Finite Group Databasebeta

← C3×#29785 #10648392

Subgroups — C3×#29785 #10648392

Order 768.

838 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#469C211yesmaximal, characteristic
2C2×C3×#424C211yesmaximal, characteristic, complemented
3C3×#2624C212yesmaximal, complemented
4C3×#2088C212yesmaximal, complemented
5C3×#2067C212yesmaximal, complemented
6C3×#2065C212yesmaximal, complemented
7C3×#2054C212yesmaximal, complemented
8C3×#1893C212yesmaximal, complemented
9C3×#1809C211yesmaximal, characteristic, complemented
10#29785C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×D8C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×C3×#117C2211yescharacteristic
16C3×#469—24no—
17C2×C3×#107C2211yescharacteristic
18C2×C3×#106C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C3×#424—24no—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C3×#399—22no—
27C3×#397—22no—
28C3×#394—22no—
29C3×#394—22no—
30C3×#390C2212yes—
31C3×#390C2212yes—
32C3×#389C2212yes—
33C3×#389C2212yes—
34C3×#380C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C2×#469C2×C311yescharacteristic
39C2×#424C2×C311yescharacteristic, complemented
40#2624C2×C312yescomplemented
41#2088C2×C312yescomplemented
42#2067C2×C312yescomplemented
43#2065C2×C312yescomplemented
44#2054C2×C312yescomplemented
45#1893C2×C312yescomplemented
46#1809C2×C311yescharacteristic, complemented
47C22×C3×D4—22no—
48C22×C3×D4C2312yes—
49C23×C3×C4C2311yescharacteristic
50C2×C3×D8—22no—
51C2×C3×D8—41no—
52C2×C3×D8—41no—
53C2×C3×M16—41no—
54C2×C3×M16—41no—
55C2×C3×M16D414yes—
56C22×C3×C8—22no—
57C22×C3×C8—41no—
58C22×C3×C8—41no—
59C22×C3×C8C2312yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×#121—24no—
63C3×#121—24no—
64C3×#117—22no—
65C3×#117—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
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×C42C2312yes—
81C3×#107—41no—
82C3×#106—22no—
83C3×#106—41no—
84C3×C4⋊C8—24no—
85C3×C4⋊C8—24no—
86C3×#102—24no—
87C3×#102—42no—
88C3×#102—42no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—22no—
92C22×D8C22×C311yescharacteristic
93C22×M16C22×C311yescharacteristic
94C23×C8C22×C311yescharacteristic
95C2×#121C22×C312yes—
96C2×#117C22×C311yescharacteristic
97#469—24no—
98C2×#107C22×C311yescharacteristic
99C2×#106C22×C311yescharacteristic
100C2×#106C22×C311yescharacteristic
101#424—24no—
102C2×C4⋊C8C22×C312yes—
103C2×C4⋊C8C22×C312yes—
104C2×#102C22×C312yes—
105C2×#102C22×C312yes—
106C2×#102C22×C312yes—
107#399—22no—
108#397—22no—
109#394—22no—
110#394—22no—
111#390C22×C312yes—
112#390C22×C312yes—
113#389C22×C312yes—
114#389C22×C312yes—
115#380C22×C312yes—
116#340C22×C312yes—
117#340C22×C312yes—
118#340C22×C312yes—
119C24×C3—22no—
120C24×C3C2×D411yescharacteristic
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—44no—
128C2×C3×D4—82no—
129C2×C3×D4—82no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D412yes—
141C22×C3×C4#4112yes—
142C22×C3×C4C2411yescharacteristic
143C3×D8—44no—
144C3×M16—44no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—44no—
153C2×C3×C8C2×D412yes—
154C2×C3×C8C2×D412yes—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—82no—
163C3×C4⋊C4—82no—
164C3×C22⋊C4—22no—
165C3×C22⋊C4—22no—
166C3×C22⋊C4—44no—
167C3×C22⋊C4—82no—
168C3×C22⋊C4—82no—
169C3×C22⋊C4—82no—
170C3×C22⋊C4—82no—
171C3×C22⋊C4—82no—
172C3×C42—22no—
173C3×C42—22no—
174C22×D4—22no—
175C22×D4C23×C312yes—
176C23×C4C23×C311yescharacteristic
177C2×D8—22no—
178C2×D8—41no—
179C2×D8—41no—
180C2×M16—41no—
181C2×M16—41no—
182C2×M16C3×D414yes—
183C22×C8—22no—
184C22×C8—41no—
185C22×C8—41no—
186C22×C8C23×C312yes—
187C22×C8C23×C311yescharacteristic
188C22×C8C23×C311yescharacteristic
189#121—24no—
190#121—24no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C3×D412yes—
196C2×C4⋊C4C23×C312yes—
197C2×C4⋊C4C23×C312yes—
