Finite Group Databasebeta

← #10592509⋊C3 #13141931

Subgroups — #10592509⋊C3 #13141931

Order 1728.

609 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 100 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×#88722C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592509C311yesmaximal, characteristic, complemented
9C3×#4810—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722—22no—
12#88722—22no—
13C2×C4⋊((C3⋊D3)⋊C3)—21no—
14C2×C4⋊((C3⋊D3)⋊C3)—21no—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3—21no—
18C2×C4×(C3⋊D3)⋊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#3745⋊C3—22no—
25C2×#3832C2×C311yescharacteristic, complemented
26C2×C3×#239—31no—
27C2×#3755C2×C311yescharacteristic, complemented
28C2×#3745C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic, complemented
30C2×C3×#215—31no—
31C2×C3×D24—31no—
32C3×#895—31no—
33C32×#103—21no—
34C32×#103D311yescharacteristic, complemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63636C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#809—31no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3D411yescharacteristic
43Q8×H(3,𝔽3)—21no—
44Q8×H(3,𝔽3)D412yescomplemented
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—24no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)—42no—
54C4⋊((C3⋊D3)⋊C3)D412yescomplemented
55C4×(C3⋊D3)⋊C3—24no—
56C4×(C3⋊D3)⋊C3—21no—
57C4×(C3⋊D3)⋊C3—21no—
58C8×H(3,𝔽3)—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60C3×#449—91no—
61#4810—31no—
62#4810—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C22×C311yescharacteristic
66#3832—22no—
67#3832—22no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C4×C3⋊D3—21no—
73C2×C4×C3⋊D3C22×C311yescharacteristic
74C3×#239—62no—
75C3×#239—62no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×D12—61no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—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#3745—22no—
92C3×#215—62no—
93C3×(C2×D3)⋊C4—61no—
94C2×C3×C3⋊C8—31no—
95C3×D24—62no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—21no—
100C2×(C3⋊D3)⋊C3—21no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3—41no—
103C2×(C3⋊D3)⋊C3—41no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C4×H(3,𝔽3)—22no—
107C4×H(3,𝔽3)—41no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110(C3⋊Dic3)⋊C3—21no—
111C2×C3×#41—91no—
112C2×#239—31no—
113C2×#239—91no—
114C2×C3×SD16—91no—
115C2×C3×D8—91no—
116C3×#121—91no—
117C2×#215—31no—
118C2×#215—91no—
119C2×D24—31no—
120C2×D24—91no—
121#895—31no—
122#895—91no—
123C3×#103—31no—
124C3×#103—31no—
125C3×#103—61no—
126C3×#103C3×D311yescharacteristic
127C3×#102—91no—
128C3×C22⋊C8—91no—
129(C2×D3)⋊C8—31no—
130(C2×D3)⋊C8—91no—
131#809—31no—
132#809—91no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3C3×D411yescharacteristic
136C22×C3×D3—31no—
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C32×Q8—21no—
140C32×Q8—21no—
141C32×Q8—22no—
142C32×Q8—41no—
143C32×Q8D1212yescomplemented
144C32×Q8C3×D412yescomplemented
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154C2×C3⋊Dic3C3×D411yescharacteristic
155C4⋊(C3⋊D3)—24no—
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—42no—
160C4⋊(C3⋊D3)C3×D412yescomplemented
161C4×C3⋊D3—24no—
162C4×C3⋊D3—21no—
163C4×C3⋊D3—21no—
164C2×C3×Dic3—31no—
165C3×D12—32no—
166C3×D12—64no—
167C3×D12—61no—
168C3×D12—61no—
169C3×D12—61no—
170C3×D12—122no—
171C3×C4×D3—64no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—41no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—61no—
179#449—91noSylow
180C2×H(3,𝔽3)#12011yescharacteristic
181C2×H(3,𝔽3)#13511yescharacteristic
182C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
