Finite Group Databasebeta

← C3×#29343 #10647950

Subgroups — C3×#29343 #10647950

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2619C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic, complemented
3C3×#2101C212yesmaximal, complemented
4C3×#2041C212yesmaximal, complemented
5C3×#1936C212yesmaximal, complemented
6C3×#1934C212yesmaximal, complemented
7C3×#1875C212yesmaximal, complemented
8C3×#1874C212yesmaximal, complemented
9C3×#1750C211yesmaximal, characteristic, complemented
10#29343C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#121C2212yes—
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×C3×#102C2212yes—
17C2×C3×#102C2212yes—
18C2×C3×#102C2212yes—
19C2×C3×#102C2212yes—
20C2×C3×C22⋊C8C2212yes—
21C2×C3×#97C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×#394—21no—
24C3×#394—21no—
25C3×#392—21no—
26C3×#392—21no—
27C3×#390C2212yes—
28C3×#389C2212yes—
29C3×#389C2212yes—
30C3×#385—21no—
31C3×#385—21no—
32C3×#384—21no—
33C3×#382—21no—
34C3×#382C2211yescharacteristic
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2212yes—
40#2619C2×C311yescharacteristic
41#2317C2×C311yescharacteristic, complemented
42#2101C2×C312yescomplemented
43#2041C2×C312yescomplemented
44#1936C2×C312yescomplemented
45#1934C2×C312yescomplemented
46#1875C2×C312yescomplemented
47#1874C2×C312yescomplemented
48#1750C2×C311yescharacteristic, complemented
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C23×C3×C4—21no—
54C23×C3×C4C2311yescharacteristic
55C22×C3×C8C2312yes—
56C22×C3×C8C2312yes—
57C3×#127—41no—
58C3×#127—41no—
59C3×#121—24no—
60C3×#121—24no—
61C3×#117—24no—
62C3×#117—41no—
63C3×#117—41no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—41no—
68C2×C3×C4⋊C4—41no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4D411yescharacteristic
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#102—24no—
84C3×#102—24no—
85C3×#102—42no—
86C3×#102—42no—
87C3×C22⋊C8—24no—
88C3×#97—41no—
89C3×C4×C8—41no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—21no—
93C3×#95—21no—
94C2×#127C22×C311yescharacteristic
95C2×#121C22×C312yes—
96C2×#117C22×C311yescharacteristic
97C2×#117C22×C311yescharacteristic
98C22×C4⋊C4C22×C311yescharacteristic
99C2×#102C22×C312yes—
100C2×#102C22×C312yes—
101C2×#102C22×C312yes—
102C2×#102C22×C312yes—
103C2×C22⋊C8C22×C312yes—
104C2×#97C22×C311yescharacteristic
105C2×C4×C8C22×C311yescharacteristic
106#394—21no—
107#394—21no—
108#392—21no—
109#392—21no—
110#390C22×C312yes—
111#389C22×C312yes—
112#389C22×C312yes—
113#385—21no—
114#385—21no—
115#384—21no—
116#382—21no—
117#382C22×C311yescharacteristic
118#340C22×C312yes—
119#340C22×C312yes—
120#340C22×C312yes—
121#340C22×C312yes—
122#336C22×C312yes—
123C24×C3—22no—
124C24×C3C2×D411yescharacteristic
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—22no—
130C2×C3×D4—22no—
131C2×C3×D4—42no—
132C2×C3×D4—42no—
133C2×C3×D4—44no—
134C2×C3×D4—82no—
135C2×C3×D4—82no—
136C2×C3×D4—82no—
137C2×C3×D4—82no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—42no—
145C22×C3×C4—82no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4#4114yes—
152C22×C3×C4C2411yescharacteristic
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4—41no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—82no—
168C3×C4⋊C4—82no—
169C3×C4⋊C4#4114yes—
170C3×C22⋊C4—24no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—44no—
174C3×C22⋊C4—82no—
175C3×C22⋊C4—82no—
176C3×C42—24no—
177C3×C42—21no—
178C3×C42—21no—
179C3×C42—21no—
180C3×C42—21no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C23×C312yes—
185C23×C4—21no—
186C23×C4C23×C311yescharacteristic
187C22×C8C23×C312yes—
188C22×C8C23×C312yes—
189#127—41no—
190#127—41no—
191#121—24no—
