Finite Group Databasebeta

← C3×#15173 #10633780

Subgroups — C3×#15173 #10633780

Order 768.

1530 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×#398C212yesmaximal
2C2×C3×#395C212yesmaximal
3C2×C3×#391C211yesmaximal, characteristic
4C2×C3×#389C211yesmaximal, characteristic
5C3×#1444C212yesmaximal
6C3×#1444C212yesmaximal
7C3×C4⋊#95C214yesmaximal
8C3×C4×#95C211yesmaximal, characteristic
9#15173C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2214yes—
15C22×C3×C4⋊C4C2214yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×C3×C42C2212yes—
19C3×#398—28no—
20C3×#395—28no—
21C3×#391—24no—
22C3×#389—24no—
23C2×C3×#95C2214yes—
24C2×C3×#95C2214yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×#398C2×C312yes—
31C2×#395C2×C312yes—
32C2×#391C2×C311yescharacteristic
33C2×#389C2×C311yescharacteristic
34#1444C2×C312yes—
35#1444C2×C312yes—
36C4⋊#95C2×C314yes—
37C4×#95C2×C311yescharacteristic
38C22×C3×Q8—22no—
39C22×C3×Q8—24no—
40C22×C3×Q8—24no—
41C22×C3×Q8—42no—
42C23×C3×C4C2312yes—
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2314yes—
49C23×C3×C4C2311yescharacteristic
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—28no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—28no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—28no—
61C2×C3×C4⋊C4—28no—
62C2×C3×C4⋊C4—44no—
63C2×C3×C4⋊C4—44no—
64C2×C3×C4⋊C4—42no—
65C2×C3×C42—42no—
66C2×C3×C42—42no—
67C2×C3×C42—42no—
68C2×C3×C42D414yes—
69C2×C3×C42Q818yescomplemented
70C3×#95—24no—
71C3×#95—24no—
72C3×#95—24no—
73C3×#95—28no—
74C3×#95—28no—
75C3×#95D418yes—
76C3×#95Q818yescomplemented
77C23×Q8C22×C311yescharacteristic
78C22×C4⋊C4C22×C312yes—
79C22×C4⋊C4C22×C311yescharacteristic
80C22×C4⋊C4C22×C311yescharacteristic
81C22×C4⋊C4C22×C314yes—
82C22×C4⋊C4C22×C314yes—
83C22×C4⋊C4C22×C311yescharacteristic
84C22×C42C22×C311yescharacteristic
85C22×C42C22×C312yes—
86#398—28no—
87#395—28no—
88#391—24no—
89#389—24no—
90C2×#95C22×C314yes—
91C2×#95C22×C314yes—
92C2×#95C22×C312yes—
93C2×#95C22×C312yes—
94C2×#95C22×C312yes—
95C2×#95C22×C312yes—
96C2×#95C22×C312yes—
97C24×C3C2411yescharacteristic
98C2×C3×Q8—28no—
99C2×C3×Q8—416no—
100C2×C3×Q8—44no—
101C2×C3×Q8—44no—
102C22×C3×C4—22no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—24no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—24no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—22no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4C2×D414yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D418yes—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×Q814yes—
139C22×C3×C4C2×Q818yes—
140C22×C3×C4C2×Q818yes—
141C22×C3×C4C2×Q814yes—
142C22×C3×C4#4114yes—
143C22×C3×C4#4114yes—
144C3×C4⋊C4—28no—
145C3×C4⋊C4—416no—
146C3×C4⋊C4—416no—
147C3×C4⋊C4—44no—
148C3×C4⋊C4—48no—
149C3×C4⋊C4—44no—
150C3×C42—44no—
151C3×C42—48no—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—42no—
156C23×C4C23×C312yes—
157C23×C4C23×C312yes—
158C23×C4C23×C312yes—
159C23×C4C23×C312yes—
160C23×C4C23×C311yescharacteristic
161C23×C4C23×C314yes—
162C23×C4C23×C311yescharacteristic
163C23×C4C23×C311yescharacteristic
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42C3×D414yes—
183C2×C42C3×Q818yescomplemented
184#95—24no—
185#95—24no—
186#95—24no—
187#95—28no—
188#95—28no—
189#95C3×D418yes—
190#95C3×Q818yescomplemented
191C23×C3C22×D411yescharacteristic
192C23×C3C22×D412yes—
193C23×C3C22×Q812yes—
194C23×C3C22×Q812yes—
195C23×C3C2×#4112yes—
196C23×C3C2×#4112yes—
197C23×C3H(5,𝔽2)12yes—
198C23×C3H(5,𝔽2)11yescharacteristic
199C23×C32−1+411yescharacteristic
