Finite Group Databasebeta

← C3×#33010 #10651617

Subgroups — C3×#33010 #10651617

Order 768.

1314 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic
3C2×C3×#451C211yesmaximal, characteristic
4C3×#2855C211yesmaximal, characteristic
5C3×#1878C214yesmaximal, complemented
6C3×#1863C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic
8C3×#1817C211yesmaximal, characteristic
9C3×#1787C211yesmaximal, characteristic
10#33010C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×Q16C2212yes—
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8—24no—
15C2×C3×C4×Q8—24no—
16C2×C3×C4×Q8—22no—
17C2×C3×C4×Q8C2214yes—
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C42—21no—
21C22×C3×C42C2211yescharacteristic
22C3×#484—24no—
23C3×#451—24no—
24C2×C3×#107C2212yes—
25C2×C3×#103C2214yes—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#391C2212yes—
31C3×#389C2212yes—
32C3×#387—22no—
33C3×#385—21no—
34C3×#381—21no—
35C3×#380C2211yescharacteristic
36C3×#340C2214yes—
37C3×#336C2214yes—
38C22×C4×Q8C2×C311yescharacteristic
39C2×#484C2×C311yescharacteristic
40C2×#451C2×C311yescharacteristic
41#2855C2×C311yescharacteristic
42#1878C2×C314yescomplemented
43#1863C2×C314yescomplemented
44#1824C2×C311yescharacteristic
45#1817C2×C311yescharacteristic
46#1787C2×C311yescharacteristic
47C22×C3×Q8—28no—
48C22×C3×Q8—22no—
49C22×C3×Q8—21no—
50C22×C3×Q8D414yes—
51C22×C3×Q8C2311yescharacteristic
52C22×C3×Q8C2312yes—
53C23×C3×C4—21no—
54C23×C3×C4—21no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C2×C3×Q16—24no—
59C2×C3×Q16—28no—
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×C4×Q8—216no—
65C3×C4×Q8—28no—
66C3×C4×Q8—28no—
67C2×C3×C4⋊C4—28no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D414yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—28no—
80C2×C3×C42—24no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42D412yes—
84C2×C3×C42C2312yes—
85C3×#107—24no—
86C3×#103—24no—
87C3×#103—28no—
88C3×#103—24no—
89C3×C22⋊C8—22no—
90C3×C22⋊C8—22no—
91C3×#95—21no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—22no—
95C23×Q8C22×C311yescharacteristic
96C22×Q16C22×C312yes—
97C2×C22⋊Q8C22×C311yescharacteristic
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—22no—
101C2×C4×Q8C22×C314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C42—21no—
105C22×C42C22×C311yescharacteristic
106#484—24no—
107#451—24no—
108C2×#107C22×C312yes—
109C2×#103C22×C312yes—
110C2×#103C22×C312yes—
111C2×#103C22×C314yes—
112C2×C22⋊C8C22×C311yescharacteristic
113C2×C22⋊C8C22×C311yescharacteristic
114#391C22×C312yes—
115#389C22×C312yes—
116#387—22no—
117#385—21no—
118#381—21no—
119#380C22×C311yescharacteristic
120#340C22×C314yes—
121#336C22×C314yes—
122C24×C3C2×D411yescharacteristic
123C2×C3×Q8—216no—
124C2×C3×Q8—28no—
125C2×C3×Q8—216no—
126C2×C3×Q8—24no—
127C2×C3×Q8—22no—
128C2×C3×Q8—24no—
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—24no—
132C2×C3×Q8C2×D414yes—
133C2×C3×Q8#4118yes—
134C22×C3×C4—28no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—42no—
147C22×C3×C4—44no—
148C22×C3×C4Q1614yescomplemented
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2411yescharacteristic
155C3×Q16—48no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—24no—
159C2×C3×C8—24no—
160C3×C4⋊C4—216no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—44no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4C2×D414yes—
169C3×C22⋊C4—42no—
170C3×C22⋊C4—42no—
171C3×C42—216no—
172C3×C42—24no—
173C3×C42—22no—
174C3×C42—22no—
175C22×Q8—28no—
176C22×Q8—22no—
177C22×Q8—21no—
178C22×Q8C3×D414yes—
179C22×Q8C23×C311yescharacteristic
180C22×Q8C23×C312yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C3×D411yescharacteristic
184C23×C4C3×D411yescharacteristic
185C23×C4C23×C311yescharacteristic
186C2×Q16—24no—
187C2×Q16—28no—
188C22×C8C23×C312yes—
189C22×C8C23×C312yes—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C4×Q8—216no—
