Finite Group Databasebeta

← C3×#16603 #10635210

Subgroups — C3×#16603 #10635210

Order 768.

1598 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#395C212yesmaximal
3C2×C3×#387C211yesmaximal, characteristic
4C3×#1446C212yesmaximal
5C3×#1444C212yesmaximal
6C3×#1444C212yesmaximal
7C3×#1442C212yesmaximal
8C3×#1441C212yesmaximal
9C3×#1434C211yesmaximal, characteristic
10#16603C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2212yes—
12C2×C3×C4⋊Q8—28no—
13C2×C3×C4⋊Q8—24no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2214yes—
19C22×C3×C42C2211yescharacteristic
20C3×#395—28no—
21C3×#387—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2214yes—
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2211yescharacteristic
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2214yes—
30C2×C3×#95C2214yes—
31C2×C3×#95C2212yes—
32C22×C4⋊Q8C2×C311yescharacteristic
33C2×#395C2×C312yes—
34C2×#387C2×C311yescharacteristic
35#1446C2×C312yes—
36#1444C2×C312yes—
37#1444C2×C312yes—
38#1442C2×C312yes—
39#1441C2×C312yes—
40#1434C2×C311yescharacteristic
41C22×C3×Q8—24no—
42C22×C3×Q8—24no—
43C22×C3×Q8—44no—
44C22×C3×Q8—44no—
45C22×C3×Q8D418yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2314yes—
49C23×C3×C4C2314yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C3×C4⋊Q8—232no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—28no—
58C2×C3×C4⋊C4—28no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—28no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—44no—
66C2×C3×C4⋊C4—44no—
67C2×C3×C4⋊C4Q814yescomplemented
68C2×C3×C42—24no—
69C2×C3×C42D414yes—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—22no—
73C3×#95—22no—
74C3×#95—28no—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—44no—
78C3×#95—44no—
79C3×#95Q818yescomplemented
80C23×Q8C22×C312yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—24no—
83C22×C4⋊C4C22×C311yescharacteristic
84C22×C4⋊C4C22×C312yes—
85C22×C4⋊C4C22×C312yes—
86C22×C4⋊C4C22×C311yescharacteristic
87C22×C4⋊C4C22×C314yes—
88C22×C42C22×C311yescharacteristic
89#395—28no—
90#387—24no—
91C2×#95C22×C312yes—
92C2×#95C22×C314yes—
93C2×#95C22×C312yes—
94C2×#95C22×C311yescharacteristic
95C2×#95C22×C311yescharacteristic
96C2×#95C22×C311yescharacteristic
97C2×#95C22×C311yescharacteristic
98C2×#95C22×C314yes—
99C2×#95C22×C314yes—
100C2×#95C22×C312yes—
101C24×C3C2411yescharacteristic
102C2×C3×Q8—216no—
103C2×C3×Q8—416no—
104C2×C3×Q8—416no—
105C2×C3×Q8—48no—
106C2×C3×Q8—48no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—22no—
115C22×C3×C4—24no—
116C22×C3×C4—21no—
117C22×C3×C4—24no—
118C22×C3×C4—24no—
119C22×C3×C4—24no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—24no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—24no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D418yes—
142C22×C3×C4C2×D414yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×Q818yes—
147C22×C3×C4C2×Q812yes—
148C22×C3×C4C2×Q812yes—
149C22×C3×C4C2×Q812yes—
150C22×C3×C4C2×Q814yes—
151C22×C3×C4#4112yes—
152C3×C4⋊C4—216no—
153C3×C4⋊C4—216no—
154C3×C4⋊C4—28no—
155C3×C4⋊C4—416no—
156C3×C4⋊C4—44no—
157C3×C42—28no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8C3×D418yes—
163C23×C4C23×C311yescharacteristic
164C23×C4C23×C311yescharacteristic
165C23×C4C23×C314yes—
166C23×C4C23×C311yescharacteristic
167C23×C4C23×C314yes—
168C23×C4C23×C311yescharacteristic
169C23×C4C23×C311yescharacteristic
170C23×C4C23×C311yescharacteristic
171C23×C4C23×C311yescharacteristic
172C4⋊Q8—232no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C3×Q814yescomplemented
185C2×C42—24no—
186C2×C42C3×D414yes—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—28no—
192#95—24no—
193#95—24no—
194#95—44no—
195#95—44no—
196#95C3×Q818yescomplemented
197C23×C3C22×D411yescharacteristic
198C23×C3C22×D412yes—
199C23×C3C22×Q811yescharacteristic
