Finite Group Databasebeta

← C3×#21877 #10640484

Subgroups — C3×#21877 #10640484

Order 768.

1098 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#539C211yesmaximal, characteristic
2C3×#2316C211yesmaximal, characteristic, complemented
3C3×#2313C211yesmaximal, characteristic, complemented
4C3×#1996C212yesmaximal, complemented
5C3×#1964C212yesmaximal
6C3×#1899C214yesmaximal, complemented
7C3×#1799C214yesmaximal, complemented
8#21877C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#539—24no—
10C3×#539—24no—
11C2×C3×#126C412yescomplemented
12C2×C3×#125C412yescomplemented
13C2×C3×#123C412yescomplemented
14C2×C3×#123C2212yes—
15C2×C3×C22⋊Q8C412yescomplemented
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4C2212yes—
19C2×C3×#117C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×C3×#99C2218yes—
22C3×#389—24no—
23C3×#387—24no—
24C3×#381C2212yes—
25C3×#380C2212yes—
26C3×#380C2212yes—
27C3×#380C2212yes—
28C3×C4×C22⋊C4C2212yes—
29C3×#342C2218yes—
30C2×#539C2×C311yescharacteristic
31#2316C2×C311yescharacteristic, complemented
32#2313C2×C311yescharacteristic, complemented
33#1996C2×C312yescomplemented
34#1964C2×C312yes—
35#1899C2×C314yescomplemented
36#1799C2×C314yescomplemented
37C22×C3×Q8C2×C411yescharacteristic
38C22×C3×D4C2×C411yescharacteristic
39C22×C3×D4C2312yes—
40C23×C3×C4C2×C412yes—
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C3×#126—28no—
45C3×#125—28no—
46C3×#123—24no—
47C3×#123—24no—
48C3×#123—28no—
49C3×C22⋊Q8—22no—
50C3×C22⋊Q8—22no—
51C3×C22⋊Q8—28no—
52C3×#121—22no—
53C3×#121—22no—
54C3×C4×D4—24no—
55C3×C4×D4—24no—
56C3×#117—44no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4C2×C412yes—
62C2×C3×C4⋊C4C2×C411yescharacteristic
63C2×C3×C4⋊C4C2×C412yes—
64C2×C3×C4⋊C4C2×C412yes—
65C2×C3×C4⋊C4C2×C411yescharacteristic
66C2×C3×C4⋊C4C2×C412yes—
67C2×C3×C4⋊C4C2×C412yes—
68C2×C3×C4⋊C4C2×C412yes—
69C2×C3×C4⋊C4D414yes—
70C2×C3×C4⋊C4D414yes—
71C2×C3×C22⋊C4C2×C412yes—
72C2×C3×C22⋊C4C2×C412yes—
73C2×C3×C22⋊C4C2×C412yes—
74C2×C3×C22⋊C4C2×C412yes—
75C2×C3×C22⋊C4C2314yes—
76C2×C3×C22⋊C4C2314yes—
77C2×C3×C22⋊C4C2312yes—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42C2×C412yes—
81C3×#99—48no—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—24no—
87#539—24no—
88#539—24no—
89C2×#126C3×C412yescomplemented
90C2×#125C3×C412yescomplemented
91C2×#123C3×C412yescomplemented
92C2×#123C22×C312yes—
93C2×C22⋊Q8C3×C412yescomplemented
94C2×C22⋊Q8C22×C311yescharacteristic
95C2×#121C22×C311yescharacteristic
96C2×C4×D4C22×C312yes—
97C2×#117C22×C312yes—
98C22×C4⋊C4C22×C311yescharacteristic
99C2×#99C22×C318yes—
100#389—24no—
101#387—24no—
102#381C22×C312yes—
103#380C22×C312yes—
104#380C22×C312yes—
105#380C22×C312yes—
106C4×C22⋊C4C22×C312yes—
107#342C22×C318yes—
108C24×C3C22×C412yes—
109C24×C3C2411yescharacteristic
110C2×C3×Q8—22no—
111C2×C3×Q8—81no—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—24no—
115C2×C3×D4—24no—
116C2×C3×D4—22no—
117C2×C3×D4—81no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—24no—
121C22×C3×C4—24no—
122C22×C3×C4—24no—
123C22×C3×C4—81no—
124C22×C3×C4—81no—
125C22×C3×C4—82no—
126C22×C3×C4C22×C412yes—
127C22×C3×C4C22×C412yes—
128C22×C3×C4C22×C412yes—
129C22×C3×C4C22×C412yes—
130C22×C3×C4C22×C412yes—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D412yes—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—44no—
146C3×C4⋊C4—82no—
147C3×C4⋊C4—82no—
148C3×C4⋊C4—82no—
149C3×C4⋊C4C22⋊C418yes—
150C3×C4⋊C4C22⋊C418yes—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—22no—
153C3×C22⋊C4—24no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—28no—
159C3×C22⋊C4—24no—
160C3×C42—24no—
161C3×C42—44no—
162C3×C42—82no—
163C22×Q8C2×C3×C411yescharacteristic
164C22×D4C2×C3×C411yescharacteristic
165C22×D4C23×C312yes—
166C23×C4C2×C3×C412yes—
167C23×C4C23×C311yescharacteristic
168C23×C4C23×C311yescharacteristic
169C23×C4C23×C311yescharacteristic
170#126—28no—
171#125—28no—
172#123—24no—
173#123—24no—
174#123—28no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—28no—
178#121—22no—
179#121—22no—
180C4×D4—24no—
181C4×D4—24no—
182#117—44no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C2×C3×C412yes—
