Finite Group Databasebeta

← C3×#33756 #10652363

Subgroups — C3×#33756 #10652363

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3050C211yesmaximal, characteristic
2C2×C3×#467C211yesmaximal, characteristic, complemented
3C3×#2237C212yesmaximal, complemented
4C3×#2235C211yesmaximal, characteristic, complemented
5C3×#2232C211yesmaximal, characteristic, complemented
6C3×#2217C212yesmaximal, complemented
7C3×#2216C212yesmaximal, complemented
8C3×#2189C212yesmaximal, complemented
9C3×#2188C211yesmaximal, characteristic, complemented
10C3×#2185C211yesmaximal, characteristic, complemented
11C2×C3×C22⋊C16C211yesmaximal, characteristic, complemented
12#33756C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#135C2211yescharacteristic
14C22×C3×Q16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C3×#525—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×Q32C2212yescomplemented
19C2×C3×SD32C2212yescomplemented
20C22×C3×C16C2212yes—
21C3×#467—21no—
22C3×#467—21no—
23C3×#467C2212yes—
24C3×#467C2212yes—
25C3×#467C2211yescharacteristic
26C3×#467C2211yescharacteristic
27C3×#466C2211yescharacteristic
28C3×#466C2211yescharacteristic
29C3×#465—21no—
30C3×#465—21no—
31C3×#427C2211yescharacteristic
32C2×C3×#107C2211yescharacteristic
33C3×#416—21no—
34C2×C3×#103—21no—
35C3×#367C2212yescomplemented
36C3×#366C2212yescomplemented
37C3×#358C2212yescomplemented
38C3×#358C2212yescomplemented
39C3×#358C2212yescomplemented
40C3×#357C2212yescomplemented
41C3×C22⋊C16C2211yescharacteristic
42C3×C22⋊C16C2211yescharacteristic
43C3×C22⋊C16C2212yes—
44#3050C2×C311yescharacteristic
45C2×#467C2×C311yescharacteristic, complemented
46#2237C2×C312yescomplemented
47#2235C2×C311yescharacteristic, complemented
48#2232C2×C311yescharacteristic, complemented
49#2217C2×C312yescomplemented
50#2216C2×C312yescomplemented
51#2189C2×C312yescomplemented
52#2188C2×C311yescharacteristic, complemented
53#2185C2×C311yescharacteristic, complemented
54C2×C22⋊C16C2×C311yescharacteristic, complemented
55C2×C3×#41—21no—
56C22×C3×Q8—21no—
57C23×C3×C4D411yescharacteristic
58C3×#135—42no—
59C2×C3×Q16—21no—
60C2×C3×Q16—42no—
61C2×C3×Q16C2312yes—
62C2×C3×Q16C2311yescharacteristic
63C2×C3×SD16—21no—
64C2×C3×D8C2311yescharacteristic
65C22×C3×C8—42no—
66C22×C3×C8D411yescharacteristic
67C22×C3×C8D411yescharacteristic
68C22×C3×C8D411yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×#123—41no—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8—21no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—41no—
78C3×C22⋊Q8—41no—
79C3×#121—21no—
80C3×#121—21no—
81C3×#120—41no—
82C3×C4×D4—41no—
83C3×C4×D4—41no—
84C3×#117—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—41no—
87C2×C3×C22⋊C4—41no—
88C3×Q32—42no—
89C3×SD32—42no—
90C2×C3×C16—42no—
91C2×C3×C16C2312yes—
92C2×C3×C16C2312yes—
93C3×#107—21no—
94C3×#107C2311yescharacteristic
95C3×#107C2312yes—
96C3×#107C2311yescharacteristic
97C3×#106—21no—
98C3×#103—22no—
99C3×#103—22no—
100C3×#103—21no—
101C3×#103—21no—
102C3×#102—21no—
103C3×#102—21no—
104C2×#135C22×C311yescharacteristic
105C22×Q16C22×C311yescharacteristic
106C23×C8C22×C311yescharacteristic
107#525—21no—
108C2×C22⋊Q8—21no—
109C2×Q32C22×C312yescomplemented
110C2×SD32C22×C312yescomplemented
111C22×C16C22×C312yes—
112#467—21no—
113#467—21no—
114#467C22×C311yescharacteristic
115#467C22×C312yes—
116#467C22×C311yescharacteristic
117#467C22×C312yes—
118#466C22×C311yescharacteristic
119#466C22×C311yescharacteristic
120#465—21no—
121#465—21no—
122#427C22×C311yescharacteristic
123C2×#107C22×C311yescharacteristic
124#416—21no—
125C2×#103—21no—
126#367C22×C312yescomplemented
127#366C22×C312yescomplemented
128#358C22×C312yescomplemented
129#358C22×C312yescomplemented
130#358C22×C312yescomplemented
131#357C22×C312yescomplemented
132C22⋊C16C22×C311yescharacteristic
133C22⋊C16C22×C311yescharacteristic
134C22⋊C16C22×C312yes—
135C24×C3D811yescharacteristic
136C3×#41—42no—
137C3×#41—42no—
138C2×C3×Q8—21no—
139C2×C3×Q8—22no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×Q8—41no—
143C2×C3×D4—21no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4D811yescharacteristic
153C22×C3×C4D811yescharacteristic
154C22×C3×C4D811yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C3×Q16—22no—
159C3×Q16—22no—
160C3×Q16—22no—
161C3×Q16—42no—
162C3×SD16—42no—
163C3×D8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—41no—
168C2×C3×C8—41no—
169C2×C3×C8—41no—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2411yescharacteristic
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4—41no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—41no—
189C3×C42—41no—
190C3×C16—22no—
191C3×C16—22no—
192C2×#41—21no—
193C22×Q8—21no—
194C23×C4C3×D411yescharacteristic
195#135—42no—
196C2×Q16—21no—