200C23×C3C22×Q812yes—
201C23×C3C2×#4111yescharacteristic
202C23×C3H(5,𝔽2)12yes—
203C23×C3H(5,𝔽2)12yes—
204C23×C32−1+412yes—
205C23×C32−1+412yes—
206C3×Q8—432no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—28no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—28no—
217C2×C3×C4—28no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—28no—
221C2×C3×C4—28no—
222C2×C3×C4—24no—
223C2×C3×C4—28no—
224C2×C3×C4—24no—
225C2×C3×C4—28no—
226C2×C3×C4—28no—
227C2×C3×C4—41no—
228C2×C3×C4—44no—
229C2×C3×C4—44no—
230C2×C3×C4—44no—
231C2×C3×C4—41no—
232C2×C3×C4—44no—
233C2×C3×C4—44no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—44no—
238C2×C3×C4—42no—
239C2×C3×C4—44no—
240C2×C3×C4—44no—
241C2×C3×C4—41no—
242C2×C3×C4—44no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—44no—
249C2×C3×C4#12014yes—
250C2×C3×C4C22⋊Q814yes—
251C2×C3×C4C4⋊Q814yes—
252C24C24×C311yescharacteristic
253C2×Q8—216no—
254C2×Q8—416no—
255C2×Q8—416no—
256C2×Q8—48no—
257C2×Q8—48no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4C2×C3×D418yes—
293C22×C4C2×C3×D414yes—
294C22×C4C2×C3×D412yes—
295C22×C4C2×C3×D412yes—
296C22×C4C2×C3×D412yes—
297C22×C4C2×C3×Q818yes—
298C22×C4C2×C3×Q812yes—
299C22×C4C2×C3×Q812yes—
300C22×C4C2×C3×Q812yes—
301C22×C4C2×C3×Q814yes—
302C22×C4C3×#4112yes—
303C4⋊C4—216no—
304C4⋊C4—216no—
305C4⋊C4—28no—
306C4⋊C4—416no—
307C4⋊C4—44no—
308C42—28no—
309C22×C3C2×#12011yescharacteristic
310C22×C3C2×C22⋊Q811yescharacteristic
311C22×C3C2×C4⋊Q811yescharacteristic
312C22×C3#53414yes—
313C22×C3#53511yescharacteristic
314C22×C3#53511yescharacteristic
315C22×C3#53611yescharacteristic
316C22×C3#53611yescharacteristic
317C22×C3#54212yes—
318C22×C3#54412yes—
319C22×C3#54414yes—
320C22×C3#54812yes—
321C22×C3D4×Q812yes—
322C22×C3D4×Q814yes—
323C22×C3#55412yes—
324C22×C3Q8212yes—
325C22×C3#56314yes—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—24no—
329C3×C4—48no—
330C3×C4—42no—
331C3×C4—42no—
332C3×C4—48no—
333C3×C4—81no—
334C3×C4—81no—
335C23C22×C3×D411yescharacteristic
336C23C22×C3×D412yes—
337C23C22×C3×Q811yescharacteristic
338C23C22×C3×Q812yes—
339C23C2×C3×#4111yescharacteristic
340C23C3×H(5,𝔽2)12yes—
341C23C3×H(5,𝔽2)12yes—
342C23C3×2−1+412yes—
343C23C3×2−1+412yes—
344Q8—432no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4C3×#12014yes—
388C2×C4C3×C22⋊Q814yes—
389C2×C4C3×C4⋊Q814yes—
390C2×C3#240911yescharacteristic
391C2×C3#243411yescharacteristic
392C2×C3#256711yescharacteristic
393C2×C3#261311yescharacteristic
394C2×C3#262811yescharacteristic
395C2×C3#266011yescharacteristic
396C2×C3#272914yes—
397C2×C3#273414yes—
398C2×C3#282111yescharacteristic
399C22C2×C3×#12011yescharacteristic
400C22C2×C3×C22⋊Q811yescharacteristic
401C22C2×C3×C4⋊Q811yescharacteristic
402C22C3×#53414yes—
403C22C3×#53511yescharacteristic
404C22C3×#53511yescharacteristic
405C22C3×#53611yescharacteristic
406C22C3×#53611yescharacteristic
407C22C3×#54212yes—
408C22C3×#54412yes—
409C22C3×#54414yes—
410C22C3×#54812yes—
411C22C3×D4×Q812yes—
412C22C3×D4×Q814yes—
413C22C3×#55412yes—
414C22C3×Q8212yes—
415C22C3×#56314yes—
416C4—24no—
417C4—24no—
418C4—24no—
419C4—42no—
420C4—42no—
421C4—48no—
422C4—48no—
423C4—81no—
424C4—81no—
425C3#1660311yescharacteristic, direct product factor, Sylow
426C2C3×#240911yescharacteristic
427C2C3×#243411yescharacteristic
428C2C3×#256711yescharacteristic
429C2C3×#261311yescharacteristic
430C2C3×#262811yescharacteristic
431C2C3×#266011yescharacteristic
432C2C3×#272914yes—
433C2C3×#273414yes—
434C2C3×#282111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
10. #16603C3direct (N × H)
67. C2×C3×C4⋊C4Q8semidirect
79. C3×#95Q8semidirect
184. C2×C4⋊C4C3×Q8semidirect
196. #95C3×Q8semidirect
425. C3#16603direct (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×#3952
C2×C3×#3872
C3×#14462
C3×#14442
C3×#14422
C3×#14412
C3×#14342
#166033