Finite Group Databasebeta

← C3×#10058 #10628665

Subgroups — C3×#10058 #10628665

Order 768.

602 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic
4C3×#1987C211yesmaximal, characteristic, complemented
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×D4⋊C8C211yesmaximal, characteristic
7C3×#1279C211yesmaximal, characteristic, complemented
8#10058C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#124—21no—
12C2×C3×C4×D4C2211yescharacteristic
13C3×#487—22no—
14C3×#487—22no—
15C3×#476—22no—
16C3×#476—22no—
17C2×C3×#106—21no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#103—22no—
21C2×C3×#102—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×#487C2×C311yescharacteristic, complemented
31C2×#476C2×C311yescharacteristic
32#2547C2×C311yescharacteristic
33#1987C2×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—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8D412yes—
52C3×#124—42no—
53C3×#124—42no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D412yes—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D412yes—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C3×#106—24no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#103—28no—
74C3×#102—24no—
75C3×C4×C8—21no—
76C3×C4×C8D412yes—
77C3×#95—24no—
78C3×#95—21no—
79C2×C4⋊Q8C22×C311yescharacteristic
80C2×C4⋊Q8C22×C311yescharacteristic
81C2×#124—21no—
82C2×C4×D4C22×C311yescharacteristic
83#487—22no—
84#487—22no—
85#476—22no—
86#476—22no—
87C2×#106—21no—
88C2×C4⋊C8C22×C311yescharacteristic
89C2×C4⋊C8C22×C311yescharacteristic
90C2×#103—22no—
91C2×#102—21no—
92C2×C4×C8C22×C311yescharacteristic
93#397—22no—
94#391—21no—
95#380—22no—
96C4×C4⋊C4C22×C311yescharacteristic
97#340—22no—
98#332—22no—
99D4⋊C8—22no—
100C24×C3—21no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C2×C3×Q8—41no—
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×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4SD1612yes—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—42no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—48no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155C22×Q8—21no—
156C22×Q8—21no—
157C22×D4C3×D411yescharacteristic
158C23×C4—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C3×D412yes—
170#124—42no—
171#124—42no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C3×D411yescharacteristic
180C2×C4⋊C4C3×D412yes—
181C2×C4⋊C4C3×D412yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—21no—
184C2×C42—21no—
185C2×C42C23×C311yescharacteristic
186#106—24no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8C3×D412yes—
191#103—28no—
192#102—24no—
193C4×C8—21no—
194C4×C8C3×D412yes—
195#95—24no—
196#95—21no—
197C23×C3—22no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3—41no—
201C23×C3—42no—
202C23×C3#12011yescharacteristic
203C3×Q8—44no—
204C3×Q8—82no—
205C3×Q8—81no—
206C3×Q8—81no—
207C3×Q8—81no—
208C3×Q8—81no—
209C3×D4—22no—
210C3×D4—22no—
211C3×D4—41no—
212C3×D4—41no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—82no—
224C2×C3×C4—82no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×SD1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4#13512yes—
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4#13711yescharacteristic
238C2×C3×C4#13712yes—
239C3×C8—42no—
240C3×C8—42no—
241C3×C8—81no—
242C24—21no—
243C2×Q8—22no—
244C2×Q8—44no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4C3×SD1612yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4C2×C3×D411yescharacteristic
267C22×C4C2×C3×D411yescharacteristic
268C22×C4C2×C3×D411yescharacteristic
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—48no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—41no—
292C42—41no—
293C42C2×C3×D411yescharacteristic
294C42C2×C3×D411yescharacteristic
295C42C2×C3×D411yescharacteristic
296C42C3×#4111yescharacteristic
297C22×C3—44no—
298C22×C3—42no—
299C22×C3—82no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3#39211yescharacteristic
303C22×C3#45011yescharacteristic
304C22×C3#45212yes—
305C22×C3#45211yescharacteristic
306C22×C3#45412yes—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—82no—
310C3×C4—82no—
311C3×C4—82no—
312C3×C4—81no—
313C3×C4#44811yescharacteristic
314C3×C4#44911yescharacteristic
315C3×C4#46412yes—
316C3×C4#46511yescharacteristic
317C3×C4#46811yescharacteristic
318C3×C4#49511yescharacteristic
319C3×C4#49711yescharacteristic
320C23—22no—
321C23—41no—
322C23—42no—
323C23—41no—
324C23—42no—
325C23C3×#12011yescharacteristic
326Q8—44no—
327Q8—82no—
328Q8—81no—
329Q8—81no—
330Q8—81no—
331Q8—81no—
332D4—22no—
333D4—22no—
334D4—41no—
335D4—41no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4C3×#12011yescharacteristic
351C2×C4C3×#12011yescharacteristic
352C2×C4C3×#12111yescharacteristic
353C2×C4C3×#12111yescharacteristic
354C2×C4C3×#12111yescharacteristic
355C2×C4C3×#12711yescharacteristic
356C2×C4C2×C3×SD1611yescharacteristic
357C2×C4C3×#13511yescharacteristic
358C2×C4C3×#13512yes—
359C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
360C2×C4C3×#13711yescharacteristic
361C2×C4C3×#13712yes—
362C8—42no—
363C8—42no—
364C8—81no—
365C2×C3—82no—
366C2×C3#163512yes—
367C2×C3#163812yes—
368C2×C3#200011yescharacteristic
369C2×C3#200311yescharacteristic
370C2×C3#201511yescharacteristic
371C22—42no—
372C22—44no—
373C22—81no—
374C22—81no—
375C22—82no—
376C22C3×#39211yescharacteristic
377C22C3×#45011yescharacteristic
378C22C3×#45211yescharacteristic
379C22C3×#45212yes—
380C22C3×#45412yes—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—82no—
385C4—81no—
386C4—82no—
387C4C3×#44811yescharacteristic
388C4C3×#44911yescharacteristic
389C4C3×#46412yes—
390C4C3×#46511yescharacteristic
391C4C3×#46811yescharacteristic
392C4C3×#49511yescharacteristic
393C4C3×#49711yescharacteristic
394C3#1005811yescharacteristic, direct product factor, Sylow
395C2—82no—
396C2C3×#163512yes—
397C2C3×#163812yes—
398C2C3×#200011yescharacteristic
399C2C3×#200311yescharacteristic
400C2C3×#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×#487C2semidirect
4. C3×#1987C2semidirect
5. C2×C3×#332C2semidirect
7. C3×#1279C2semidirect
8. #10058C3direct (N × H)
30. C2×#487C2×C3semidirect
33. #1987C2×C3semidirect
34. C2×#332C2×C3semidirect
36. #1279C2×C3semidirect
394. C3#10058direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×C3×#4762
C3×#25472
C3×#19872
C2×C3×#3322
C2×C3×D4⋊C82
C3×#12792
#100583