193C4×Q8—28no—
194C4×Q8—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C3×D414yes—
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C312yes—
204C2×C4⋊C4C23×C312yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—28no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42C3×D412yes—
212C2×C42C23×C312yes—
213#107—24no—
214#103—24no—
215#103—28no—
216#103—24no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—22no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3#12012yes—
226C23×C3C2×Q1612yes—
227C23×C3#13712yes—
228C23×C3C22×D411yescharacteristic
229C3×Q8—216no—
230C3×Q8—24no—
231C3×Q8—28no—
232C3×Q8—42no—
233C3×Q8—42no—
234C3×Q8#12318yes—
235C2×C3×C4—216no—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—28no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—28no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—44no—
249C2×C3×C4—48no—
250C2×C3×C4—44no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4#12012yes—
259C2×C3×C4C2×Q1614yes—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C2×#4114yes—
263C3×C8—42no—
264C3×C8—42no—
265C24C2×C3×D411yescharacteristic
266C2×Q8—216no—
267C2×Q8—28no—
268C2×Q8—216no—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—24no—
275C2×Q8C2×C3×D414yes—
276C2×Q8C3×#4118yes—
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4C3×Q1614yescomplemented
292C22×C4C2×C3×D412yes—
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C2×C3×D412yes—
295C22×C4C2×C3×D412yes—
296C22×C4C2×C3×D412yes—
297C22×C4C24×C311yescharacteristic
298Q16—48no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C4⋊C4—216no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—24no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4C2×C3×D414yes—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C42—216no—
315C42—24no—
316C42—22no—
317C42—22no—
318C22×C3—24no—
319C22×C3—22no—
320C22×C3—24no—
321C22×C3—22no—
322C22×C3#45114yes—
323C22×C3C2×#12011yescharacteristic
324C22×C3#52512yes—
325C22×C3C22×Q1611yescharacteristic
326C22×C3C2×#13711yescharacteristic
327C22×C3#57512yes—
328C3×C4—28no—
329C3×C4—22no—
330C3×C4—24no—
331C3×C4—42no—
332C3×C4—44no—
333C3×C4—81no—
334C3×C4—82no—
335C3×C4—81no—
336C3×C4C2×#12312yes—
337C3×C4#52512yes—
338C23—22no—
339C23—22no—
340C23C3×#12012yes—
341C23C2×C3×Q1612yes—
342C23C3×#13712yes—
343C23C22×C3×D411yescharacteristic
344Q8—216no—
345Q8—24no—
346Q8—28no—
347Q8—42no—
348Q8—42no—
349Q8C3×#12318yes—
350C2×C4—216no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4C3×#12012yes—
374C2×C4C2×C3×Q1614yes—
375C2×C4C22×C3×D411yescharacteristic
376C2×C4C22×C3×D411yescharacteristic
377C2×C4C2×C3×#4114yes—
378C8—42no—
379C8—42no—
380C2×C3—24no—
381C2×C3#284911yescharacteristic
382C2×C3C2×#45111yescharacteristic
383C2×C3#300511yescharacteristic
384C2×C3#310412yes—
385C2×C3#311712yes—
386C22—22no—
387C22—24no—
388C22—22no—
389C22—24no—
390C22C3×#45114yes—
391C22C2×C3×#12011yescharacteristic
392C22C3×#52512yes—
393C22C22×C3×Q1611yescharacteristic
394C22C2×C3×#13711yescharacteristic
395C22C3×#57512yes—
396C4—24no—
397C4—28no—
398C4—22no—
399C4—42no—
400C4—44no—
401C4—81no—
402C4—81no—
403C4—82no—
404C4C2×C3×#12312yes—
405C4C3×#52512yes—
406C3#3301011yescharacteristic, direct product factor, Sylow
407C2—24no—
408C2C3×#284911yescharacteristic
409C2C2×C3×#45111yescharacteristic
410C2C3×#300511yescharacteristic
411C2C3×#310412yes—
412C2C3×#311712yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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
5. C3×#1878C2semidirect
6. C3×#1863C2semidirect
10. #33010C3direct (N × H)
42. #1878C2×C3semidirect
43. #1863C2×C3semidirect
148. C22×C3×C4Q16semidirect
291. C22×C4C3×Q16semidirect
406. C3#33010direct (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×#4842
C2×C3×#4512
C3×#28552
C3×#18782
C3×#18632
C3×#18242
C3×#18172
C3×#17872
#330103