198C2×C4⋊C4C23×C312yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4C3×D411yescharacteristic
206C2×C22⋊C4C3×D411yescharacteristic
207C2×C42C23×C312yes—
208#107—41no—
209#106—22no—
210#106—41no—
211C4⋊C8—24no—
212C4⋊C8—24no—
213#102—24no—
214#102—42no—
215#102—42no—
216#95—22no—
217#95—22no—
218#95—22no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3#12112yes—
231C23×C3C22×D411yescharacteristic
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—88no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4#12112yes—
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4H(5,𝔽2)11yescharacteristic
262C2×C3×C4H(5,𝔽2)11yescharacteristic
263C2×C3×C42−1+411yescharacteristic
264C2×C3×C42−1+411yescharacteristic
265C3×C8—22no—
266C3×C8—22no—
267C3×C8—42no—
268C24—22no—
269C24C2×C3×D411yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—44no—
277C2×D4—82no—
278C2×D4—82no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4C2×C3×D412yes—
287C22×C4C2×C3×D412yes—
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D412yes—
290C22×C4C3×#4112yes—
291C22×C4C24×C311yescharacteristic
292D8—44no—
293M16—44no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—44no—
302C2×C8C2×C3×D412yes—
303C2×C8C2×C3×D412yes—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C42—22no—
322C42—22no—
323C22×C3—22no—
324C22×C3—24no—
325C22×C3—42no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3—82no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3C2×#12111yescharacteristic
336C22×C3#53711yescharacteristic
337C22×C3#53711yescharacteristic
338C22×C3#57612yes—
339C22×C3#57712yes—
340C3×C4—22no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—82no—
344C3×C4—82no—
345C3×C4—82no—
346C3×C4#53411yescharacteristic
347C3×C4#53611yescharacteristic
348C3×C4#53911yescharacteristic
349C3×C4#53911yescharacteristic
350C23—22no—
351C23—22no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23C3×#12112yes—
362C23C22×C3×D411yescharacteristic
363D4—42no—
364D4—42no—
365D4—42no—
366D4—42no—
367D4—88no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4C3×#12112yes—
390C2×C4C22×C3×D411yescharacteristic
391C2×C4C2×C3×#4111yescharacteristic
392C2×C4C3×H(5,𝔽2)11yescharacteristic
393C2×C4C3×H(5,𝔽2)11yescharacteristic
394C2×C4C3×2−1+411yescharacteristic
395C2×C4C3×2−1+411yescharacteristic
396C8—22no—
397C8—22no—
398C8—42no—
399C2×C3—42no—
400C2×C3—162no—
401C2×C3#261811yescharacteristic
402C2×C3#306212yes—
403C2×C3#325712yes—
404C2×C3#325911yescharacteristic
405C2×C3#326311yescharacteristic
406C22—22no—
407C22—24no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22C2×C3×#12111yescharacteristic
419C22C3×#53711yescharacteristic
420C22C3×#53711yescharacteristic
421C22C3×#57612yes—
422C22C3×#57712yes—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4C3×#53411yescharacteristic
430C4C3×#53611yescharacteristic
431C4C3×#53911yescharacteristic
432C4C3×#53911yescharacteristic
433C3#2978511yescharacteristic, direct product factor, Sylow
434C2—42no—
435C2—162no—
436C2C3×#261811yescharacteristic
437C2C3×#306212yes—
438C2C3×#325712yes—
439C2C3×#325911yescharacteristic
440C2C3×#326311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#469C2
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. C2×C3×#424C2semidirect
3. C3×#2624C2semidirect
4. C3×#2088C2semidirect
5. C3×#2067C2semidirect
6. C3×#2065C2semidirect
7. C3×#2054C2semidirect
8. C3×#1893C2semidirect
9. C3×#1809C2semidirect
10. #29785C3direct (N × H)
39. C2×#424C2×C3semidirect
40. #2624C2×C3semidirect
41. #2088C2×C3semidirect
42. #2067C2×C3semidirect
43. #2065C2×C3semidirect
44. #2054C2×C3semidirect
45. #1893C2×C3semidirect
46. #1809C2×C3semidirect
433. C3#29785direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4692
C2×C3×#4242
C3×#26242
C3×#20882
C3×#20672
C3×#20652
C3×#20542
C3×#18932
C3×#18092
#297853