42C2×C3×D411yescharacteristic
310C42C2×C3×D411yescharacteristic
311C42C3×#4111yescharacteristic
312C22×C3—44no—
313C22×C3—42no—
314C22×C3—82no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3#39211yescharacteristic
318C22×C3#44711yescharacteristic
319C22×C3#44911yescharacteristic
320C22×C3#45212yes—
321C22×C3#45612yes—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—82no—
325C3×C4—82no—
326C3×C4—82no—
327C3×C4—81no—
328C3×C4#44711yescharacteristic
329C3×C4#44911yescharacteristic
330C3×C4#46412yes—
331C3×C4#46611yescharacteristic
332C3×C4#46911yescharacteristic
333C3×C4#49511yescharacteristic
334C3×C4#49711yescharacteristic
335C23—22no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23C3×#12011yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349D4—22no—
350D4—22no—
351D4—41no—
352D4—41no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4C3×#12011yescharacteristic
369C2×C4C3×#12011yescharacteristic
370C2×C4C3×#12111yescharacteristic
371C2×C4C3×#12111yescharacteristic
372C2×C4C3×#12111yescharacteristic
373C2×C4C3×#12711yescharacteristic
374C2×C4C2×C3×D811yescharacteristic
375C2×C4C3×#13511yescharacteristic
376C2×C4C3×#13512yes—
377C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
378C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
379C2×C4C3×#13712yes—
380C8—24no—
381C8—42no—
382C8—81no—
383C2×C3—82no—
384C2×C3#164712yes—
385C2×C3#165012yes—
386C2×C3#199911yescharacteristic
387C2×C3#200511yescharacteristic
388C2×C3#201511yescharacteristic
389C22—42no—
390C22—44no—
391C22—81no—
392C22—81no—
393C22—82no—
394C22C3×#39211yescharacteristic
395C22C3×#44711yescharacteristic
396C22C3×#44911yescharacteristic
397C22C3×#45212yes—
398C22C3×#45612yes—
399C4—21no—
400C4—21no—
401C4—82no—
402C4—82no—
403C4—81no—
404C4—82no—
405C4C3×#44711yescharacteristic
406C4C3×#44911yescharacteristic
407C4C3×#46412yes—
408C4C3×#46611yescharacteristic
409C4C3×#46911yescharacteristic
410C4C3×#49511yescharacteristic
411C4C3×#49711yescharacteristic
412C3#1006611yescharacteristic, direct product factor, Sylow
413C2—82no—
414C2C3×#164712yes—
415C2C3×#165012yes—
416C2C3×#199911yescharacteristic
417C2C3×#200511yescharacteristic
418C2C3×#201511yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
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×#500C2semidirect
4. C3×#1988C2semidirect
5. C2×C3×#332C2semidirect
7. C3×#1279C2semidirect
8. #10066C3direct (N × H)
30. C2×#500C2×C3semidirect
33. #1988C2×C3semidirect
34. C2×#332C2×C3semidirect
36. #1279C2×C3semidirect
412. C3#10066direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5002
C2×C3×#4612
C3×#25472
C3×#19882
C2×C3×#3322
C2×C3×D4⋊C82
C3×#12792
#100663