Finite Group Databasebeta

← #10592505⋊C3 #13141919

Subgroups — #10592505⋊C3 #13141919

Order 1728.

595 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592505C311yesmaximal, characteristic, complemented
9C3×#4806—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3—22no—
12Q8×(C3⋊D3)⋊C3—22no—
13C2×C4⋊((C3⋊D3)⋊C3)—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3—21no—
18C2×#634⋊C3C2211yescharacteristic
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21#3755⋊C3—22no—
22#3752⋊C3—21no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24#3744⋊C3—22no—
25C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D3×Q8—31no—
27C2×#3755C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic
30C2×C3×#215—31no—
31C2×C3×#204—31no—
32C3×#895—31no—
33C32×#103—21no—
34C32×#103D311yescharacteristic, complemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#810—31no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)D412yescomplemented
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—42no—
52C4×(C3⋊D3)⋊C3—24no—
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3—21no—
55#634⋊C3—24no—
56#634⋊C3—21no—
57#634⋊C3D412yescomplemented
58C8×H(3,𝔽3)—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60C3×#448—91no—
61#4806—31no—
62#4806—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C22×C311yescharacteristic
66Q8×C3⋊D3—22no—
67Q8×C3⋊D3—22no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C4×C3⋊D3—21no—
71C2×C4×C3⋊D3C22×C311yescharacteristic
72C2×#634—21no—
73C2×#634C22×C311yescharacteristic
74C3×D3×Q8—62no—
75C3×D3×Q8—62no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—61no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×C32×C8—21no—
83C2×C32×C8C2×D311yescharacteristic
84C2×C32×C8C22×C311yescharacteristic
85C32×C4⋊C4—21no—
86C32×C4⋊C4C2×D311yescharacteristic
87C32×C4⋊C4C22×C311yescharacteristic
88#3755—22no—
89#3752—21no—
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91#3744—22no—
92C3×#215—62no—
93C3×(C2×D3)⋊C4—61no—
94C2×C3×C3⋊C8—31no—
95C3×#204—62no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—21no—
99C2×(C3⋊D3)⋊C3—41no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3SD1612yescomplemented
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)—41no—
105C4×H(3,𝔽3)C2×D411yescharacteristic
106C4×H(3,𝔽3)C2×D411yescharacteristic
107(C3⋊Dic3)⋊C3—21no—
108(C3⋊Dic3)⋊C3—22no—
109C22×C3×Q8—91no—
110C2×D3×Q8—31no—
111C2×D3×Q8—91no—
112C2×C3×SD16—91no—
113C2×C3×SD16—91no—
114C3×#121—91no—
115C2×#215—31no—
116C2×#215—91no—
117C2×#204—31no—
118C2×#204—91no—
119#895—31no—
120#895—91no—
121C3×#103—31no—
122C3×#103—31no—
123C3×#103—61no—
124C3×#103—91no—
125C3×#103C3×D311yescharacteristic
126C3×C22⋊C8—91no—
127(C2×D3)⋊C8—31no—
128(C2×D3)⋊C8—91no—
129#810—31no—
130#810—91no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3C3×D411yescharacteristic
133C22×C3×D3—31no—
134C22×C3×D3—61no—
135C32×Q8—21no—
136C32×Q8—21no—
137C32×Q8—22no—
138C32×Q8—41no—
139C32×Q8D1212yescomplemented
140C32×Q8C3×D412yescomplemented
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C23×C311yescharacteristic
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3C3×D411yescharacteristic
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—42no—
155C4×C3⋊D3—24no—
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
158#634—24no—
159#634—21no—
160#634C3×D412yescomplemented
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C3×D12—61no—
