Finite Group Databasebeta

← C3×#29204 #10647811

Subgroups — C3×#29204 #10647811

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2647C211yesmaximal, characteristic, complemented
2C3×#2328C211yesmaximal, characteristic
3C3×#2067C212yesmaximal, complemented
4C3×#2065C212yesmaximal, complemented
5C3×#1961C211yesmaximal, characteristic
6C3×#1957C211yesmaximal, characteristic
7C3×#1864C212yesmaximal, complemented
8C3×#1862C212yesmaximal, complemented
9C3×#1847C211yesmaximal, characteristic, complemented
10C3×#1835C211yesmaximal, characteristic, complemented
11C3×#1768C211yesmaximal, characteristic, complemented
12#29204C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#107C2212yes—
17C2×C3×#106C2212yes—
18C2×C3×C4⋊C8C2212yes—
19C2×C3×#102C2212yes—
20C2×C3×#102C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#97C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#400—22no—
26C3×#398—22no—
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#389C2212yes—
30C3×#389C2212yes—
31C3×#386—22no—
32C3×#386—22no—
33C3×#384C2211yescharacteristic
34C3×#382C2211yescharacteristic
35C3×#382C2211yescharacteristic
36C3×#376C2211yescharacteristic
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2212yes—
40#2647C2×C311yescharacteristic, complemented
41#2328C2×C311yescharacteristic
42#2067C2×C312yescomplemented
43#2065C2×C312yescomplemented
44#1961C2×C311yescharacteristic
45#1957C2×C311yescharacteristic
46#1864C2×C312yescomplemented
47#1862C2×C312yescomplemented
48#1847C2×C311yescharacteristic, complemented
49#1835C2×C311yescharacteristic, complemented
50#1768C2×C311yescharacteristic, complemented
51C22×C3×D4—22no—
52C22×C3×D4C2311yescharacteristic
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C22×C3×C8C2312yes—
57C22×C3×C8C2312yes—
58C3×#127—22no—
59C3×#127Q814yes—
60C3×C4×D4—42no—
61C3×C4×D4—42no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C3×#107—42no—
77C3×#106—42no—
78C3×C4⋊C8—24no—
79C3×#102—24no—
80C3×#102—24no—
81C3×#102—42no—
82C3×#102—42no—
83C3×#97—22no—
84C3×C4×C8—22no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95D412yes—
90C3×#95D412yes—
91C2×#127C22×C311yescharacteristic
92C2×C4×D4C22×C311yescharacteristic
93C2×C4×D4C22×C311yescharacteristic
94C2×#107C22×C312yes—
95C2×#106C22×C312yes—
96C2×C4⋊C8C22×C312yes—
97C2×#102C22×C312yes—
98C2×#102C22×C312yes—
99C2×#102C22×C312yes—
100C2×#102C22×C312yes—
101C2×#97C22×C311yescharacteristic
102C2×C4×C8C22×C311yescharacteristic
103#400—22no—
104#398—22no—
105#390C22×C311yescharacteristic
106#390C22×C311yescharacteristic
107#389C22×C312yes—
108#389C22×C312yes—
109#386—22no—
110#386—22no—
111#384C22×C311yescharacteristic
112#382C22×C311yescharacteristic
113#382C22×C311yescharacteristic
114#376C22×C311yescharacteristic
115#340C22×C312yes—
116#340C22×C312yes—
117#336C22×C312yes—
118C24×C3—21no—
119C24×C3—21no—
120C2×C3×D4—28no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—82no—
128C2×C3×D4—82no—
129C2×C3×D4C2×Q812yes—
130C2×C3×D4C2×Q812yes—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4—82no—
138C22×C3×C4—82no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2411yescharacteristic
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8#4114yes—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—82no—
157C3×C4⋊C4—82no—
158C3×C4⋊C4—82no—
159C3×C22⋊C4—42no—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—82no—
162C3×C22⋊C4—82no—
163C3×C42—22no—
164C3×C42—22no—
165C3×C42C2×Q812yes—
166C3×C42#4112yes—
167C22×D4—22no—
168C22×D4C23×C311yescharacteristic
169C22×D4C23×C311yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C22×C8C23×C312yes—
173C22×C8C23×C312yes—
174#127—22no—
175#127C3×Q814yes—
176C4×D4—42no—
