Finite Group Databasebeta

← C3×#10051 #10628658

Subgroups — C3×#10051 #10628658

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic
3C3×#2546C211yesmaximal, characteristic, complemented
4C3×#1982C211yesmaximal, characteristic
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10051C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8—21no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic
14C3×#486—22no—
15C3×#486—22no—
16C3×#460—22no—
17C3×#460—22no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#102—22no—
21C2×C3×#102—21no—
22C2×C3×#102—22no—
23C2×C3×C4×C8C2211yescharacteristic
24C3×#398—22no—
25C3×#391—21no—
26C3×#389—22no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C3×#332—22no—
29C3×Q8⋊C8—22no—
30C2×#486C2×C311yescharacteristic
31C2×#460C2×C311yescharacteristic
32#2546C2×C311yescharacteristic, complemented
33#1982C2×C311yescharacteristic
34C2×#332C2×C311yescharacteristic, complemented
35C2×Q8⋊C8C2×C311yescharacteristic, complemented
36#1279C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×C3×SD16—24no—
43C2×C3×SD16—42no—
44C2×C3×SD16—42no—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—42no—
53C3×C4⋊Q8—42no—
54C3×#127—21no—
55C3×#127—21no—
56C3×#127—21no—
57C3×#127D412yes—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×Q8—21no—
61C3×C4×Q8—21no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D412yes—
67C2×C3×C4⋊C4D412yes—
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×#102—24no—
76C3×#102—28no—
77C3×#102—44no—
78C3×C4×C8—21no—
79C3×C4×C8D412yes—
80C3×#95—24no—
81C3×#95—21no—
82C22×SD16—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C22×C311yescharacteristic
85C2×#127C22×C311yescharacteristic
86C2×C4×Q8C22×C311yescharacteristic
87#486—22no—
88#486—22no—
89#460—22no—
90#460—22no—
91C2×C4⋊C8C22×C311yescharacteristic
92C2×C4⋊C8C22×C311yescharacteristic
93C2×#102—22no—
94C2×#102—21no—
95C2×#102—22no—
96C2×C4×C8C22×C311yescharacteristic
97#398—22no—
98#391—21no—
99#389—22no—
100C4×C4⋊C4C22×C311yescharacteristic
101#332—22no—
102Q8⋊C8—22no—
103C24×C3—41no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
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×Q8D812yes—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—44no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—42no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—42no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C3×SD16—48no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—42no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C2×C3×C8—42no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—48no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4SD1614yescomplemented
155C3×C42—41no—
156C3×C42—41no—
157C3×C42—41no—
158C3×C42—41no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163C22×Q8—21no—
164C22×Q8C3×D411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C2×SD16—24no—
169C2×SD16—42no—
170C2×SD16—42no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180#127—21no—
181#127—21no—
182#127—21no—
183#127C3×D412yes—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C3×D411yescharacteristic
192C2×C4⋊C4C3×D412yes—
193C2×C4⋊C4C3×D412yes—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C23×C311yescharacteristic
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8C3×D412yes—
201#102—24no—
202#102—28no—
203#102—44no—
204C4×C8—21no—
205C4×C8C3×D412yes—
206#95—24no—
207#95—21no—
208C23×C3—42no—
209C23×C3—81no—
210C23×C3—82no—
211C23×C3—81no—
212C23×C3—82no—
213C23×C3#12011yescharacteristic
214C3×Q8—22no—
215C3×Q8—22no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×Q8—81no—
219C3×Q8—81no—
220C3×Q8—81no—
221C3×Q8—81no—
222C3×D4—44no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—44no—
226C3×D4—82no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—81no—
230C3×D4—81no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12711yescharacteristic
249C2×C3×C4C2×D811yescharacteristic
250C2×C3×C4C2×SD1612yes—
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4#13712yes—
255C3×C8—42no—
256C3×C8—42no—
257C3×C8—81no—
258C24—41no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8C3×D812yes—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291SD16—48no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—48no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4C3×SD1614yescomplemented
310C42—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42C2×C3×D411yescharacteristic
315C42C2×C3×D411yescharacteristic
316C42C2×C3×D411yescharacteristic
317C42C3×#4111yescharacteristic
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—162no—
322C22×C3—162no—
323C22×C3#39211yescharacteristic
324C22×C3#44711yescharacteristic
325C22×C3#44812yes—
326C22×C3#44911yescharacteristic
327C22×C3#45412yes—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—44no—
331C3×C4—42no—
332C3×C4—82no—
333C3×C4—81no—
334C3×C4#44711yescharacteristic
335C3×C4#44911yescharacteristic
336C3×C4#46112yes—
337C3×C4#46611yescharacteristic
338C3×C4#46911yescharacteristic
339C3×C4#49211yescharacteristic
340C3×C4#49711yescharacteristic
341C23—42no—
342C23—81no—
343C23—82no—
344C23—81no—
345C23—82no—
346C23C3×#12011yescharacteristic
347Q8—22no—
348Q8—22no—
349Q8—41no—
350Q8—41no—
351Q8—81no—
352Q8—81no—
353Q8—81no—
354Q8—81no—
355D4—44no—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—82no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4C3×#12011yescharacteristic
377C2×C4C3×#12011yescharacteristic
378C2×C4C3×#12111yescharacteristic
379C2×C4C3×#12111yescharacteristic
380C2×C4C3×#12111yescharacteristic
381C2×C4C3×#12711yescharacteristic
382C2×C4C2×C3×D811yescharacteristic
383C2×C4C2×C3×SD1612yes—
384C2×C4C3×#13511yescharacteristic
385C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
386C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
387C2×C4C3×#13712yes—
388C8—42no—
389C8—42no—
390C8—81no—
391C2×C3—162no—
392C2×C3#162612yes—
393C2×C3#162712yes—
394C2×C3#199911yescharacteristic
395C2×C3#200311yescharacteristic
396C2×C3#201311yescharacteristic
397C22—82no—
398C22—82no—
399C22—82no—
400C22—162no—
401C22—162no—
402C22C3×#39211yescharacteristic
403C22C3×#44711yescharacteristic
404C22C3×#44812yes—
405C22C3×#44911yescharacteristic
406C22C3×#45412yes—
407C4—21no—
408C4—21no—
409C4—42no—
410C4—44no—
411C4—82no—
412C4—81no—
413C4C3×#44711yescharacteristic
414C4C3×#44911yescharacteristic
415C4C3×#46112yes—
416C4C3×#46611yescharacteristic
417C4C3×#46911yescharacteristic
418C4C3×#49211yescharacteristic
419C4C3×#49711yescharacteristic
420C3#1005111yescharacteristic, direct product factor, Sylow
421C2—162no—
422C2C3×#162612yes—
423C2C3×#162712yes—
424C2C3×#199911yescharacteristic
425C2C3×#200311yescharacteristic
426C2C3×#201311yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#127C22
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
3. C3×#2546C2semidirect
5. C2×C3×#332C2semidirect
6. C2×C3×Q8⋊C8C2semidirect
7. C3×#1279C2semidirect
8. #10051C3direct (N × H)
32. #2546C2×C3semidirect
34. C2×#332C2×C3semidirect
35. C2×Q8⋊C8C2×C3semidirect
36. #1279C2×C3semidirect
154. C3×C4⋊C4SD16semidirect
309. C4⋊C4C3×SD16semidirect
420. C3#10051direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×C3×#4602
C3×#25462
C3×#19822
C2×C3×#3322
C2×C3×Q8⋊C82
C3×#12792
#100513