197C2×Q16—42no—
198C2×Q16C23×C312yes—
199C2×Q16C23×C311yescharacteristic
200C2×SD16—21no—
201C2×D8C23×C311yescharacteristic
202C22×C8—42no—
203C22×C8C3×D411yescharacteristic
204C22×C8C3×D411yescharacteristic
205C22×C8C3×D411yescharacteristic
206C22×C8C23×C311yescharacteristic
207C22×C8C23×C311yescharacteristic
208C22×C8C23×C311yescharacteristic
209#123—41no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216#121—21no—
217#121—21no—
218#120—41no—
219C4×D4—41no—
220C4×D4—41no—
221#117—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—41no—
224C2×C22⋊C4—41no—
225Q32—42no—
226SD32—42no—
227C2×C16—42no—
228C2×C16C23×C312yes—
229C2×C16C23×C312yes—
230#107—21no—
231#107C23×C312yes—
232#107C23×C311yescharacteristic
233#107C23×C311yescharacteristic
234#106—21no—
235#103—22no—
236#103—22no—
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3C2×D811yescharacteristic
246C23×C3C2×D811yescharacteristic
247C23×C3C2×D811yescharacteristic
248C3×Q8—41no—
249C3×Q8—42no—
250C3×Q8—42no—
251C3×Q8—41no—
252C3×Q8—82no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—82no—
256C3×D4—82no—
257C3×D4—82no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—82no—
273C2×C3×C4—82no—
274C2×C3×C4—82no—
275C2×C3×C4C2×D811yescharacteristic
276C2×C3×C4C2×D811yescharacteristic
277C2×C3×C4C2×D811yescharacteristic
278C2×C3×C4C22×D411yescharacteristic
279C3×C8—21no—
280C3×C8—21no—
281C3×C8—21no—
282C3×C8H(5,𝔽2)12yes—
283C24C3×D811yescharacteristic
284#41—42no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C3×D811yescharacteristic
301C22×C4C3×D811yescharacteristic
302C22×C4C3×D811yescharacteristic
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×D411yescharacteristic
305C22×C4C2×C3×D411yescharacteristic
306Q16—22no—
307Q16—22no—
308Q16—22no—
309Q16—42no—
310SD16—42no—
311D8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8C2×C3×D411yescharacteristic
319C2×C8C2×C3×D411yescharacteristic
320C2×C8C2×C3×D411yescharacteristic
321C2×C8C24×C311yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C42—41no—
338C16—22no—
339C16—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#50812yes—
352C22×C3#51012yes—
353C22×C3C22×D811yescharacteristic
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—82no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4—82no—
362C3×C4#53411yescharacteristic
363C3×C4#57511yescharacteristic
364C23—22no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23C2×C3×D811yescharacteristic
369C23C2×C3×D811yescharacteristic
370C23C2×C3×D811yescharacteristic
371Q8—42no—
372Q8—41no—
373Q8—41no—
374Q8—42no—
375Q8—82no—
376D4—41no—
377D4—41no—
378D4—82no—
379D4—82no—
380D4—82no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4C2×C3×D811yescharacteristic
399C2×C4C2×C3×D811yescharacteristic
400C2×C4C2×C3×D811yescharacteristic
401C2×C4C22×C3×D411yescharacteristic
402C8—21no—
403C8—21no—
404C8—21no—
405C8C3×H(5,𝔽2)12yes—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3—161no—
410C2×C3#318611yescharacteristic
411C2×C3C2×#50811yescharacteristic
412C2×C3C2×#51011yescharacteristic
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22C3×#50812yes—
425C22C3×#51012yes—
426C22C22×C3×D811yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—82no—
435C4C3×#53411yescharacteristic
436C4C3×#57511yescharacteristic
437C3#3375611yescharacteristic, direct product factor, Sylow
438C2—22no—
439C2—22no—
440C2—41no—
441C2—161no—
442C2C3×#318611yescharacteristic
443C2C2×C3×#50811yescharacteristic
444C2C2×C3×#51011yescharacteristic
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC3×#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#467C2semidirect
3. C3×#2237C2semidirect
4. C3×#2235C2semidirect
5. C3×#2232C2semidirect
6. C3×#2217C2semidirect
7. C3×#2216C2semidirect
8. C3×#2189C2semidirect
9. C3×#2188C2semidirect
10. C3×#2185C2semidirect
11. C2×C3×C22⋊C16C2semidirect
12. #33756C3direct (N × H)
18. C2×C3×Q32C22semidirect
19. C2×C3×SD32C22semidirect
35. C3×#367C22semidirect
36. C3×#366C22semidirect
37. C3×#358C22semidirect
38. C3×#358C22semidirect
39. C3×#358C22semidirect
40. C3×#357C22semidirect
45. C2×#467C2×C3semidirect
46. #2237C2×C3semidirect
47. #2235C2×C3semidirect
48. #2232C2×C3semidirect
49. #2217C2×C3semidirect
50. #2216C2×C3semidirect
51. #2189C2×C3semidirect
52. #2188C2×C3semidirect
53. #2185C2×C3semidirect
54. C2×C22⋊C16C2×C3semidirect
109. C2×Q32C22×C3semidirect
110. C2×SD32C22×C3semidirect
126. #367C22×C3semidirect
127. #366C22×C3semidirect
128. #358C22×C3semidirect
129. #358C22×C3semidirect
130. #358C22×C3semidirect
131. #357C22×C3semidirect
437. C3#33756direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30502
C2×C3×#4672
C3×#22372
C3×#22352
C3×#22322
C3×#22172
C3×#22162
C3×#21892
C3×#21882
C3×#21852
C2×C3×C22⋊C162
#337563