Finite Group Databasebeta

← C3×#27675 #10646282

Subgroups — C3×#27675 #10646282

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3092C212yesmaximal
2C3×#2929C211yesmaximal, characteristic
3C22×C3×#107C211yesmaximal, characteristic, complemented
4C3×#2089C212yesmaximal, complemented
5C3×#2078C212yesmaximal, complemented
6C3×#1952C212yesmaximal, complemented
7C3×#1821C212yesmaximal, complemented
8C3×#1816C212yesmaximal, complemented
9C3×#1810C211yesmaximal, characteristic, complemented
10#27675C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#131C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C3×#525C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C3×#484—22no—
16C3×#483—22no—
17C3×#481—22no—
18C3×#480—22no—
19C3×#435—21no—
20C3×#435—21no—
21C3×C8×D4—21no—
22C3×C8×D4—21no—
23C2×C3×#108C2212yes—
24C3×#428C2214yes—
25C2×C3×#107—21no—
26C2×C3×#107—21no—
27C2×C3×#107—21no—
28C2×C3×#107—21no—
29C2×C3×#107C2212yes—
30C2×C3×#107C2212yes—
31C3×#424C2211yescharacteristic
32C3×#417C2214yes—
33C3×#408C2211yescharacteristic
34C3×#407C2211yescharacteristic
35C3×#407C2212yes—
36C3×#405C2211yescharacteristic
37C3×#343C2214yes—
38C3×#343C2214yes—
39C3×#341C2212yes—
40#3092C2×C312yes—
41#2929C2×C311yescharacteristic
42C22×#107C2×C311yescharacteristic, complemented
43#2089C2×C312yescomplemented
44#2078C2×C312yescomplemented
45#1952C2×C312yescomplemented
46#1821C2×C312yescomplemented
47#1816C2×C312yescomplemented
48#1810C2×C311yescharacteristic, complemented
49C2×C3×#41C2311yescharacteristic
50C23×C3×C4—22no—
51C23×C3×C4C2311yescharacteristic
52C3×#131—24no—
53C2×C3×M16—21no—
54C2×C3×M16C2311yescharacteristic
55C2×C3×M16C2311yescharacteristic
56C2×C3×M16C2314yes—
57C22×C3×C8—24no—
58C22×C3×C8—21no—
59C22×C3×C8D411yescharacteristic
60C22×C3×C8D411yescharacteristic
61C22×C3×C8D411yescharacteristic
62C22×C3×C8D411yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#123—21no—
66C3×C22⋊Q8D411yescharacteristic
67C3×C22⋊Q8D411yescharacteristic
68C3×#121D411yescharacteristic
69C3×#121D411yescharacteristic
70C3×#120—21no—
71C3×C4×D4—21no—
72C3×C4×D4—21no—
73C3×#117C2311yescharacteristic
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4C2314yes—
80C3×#108—24no—
81C3×#107—22no—
82C3×#107—22no—
83C3×#107—22no—
84C3×#107—22no—
85C3×#107—24no—
86C3×#106—24no—
87C3×C4⋊C8—21no—
88C3×C4⋊C8—21no—
89C3×#103—24no—
90C3×#102—24no—
91C3×C22⋊C8—21no—
92C3×C22⋊C8—21no—
93C3×C22⋊C8—22no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—21no—
96C3×C22⋊C8—21no—
97C3×#97—21no—
98C3×C4×C8—21no—
99C2×#131C22×C311yescharacteristic
100C23×C8C22×C311yescharacteristic
101#525C22×C311yescharacteristic
102C22×C4⋊C4C22×C312yes—
103#484—22no—
104#483—22no—
105#481—22no—
106#480—22no—
107#435—21no—
108#435—21no—
109C8×D4—21no—
110C8×D4—21no—
111C2×#108C22×C312yes—
112#428C22×C314yes—
113C2×#107—21no—
114C2×#107—21no—
115C2×#107—21no—
116C2×#107—21no—
117C2×#107C22×C312yes—
118C2×#107C22×C312yes—
119#424C22×C311yescharacteristic
120#417C22×C314yes—
121#408C22×C311yescharacteristic
122#407C22×C312yes—
123#407C22×C311yescharacteristic
124#405C22×C311yescharacteristic
125#343C22×C314yes—
126#343C22×C314yes—
127#341C22×C312yes—
128C24×C3#4111yescharacteristic
129C3×#41—22no—
130C3×#41—22no—
131C2×C3×Q8C2×D411yescharacteristic
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4C2×D411yescharacteristic
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—24no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—84no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C3×M16—24no—
150C3×M16—22no—
151C3×M16—22no—
152C3×M16—24no—
153C3×M16—42no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—21no—
160C2×C3×C8—24no—
161C2×C3×C8—21no—
162C2×C3×C8—42no—
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8#4114yes—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4C2×D411yescharacteristic
175C3×C4⋊C4C2×D411yescharacteristic
176C3×C22⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—21no—
180C3×C22⋊C4—21no—
181C3×C42—21no—
182C2×#41C23×C311yescharacteristic
183C23×C4—22no—
184C23×C4C23×C311yescharacteristic
185#131—24no—
186C2×M16—21no—
187C2×M16C23×C311yescharacteristic
188C2×M16C23×C314yes—
189C2×M16C23×C311yescharacteristic
190C22×C8—24no—
191C22×C8—21no—
192C22×C8C3×D411yescharacteristic