164C3×D12—61no—
165C3×D12—122no—
166C3×C4×D3—64no—
167C3×C4×D3—61no—
168C3×C4×D3—61no—
169C3×Dic6—32no—
170C3×Dic6—64no—
171C3×Dic6—61no—
172C32×C8—21no—
173C32×C8—21no—
174C32×C8—41no—
175C3⋊(C3⋊C8)—21no—
176C3×C3⋊C8—61no—
177#448—91noSylow
178C2×H(3,𝔽3)#12011yescharacteristic
179C2×H(3,𝔽3)C2×SD1611yescharacteristic
180C2×H(3,𝔽3)#13711yescharacteristic
181(C3⋊D3)⋊C3—22no—
182(C3⋊D3)⋊C3—81no—
183C2×C3×Q8—31no—
184C2×C3×Q8—31no—
185C2×C3×Q8—61no—
186C2×C3×Q8—91no—
187C2×C3×Q8—181no—
188C2×C3×Q8—182no—
189C2×C3×Q8—182no—
190C2×C3×Q8C2×C3×D311yescharacteristic
191C2×C3×D4—91no—
192C2×C3×D4—181no—
193C22×C3×C4—91no—
194C22×C3×C4—181no—
195D3×Q8—62no—
196D3×Q8—62no—
197D3×Q8—182no—
198D3×Q8—182no—
199C2×D12—31no—
200C2×D12—61no—
201C2×D12—91no—
202C2×D12—181no—
203C2×C4×D3—31no—
204C2×C4×D3—61no—
205C2×C4×D3—91no—
206C2×C4×D3—181no—
207C2×Dic6—31no—
208C2×Dic6—61no—
209C2×Dic6—91no—
210C2×Dic6—181no—
211C3×SD16—182no—
212C3×SD16—182no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—91no—
217C2×C3×C8C2×C3×D311yescharacteristic
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4C2×C3×D311yescharacteristic
222C3×C22⋊C4—181no—
223#215—62no—
224#215—182no—
225(C2×D3)⋊C4—61no—
226(C2×D3)⋊C4—181no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—91no—
229#204—62no—
230#204—182no—
231C22×C32—21no—
232C22×C32D3×D411yescharacteristic
233C22×C32C2×C3×D411yescharacteristic
234C2×C3⋊D3—21no—
235C2×C3⋊D3—21no—
236C2×C3⋊D3—41no—
237C2×C3⋊D3—41no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3C3×SD1612yescomplemented
240C2×C3×D3—32no—
241C2×C3×D3—61no—
242C2×C3×D3—61no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C32×C4—22no—
247C32×C4—22no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—42no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4—81no—
254C32×C4C2×D1211yescharacteristic
255C32×C4D3×D411yescharacteristic
256C32×C4C2×C3×D411yescharacteristic
257C32×C4C2×C3×D411yescharacteristic
258C3⋊Dic3—21no—
259C3⋊Dic3—22no—
260C3×Dic3—62no—
261C3×Dic3—61no—
262C22×Q8—91no—
263C2×SD16—91no—
264C2×SD16—91no—
265#121—91no—
266#103—91no—
267#103(C3⋊D3)⋊C311yescharacteristic, complemented
268C22⋊C8—91no—
269H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
270C23×C3—91no—
271C23×C3—181no—
272C22×D3—31no—
273C22×D3—61no—
274C22×D3—91no—
275C22×D3—181no—
276C3×Q8—21no—
277C3×Q8—32no—
278C3×Q8—32no—
279C3×Q8—61no—
280C3×Q8—61no—
281C3×Q8—62no—
282C3×Q8—92no—
283C3×Q8—121no—
284C3×Q8—181no—
285C3×Q8—184no—
286C3×Q8C3×D1212yes—
287C3×D4—181no—
288C3×D4—181no—
289C3×D4—362no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—91no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4—181no—
303C2×C3×C4—181no—
304C2×C3×C4—184no—
305C2×C3×C4—181no—
306C2×C3×C4C22×C3×D311yescharacteristic
307C2×Dic3—31no—
308C2×Dic3—61no—
309C2×Dic3—91no—
310C2×Dic3—181no—
311D12—61no—
312D12—61no—
313D12—122no—
314D12—181no—
315D12—181no—
316D12—362no—
317C4×D3—64no—
318C4×D3—61no—
319C4×D3—61no—
320C4×D3—181no—
321C4×D3—181no—
322C4×D3—184no—
323Dic6—32no—
324Dic6—64no—
325Dic6—61no—
326Dic6—92no—
327Dic6—181no—
328Dic6—184no—
329C3×C8—21no—
330C3×C8—61no—
331C3×C8—61no—
332C3×C8—121no—
333C3×C8—181no—
334C3⋊C8—61no—
335C3⋊C8—181no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32#88211yescharacteristic
340C2×C32D3×SD1611yescharacteristic
341C2×C32#91811yescharacteristic
342C2×C32C3×#12011yescharacteristic