177C4×D4—42no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C23×C311yescharacteristic
182C2×C4⋊C4C23×C311yescharacteristic
183C2×C4⋊C4C23×C312yes—
184C2×C4⋊C4C23×C312yes—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C42C23×C311yescharacteristic
190C2×C42C23×C311yescharacteristic
191C2×C42C23×C311yescharacteristic
192#107—42no—
193#106—42no—
194C4⋊C8—24no—
195#102—24no—
196#102—24no—
197#102—42no—
198#102—42no—
199#97—22no—
200C4×C8—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95C3×D412yes—
206#95C3×D412yes—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3C22×D411yescharacteristic
218C3×D4—24no—
219C3×D4—24no—
220C3×D4—42no—
221C3×D4—42no—
222C3×D4—88no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—84no—
243C2×C3×C4—84no—
244C2×C3×C4C22⋊Q814yes—
245C2×C3×C4C22×Q811yescharacteristic
246C2×C3×C4C2×#4111yescharacteristic
247C2×C3×C4C2×#4112yes—
248C2×C3×C4H(5,𝔽2)12yes—
249C3×C8—24no—
250C3×C8—42no—
251C24—21no—
252C24—21no—
253C2×D4—28no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4C2×C3×Q812yes—
263C2×D4C2×C3×Q812yes—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C2×C3×D411yescharacteristic
273C22×C4C2×C3×D412yes—
274C22×C4C2×C3×D411yescharacteristic
275C22×C4C2×C3×D412yes—
276C22×C4C24×C311yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8C3×#4114yes—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—22no—
297C42—22no—
298C42C2×C3×Q812yes—
299C42C3×#4112yes—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—44no—
303C22×C3—44no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3C2×C22⋊Q811yescharacteristic
311C22×C3#53211yescharacteristic
312C22×C3#53511yescharacteristic
313C22×C3#57612yes—
314C22×C3#57712yes—
315C3×C4—24no—
316C3×C4—42no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—82no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4#55312yes—
324C3×C4#55412yes—
325C23—22no—
326C23—22no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23C22×C3×D411yescharacteristic
336D4—24no—
337D4—24no—
338D4—42no—
339D4—42no—
340D4—88no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4C3×C22⋊Q814yes—
363C2×C4C22×C3×Q811yescharacteristic
364C2×C4C2×C3×#4111yescharacteristic
365C2×C4C2×C3×#4112yes—
366C2×C4C3×H(5,𝔽2)12yes—
367C8—24no—
368C8—42no—
369C2×C3—82no—
370C2×C3—82no—
371C2×C3#256611yescharacteristic
372C2×C3#308112yes—
373C2×C3#317912yes—
374C2×C3#322011yescharacteristic
375C2×C3#322211yescharacteristic
376C22—42no—
377C22—42no—
378C22—44no—
379C22—44no—
380C22—82no—
381C22—82no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C2×C3×C22⋊Q811yescharacteristic
387C22C3×#53211yescharacteristic
388C22C3×#53511yescharacteristic
389C22C3×#57612yes—
390C22C3×#57712yes—
391C4—24no—
392C4—42no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—81no—
397C4—82no—
398C4—81no—
399C4C3×#55312yes—
400C4C3×#55412yes—
401C3#2920411yescharacteristic, direct product factor, Sylow
402C2—82no—
403C2—82no—
404C2C3×#256611yescharacteristic
405C2C3×#308112yes—
406C2C3×#317912yes—
407C2C3×#322011yescharacteristic
408C2C3×#322211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2647C2semidirect
3. C3×#2067C2semidirect
4. C3×#2065C2semidirect
7. C3×#1864C2semidirect
8. C3×#1862C2semidirect
9. C3×#1847C2semidirect
10. C3×#1835C2semidirect
11. C3×#1768C2semidirect
12. #29204C3direct (N × H)
40. #2647C2×C3semidirect
42. #2067C2×C3semidirect
43. #2065C2×C3semidirect
46. #1864C2×C3semidirect
47. #1862C2×C3semidirect
48. #1847C2×C3semidirect
49. #1835C2×C3semidirect
50. #1768C2×C3semidirect
401. C3#29204direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26472
C3×#23282
C3×#20672
C3×#20652
C3×#19612
C3×#19572
C3×#18642
C3×#18622
C3×#18472
C3×#18352
C3×#17682
#292043