193C22×C8C3×D411yescharacteristic
194C22×C8C3×D411yescharacteristic
195C22×C8C3×D411yescharacteristic
196C22×C8C23×C311yescharacteristic
197C22×C8C23×C311yescharacteristic
198#123—21no—
199C22⋊Q8C3×D411yescharacteristic
200C22⋊Q8C3×D411yescharacteristic
201#121C3×D411yescharacteristic
202#121C3×D411yescharacteristic
203#120—21no—
204C4×D4—21no—
205C4×D4—21no—
206#117C23×C311yescharacteristic
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C23×C314yes—
213#108—24no—
214#107—22no—
215#107—22no—
216#107—22no—
217#107—22no—
218#107—24no—
219#106—24no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222#103—24no—
223#102—24no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230#97—21no—
231C4×C8—21no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—21no—
236C23×C3#12111yescharacteristic
237C23×C3#12111yescharacteristic
238C23×C3C2×#4111yescharacteristic
239C3×Q8—22no—
240C3×D4—22no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—42no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—44no—
257C2×C3×C4—84no—
258C2×C3×C4—84no—
259C2×C3×C4—84no—
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C2×#4112yes—
265C2×C3×C42−1+412yes—
266C3×C8—21no—
267C3×C8—21no—
268C3×C8—41no—
269C3×C8—41no—
270C3×C8—44no—
271C3×C8#12312yes—
272C3×C8#12312yes—
273C24C3×#4111yescharacteristic
274#41—22no—
275#41—22no—
276C2×Q8C2×C3×D411yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4C2×C3×D411yescharacteristic
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—84no—
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C2×C3×D411yescharacteristic
292C22×C4C3×#4111yescharacteristic
293C22×C4C24×C311yescharacteristic
294M16—24no—
295M16—22no—
296M16—22no—
297M16—24no—
298M16—42no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—24no—
306C2×C8—21no—
307C2×C8—42no—
308C2×C8C2×C3×D411yescharacteristic
309C2×C8C2×C3×D411yescharacteristic
310C2×C8C2×C3×D411yescharacteristic
311C2×C8C2×C3×D411yescharacteristic
312C2×C8C3×#4114yes—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4C2×C3×D411yescharacteristic
320C4⋊C4C2×C3×D411yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C42—21no—
327C22×C3—28no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3#47212yes—
338C22×C3#47312yes—
339C22×C3C2×#12111yescharacteristic
340C22×C3#55212yes—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—41no—
345C3×C4—41no—
346C3×C4—84no—
347C3×C4C2×#12311yescharacteristic
348C3×C4#53611yescharacteristic
349C3×C4#54712yes—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23C3×#12111yescharacteristic
355C23C3×#12111yescharacteristic
356C23C2×C3×#4111yescharacteristic
357Q8—22no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—42no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4C3×#12111yescharacteristic
379C2×C4C3×#12111yescharacteristic
380C2×C4C22×C3×D411yescharacteristic
381C2×C4C22×C3×D411yescharacteristic
382C2×C4C2×C3×#4112yes—
383C2×C4C3×2−1+412yes—
384C8—21no—
385C8—21no—
386C8—41no—
387C8—41no—
388C8—44no—
389C8C3×#12312yes—
390C8C3×#12312yes—
391C2×C3—22no—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—81no—
395C2×C3#248611yescharacteristic
396C2×C3C2×#47211yescharacteristic
397C2×C3C2×#47311yescharacteristic
398C22—28no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22C3×#47212yes—
409C22C3×#47312yes—
410C22C2×C3×#12111yescharacteristic
411C22C3×#55212yes—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—84no—
418C4C2×C3×#12311yescharacteristic
419C4C3×#53611yescharacteristic
420C4C3×#54712yes—
421C3#2767511yescharacteristic, direct product factor, Sylow
422C2—22no—
423C2—22no—
424C2—22no—
425C2—81no—
426C2C3×#248611yescharacteristic
427C2C2×C3×#47211yescharacteristic
428C2C2×C3×#47311yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C22×C3×#107C2semidirect
4. C3×#2089C2semidirect
5. C3×#2078C2semidirect
6. C3×#1952C2semidirect
7. C3×#1821C2semidirect
8. C3×#1816C2semidirect
9. C3×#1810C2semidirect
10. #27675C3direct (N × H)
42. C22×#107C2×C3semidirect
43. #2089C2×C3semidirect
44. #2078C2×C3semidirect
45. #1952C2×C3semidirect
46. #1821C2×C3semidirect
47. #1816C2×C3semidirect
48. #1810C2×C3semidirect
421. C3#27675direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30922
C3×#29292
C22×C3×#1072
C3×#20892
C3×#20782
C3×#19522
C3×#18212
C3×#18162
C3×#18102
#276753