343C2×C32C2×C3×SD1611yescharacteristic
344C2×C32C3×#13711yescharacteristic
345C3⋊D3—22no—
346C3⋊D3—81no—
347C3×D3—62no—
348C3×D3—241no—
349C2×Q8—91no—
350C2×Q8—181no—
351C2×Q8—182no—
352C2×Q8—182no—
353C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
354C2×D4—91no—
355C2×D4—181no—
356C22×C4—91no—
357C22×C4—181no—
358SD16—182no—
359SD16—182no—
360C2×C8—91no—
361C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
362C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
363C22⋊C4—181no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—61no—
367C22×C3—92no—
368C22×C3—181no—
369C22×C3—181no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3C3×D3×D411yescharacteristic
374C2×D3—32no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—92no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—121no—
381C2×D3—181no—
382C2×D3—181no—
383C2×D3—361no—
384C2×D3—361no—
385C2×D3—361no—
386C3×C4—22no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—31no—
391C3×C4—41no—
392C3×C4—62no—
393C3×C4—62no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—122no—
397C3×C4—121no—
398C3×C4—121no—
399C3×C4—181no—
400C3×C4—182no—
401C3×C4—241no—
402C3×C4C2×C3×D1211yescharacteristic
403C3×C4C3×D3×D411yescharacteristic
404Dic3—62no—
405Dic3—61no—
406Dic3—182no—
407Dic3—181no—
408C32—21no—
409C32#480611yescharacteristic, complemented
410C32C3×#44811yescharacteristic, complemented
411C23—91no—
412C23—181no—
413Q8—21no—
414Q8—92no—
415Q8—184no—
416Q8—181no—
417Q8C4⋊((C3⋊D3)⋊C3)12yescomplemented
418D4—181no—
419D4—181no—
420D4—362no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—91no—
424C2×C4—184no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
429C8—21no—
430C8—181no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—61no—
440C2×C3—182no—
441C2×C3—721no—
442C2×C3C3×#88211yescharacteristic
443C2×C3C3×D3×SD1611yescharacteristic
444C2×C3C3×#91811yescharacteristic
445D3—62no—
446D3—182no—
447D3—241no—
448D3—721no—
449C22—92no—
450C22—181no—
451C22—181no—
452C22—361no—
453C22—361no—
454C22—361no—
455C22D4×(C3⋊D3)⋊C311yescharacteristic
456C4—22no—
457C4—41no—
458C4—181no—
459C4—182no—
460C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
461C4D4×(C3⋊D3)⋊C311yescharacteristic
462C3—31no—
463C3—31no—
464C3—61no—
465C3C3×#480611yescharacteristic
466C2—182no—
467C2—721no—
468C2#64091⋊C311yescharacteristic
469C2SD16×(C3⋊D3)⋊C311yescharacteristic
470C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radical#64104⋊C3C2
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#448
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. C2×Q8×(C3⋊D3)⋊C3C2semidirect
2. C2×#3755⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #63647⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592505C3semidirect
25. C2×Q8×C3⋊D3C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. C2×#3744C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
36. #63647C2×C3semidirect
37. #63637C2×C3semidirect
43. Q8×H(3,𝔽3)D4semidirect
57. #634⋊C3D4semidirect
102. C2×(C3⋊D3)⋊C3SD16semidirect
139. C32×Q8D12semidirect
140. C32×Q8C3×D4semidirect
160. #634C3×D4semidirect
239. C2×C3⋊D3C3×SD16semidirect
267. #103(C3⋊D3)⋊C3semidirect
269. H(3,𝔽3)#448semidirect
409. C32#4806semidirect
410. C32C3×#448semidirect
417. Q8C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×#3755⋊C32
C2×#3744⋊C32
#64104⋊C32
H(3,𝔽3)×#1032
#63647⋊C32
#63637⋊C32
#105925053
C3×#48063