188C2×C4⋊C4C2×C3×C411yescharacteristic
189C2×C4⋊C4C2×C3×C412yes—
190C2×C4⋊C4C2×C3×C412yes—
191C2×C4⋊C4C2×C3×C411yescharacteristic
192C2×C4⋊C4C2×C3×C412yes—
193C2×C4⋊C4C2×C3×C412yes—
194C2×C4⋊C4C2×C3×C412yes—
195C2×C4⋊C4C3×D414yes—
196C2×C4⋊C4C3×D414yes—
197C2×C22⋊C4C2×C3×C412yes—
198C2×C22⋊C4C2×C3×C412yes—
199C2×C22⋊C4C2×C3×C412yes—
200C2×C22⋊C4C2×C3×C412yes—
201C2×C22⋊C4C23×C314yes—
202C2×C22⋊C4C23×C314yes—
203C2×C22⋊C4C23×C312yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C2×C3×C412yes—
207#99—48no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—24no—
213C23×C3—22no—
214C23×C3—24no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3C23×C411yescharacteristic
220C23×C3C22×D411yescharacteristic
221C23×C3C22×D411yescharacteristic
222C23×C3H(5,𝔽2)12yes—
223C23×C32−1+412yes—
224C3×Q8—82no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—44no—
228C3×D4—82no—
229C2×C3×C4—24no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—44no—
247C2×C3×C4—44no—
248C2×C3×C4—44no—
249C2×C3×C4—44no—
250C2×C3×C4—44no—
251C2×C3×C4—44no—
252C2×C3×C4—84no—
253C2×C3×C4C2×C22⋊C414yes—
254C2×C3×C4C2×C22⋊C414yes—
255C2×C3×C4C2×C22⋊C412yes—
256C2×C3×C4C2×C22⋊C412yes—
257C24C22×C3×C412yes—
258C24C24×C311yescharacteristic
259C2×Q8—22no—
260C2×Q8—81no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—81no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—82no—
275C22×C4C22×C3×C412yes—
276C22×C4C22×C3×C412yes—
277C22×C4C22×C3×C412yes—
278C22×C4C22×C3×C412yes—
279C22×C4C22×C3×C412yes—
280C22×C4C22×C3×C411yescharacteristic
281C22×C4C22×C3×C411yescharacteristic
282C22×C4C2×C3×D412yes—
283C22×C4C2×C3×D412yes—
284C22×C4C2×C3×D412yes—
285C22×C4C2×C3×D412yes—
286C22×C4C2×C3×D412yes—
287C22×C4C2×C3×D412yes—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4C3×C22⋊C418yes—
299C4⋊C4C3×C22⋊C418yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—28no—
308C22⋊C4—24no—
309C42—24no—
310C42—44no—
311C42—82no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—84no—
324C22×C3C22×C22⋊C411yescharacteristic
325C22×C3#52012yes—
326C22×C3#53511yescharacteristic
327C22×C3#53611yescharacteristic
328C22×C3#53712yes—
329C3×C4—24no—
330C3×C4—24no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—84no—
337C3×C4—84no—
338C23—22no—
339C23—24no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—42no—
344C23C23×C3×C411yescharacteristic
345C23C22×C3×D411yescharacteristic
346C23C22×C3×D411yescharacteristic
347C23C3×H(5,𝔽2)12yes—
348C23C3×2−1+412yes—
349Q8—82no—
350D4—42no—
351D4—42no—
352D4—44no—
353D4—82no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—84no—
378C2×C4C2×C3×C22⋊C414yes—
379C2×C4C2×C3×C22⋊C414yes—
380C2×C4C2×C3×C22⋊C412yes—
381C2×C4C2×C3×C22⋊C412yes—
382C2×C3—21no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—82no—
387C2×C3#231111yescharacteristic
388C2×C3#288311yescharacteristic
389C2×C3#288411yescharacteristic
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—84no—
402C22C22×C3×C22⋊C411yescharacteristic
403C22C3×#52012yes—
404C22C3×#53511yescharacteristic
405C22C3×#53611yescharacteristic
406C22C3×#53712yes—
407C4—24no—
408C4—22no—
409C4—22no—
410C4—24no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—84no—
415C4—84no—
416C3#2187711yescharacteristic, direct product factor, Sylow
417C2—21no—
418C2—22no—
419C2—22no—
420C2—21no—
421C2—82no—
422C2C3×#231111yescharacteristic
423C2C3×#288311yescharacteristic
424C2C3×#288411yescharacteristic
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2316C2semidirect
3. C3×#2313C2semidirect
4. C3×#1996C2semidirect
6. C3×#1899C2semidirect
7. C3×#1799C2semidirect
8. #21877C3direct (N × H)
11. C2×C3×#126C4semidirect
12. C2×C3×#125C4semidirect
13. C2×C3×#123C4semidirect
15. C2×C3×C22⋊Q8C4semidirect
31. #2316C2×C3semidirect
32. #2313C2×C3semidirect
33. #1996C2×C3semidirect
35. #1899C2×C3semidirect
36. #1799C2×C3semidirect
89. C2×#126C3×C4semidirect
90. C2×#125C3×C4semidirect
91. C2×#123C3×C4semidirect
93. C2×C22⋊Q8C3×C4semidirect
416. C3#21877direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5392
C3×#23162
C3×#23132
C3×#19962
C3×#19642
C3×#18992
C3×#17992
#218773