192#121—24no—
193#117—24no—
194#117—41no—
195#117—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C3×D411yescharacteristic
202C2×C4⋊C4C3×D411yescharacteristic
203C2×C4⋊C4C23×C312yes—
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C312yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C3×D411yescharacteristic
211C2×C22⋊C4C3×D411yescharacteristic
212C2×C42—21no—
213C2×C42C23×C311yescharacteristic
214C2×C42C23×C311yescharacteristic
215#102—24no—
216#102—24no—
217#102—42no—
218#102—42no—
219C22⋊C8—24no—
220#97—41no—
221C4×C8—41no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3#12412yes—
238C23×C3C22×D411yescharacteristic
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—84no—
244C3×D4—84no—
245C3×D4—88no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—84no—
268C2×C3×C4#12412yes—
269C2×C3×C4C2×#4112yes—
270C2×C3×C4C2×#4112yes—
271C2×C3×C4H(5,𝔽2)11yescharacteristic
272C2×C3×C4H(5,𝔽2)11yescharacteristic
273C3×C8—42no—
274C3×C8—42no—
275C24—22no—
276C24C2×C3×D411yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—82no—
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C2×C3×D411yescharacteristic
301C22×C4C2×C3×D411yescharacteristic
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C3×#4114yes—
304C22×C4C24×C311yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4C3×#4114yes—
322C22⋊C4—24no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C42—24no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C22×C3—22no—
334C22×C3—24no—
335C22×C3—42no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3—82no—
345C22×C3#52511yescharacteristic
346C22×C3C2×#12411yescharacteristic
347C22×C3#53411yescharacteristic
348C22×C3#57612yes—
349C22×C3#57712yes—
350C3×C4—22no—
351C3×C4—44no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—82no—
357C3×C4—82no—
358C3×C4#55312yes—
359C3×C4#55312yes—
360C23—22no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23C3×#12412yes—
372C23C22×C3×D411yescharacteristic
373D4—42no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—84no—
378D4—84no—
379D4—88no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—84no—
402C2×C4C3×#12412yes—
403C2×C4C2×C3×#4112yes—
404C2×C4C2×C3×#4112yes—
405C2×C4C3×H(5,𝔽2)11yescharacteristic
406C2×C4C3×H(5,𝔽2)11yescharacteristic
407C8—42no—
408C8—42no—
409C2×C3—42no—
410C2×C3—162no—
411C2×C3#259111yescharacteristic
412C2×C3#311912yes—
413C2×C3#313612yes—
414C2×C3#319911yescharacteristic
415C2×C3#320011yescharacteristic
416C22—24no—
417C22—22no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22C3×#52511yescharacteristic
429C22C2×C3×#12411yescharacteristic
430C22C3×#53411yescharacteristic
431C22C3×#57612yes—
432C22C3×#57712yes—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4C3×#55312yes—
442C4C3×#55312yes—
443C3#2934311yescharacteristic, direct product factor, Sylow
444C2—42no—
445C2—162no—
446C2C3×#259111yescharacteristic
447C2C3×#311912yes—
448C2C3×#313612yes—
449C2C3×#319911yescharacteristic
450C2C3×#320011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2619C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2317C2semidirect
3. C3×#2101C2semidirect
4. C3×#2041C2semidirect
5. C3×#1936C2semidirect
6. C3×#1934C2semidirect
7. C3×#1875C2semidirect
8. C3×#1874C2semidirect
9. C3×#1750C2semidirect
10. #29343C3direct (N × H)
41. #2317C2×C3semidirect
42. #2101C2×C3semidirect
43. #2041C2×C3semidirect
44. #1936C2×C3semidirect
45. #1934C2×C3semidirect
46. #1875C2×C3semidirect
47. #1874C2×C3semidirect
48. #1750C2×C3semidirect
443. C3#29343direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26192
C3×#23172
C3×#21012
C3×#20412
C3×#19362
C3×#19342
C3×#18752
C3×#18742
C3×#17502
#293433