200C3×Q8—416no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—28no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—24no—
209C2×C3×C4—28no—
210C2×C3×C4—24no—
211C2×C3×C4—28no—
212C2×C3×C4—28no—
213C2×C3×C4—28no—
214C2×C3×C4—28no—
215C2×C3×C4—28no—
216C2×C3×C4—28no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—28no—
221C2×C3×C4—24no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—44no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—44no—
236C2×C3×C4—44no—
237C2×C3×C4—44no—
238C2×C3×C4—44no—
239C2×C3×C4#12714yes—
240C2×C3×C4C4⋊Q814yes—
241C2×C3×C4C4⋊Q814yes—
242C2×C3×C4C4⋊Q8116yes—
243C24C24×C311yescharacteristic
244C2×Q8—28no—
245C2×Q8—416no—
246C2×Q8—44no—
247C2×Q8—44no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4C2×C3×D414yes—
280C22×C4C2×C3×D412yes—
281C22×C4C2×C3×D412yes—
282C22×C4C2×C3×D418yes—
283C22×C4C2×C3×D412yes—
284C22×C4C2×C3×Q818yes—
285C22×C4C2×C3×Q814yes—
286C22×C4C2×C3×Q818yes—
287C22×C4C2×C3×Q814yes—
288C22×C4C3×#4114yes—
289C22×C4C3×#4114yes—
290C4⋊C4—28no—
291C4⋊C4—416no—
292C4⋊C4—416no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C42—44no—
297C42—48no—
298C22×C3#52511yescharacteristic
299C22×C3#52511yescharacteristic
300C22×C3C2×#12711yescharacteristic
301C22×C3C2×C4⋊Q811yescharacteristic
302C22×C3C2×C4⋊Q811yescharacteristic
303C22×C3C2×C4⋊Q814yes—
304C22×C3#53314yes—
305C22×C3#53411yescharacteristic
306C22×C3#53711yescharacteristic
307C22×C3#54312yes—
308C22×C3#54412yes—
309C22×C3#54614yes—
310C22×C3#54612yes—
311C22×C3#54812yes—
312C22×C3#55414yes—
313C22×C3#55412yes—
314C22×C3#55712yes—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—216no—
319C3×C4—44no—
320C3×C4—44no—
321C3×C4—82no—
322C3×C4—82no—
323C23C22×C3×D411yescharacteristic
324C23C22×C3×D412yes—
325C23C22×C3×Q812yes—
326C23C22×C3×Q812yes—
327C23C2×C3×#4112yes—
328C23C2×C3×#4112yes—
329C23C3×H(5,𝔽2)12yes—
330C23C3×H(5,𝔽2)11yescharacteristic
331C23C3×2−1+411yescharacteristic
332Q8—416no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4C3×#12714yes—
372C2×C4C3×C4⋊Q814yes—
373C2×C4C3×C4⋊Q8116yes—
374C2×C4C3×C4⋊Q814yes—
375C2×C3#239911yescharacteristic
376C2×C3#253911yescharacteristic
377C2×C3#254011yescharacteristic
378C2×C3#254914yes—
379C2×C3#255014yes—
380C2×C3#258311yescharacteristic
381C2×C3#258411yescharacteristic
382C2×C3#265911yescharacteristic
383C2×C3#286811yescharacteristic
384C22C3×#52511yescharacteristic
385C22C3×#52511yescharacteristic
386C22C2×C3×#12711yescharacteristic
387C22C2×C3×C4⋊Q811yescharacteristic
388C22C2×C3×C4⋊Q814yes—
389C22C2×C3×C4⋊Q811yescharacteristic
390C22C3×#53314yes—
391C22C3×#53411yescharacteristic
392C22C3×#53711yescharacteristic
393C22C3×#54312yes—
394C22C3×#54412yes—
395C22C3×#54614yes—
396C22C3×#54612yes—
397C22C3×#54812yes—
398C22C3×#55412yes—
399C22C3×#55414yes—
400C22C3×#55712yes—
401C4—24no—
402C4—24no—
403C4—216no—
404C4—24no—
405C4—44no—
406C4—44no—
407C4—82no—
408C4—82no—
409C3#1517311yescharacteristic, direct product factor, Sylow
410C2C3×#239911yescharacteristic
411C2C3×#253911yescharacteristic
412C2C3×#254011yescharacteristic
413C2C3×#254914yes—
414C2C3×#255014yes—
415C2C3×#258311yescharacteristic
416C2C3×#258411yescharacteristic
417C2C3×#265911yescharacteristic
418C2C3×#286811yescharacteristic
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
9. #15173C3direct (N × H)
69. C2×C3×C42Q8semidirect
76. C3×#95Q8semidirect
183. C2×C42C3×Q8semidirect
190. #95C3×Q8semidirect
409. C3#15173direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3982
C2×C3×#3952
C2×C3×#3912
C2×C3×#3892
C3×#14442
C3×C4⋊#952
C3×C4×#952
#151733