183(C3⋊D3)⋊C3—41no—
184(C3⋊D3)⋊C3—42no—
185(C3⋊D3)⋊C3—81no—
186C3×#41—182no—
187C3×#41—182no—
188C2×C3×Q8—31no—
189C2×C3×Q8—31no—
190C2×C3×Q8—61no—
191C2×C3×Q8C2×C3×D311yescharacteristic
192C2×C3×D4—91no—
193C2×C3×D4—91no—
194C2×C3×D4—181no—
195C2×C3×D4—181no—
196C22×C3×C4—91no—
197C22×C3×C4—181no—
198#239—62no—
199#239—62no—
200#239—182no—
201#239—182no—
202C2×D12—31no—
203C2×D12—31no—
204C2×D12—61no—
205C2×D12—61no—
206C2×D12—91no—
207C2×D12—91no—
208C2×D12—181no—
209C2×D12—181no—
210C2×C4×D3—31no—
211C2×C4×D3—61no—
212C2×C4×D3—91no—
213C2×C4×D3—181no—
214C3×SD16—182no—
215C3×D8—182no—
216C2×C3×C8—31no—
217C2×C3×C8—31no—
218C2×C3×C8—61no—
219C2×C3×C8—91no—
220C2×C3×C8C2×C3×D311yescharacteristic
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4C2×C3×D311yescharacteristic
225C3×C22⋊C4—181no—
226#215—62no—
227#215—182no—
228(C2×D3)⋊C4—61no—
229(C2×D3)⋊C4—181no—
230C2×C3⋊C8—31no—
231C2×C3⋊C8—91no—
232D24—62no—
233D24—182no—
234C22×C32—21no—
235C22×C32D3×D411yescharacteristic
236C22×C32C2×C3×D411yescharacteristic
237C2×C3⋊D3—21no—
238C2×C3⋊D3—22no—
239C2×C3⋊D3—21no—
240C2×C3⋊D3—21no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3—41no—
246C2×C3×D3—62no—
247C2×C3×D3—61no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C2×C3×D3—121no—
253C2×C3×D3—121no—
254C2×C3×D3—121no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—42no—
260C32×C4—41no—
261C32×C4—41no—
262C32×C4—81no—
263C32×C4C2×D1211yescharacteristic
264C32×C4D3×D411yescharacteristic
265C32×C4C2×C3×D411yescharacteristic
266C32×C4C2×C3×D411yescharacteristic
267C3⋊Dic3—21no—
268C3×Dic3—61no—
269C2×#41—91no—
270C2×SD16—91no—
271C2×D8—91no—
272#121—91no—
273#103(C3⋊D3)⋊C311yescharacteristic, complemented
274#102—91no—
275C22⋊C8—91no—
276H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
277C23×C3—91no—
278C23×C3—181no—
279C23×C3—181no—
280C22×D3—31no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—91no—
284C22×D3—181no—
285C22×D3—181no—
286C3×Q8—21no—
287C3×Q8—32no—
288C3×Q8—32no—
289C3×Q8—61no—
290C3×Q8—61no—
291C3×Q8—62no—
292C3×Q8—121no—
293C3×Q8C3×D1212yes—
294C3×D4—92no—
295C3×D4—181no—
296C3×D4—181no—
297C3×D4—181no—
298C3×D4—184no—
299C3×D4—362no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—91no—
310C2×C3×C4—121no—
311C2×C3×C4—121no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314C2×C3×C4—184no—
315C2×C3×C4C22×C3×D311yescharacteristic
316C2×Dic3—31no—
317C2×Dic3—91no—
318D12—32no—
319D12—64no—
320D12—61no—
321D12—61no—
322D12—61no—
323D12—92no—
324D12—122no—
325D12—181no—
326D12—181no—
327D12—181no—
328D12—184no—
329D12—362no—
330C4×D3—64no—
331C4×D3—61no—
332C4×D3—61no—
333C4×D3—181no—
334C4×D3—181no—
335C4×D3—184no—
336C3×C8—21no—
337C3×C8—61no—
338C3×C8—61no—
339C3×C8—121no—
340C3×C8—181no—
341C3⋊C8—61no—
342C3⋊C8—181no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32#88211yescharacteristic
347C2×C32#91411yescharacteristic
348C2×C32#91911yescharacteristic
349C2×C32C3×#12011yescharacteristic
350C2×C32C3×#13511yescharacteristic
351C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
352C3⋊D3—41no—
353C3⋊D3—42no—
354C3⋊D3—81no—
355C3×D3—121no—
356C3×D3—122no—
357C3×D3—241no—
358#41—182no—
359#41—182no—
360C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
361C2×D4—91no—
362C2×D4—91no—
363C2×D4—181no—
364C2×D4—181no—
365C22×C4—91no—
366C22×C4—181no—
367SD16—182no—
368D8—182no—
369C2×C8—91no—
370C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
371C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
372C22⋊C4—181no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—61no—
376C22×C3—182no—
377C22×C3—181no—
378C22×C3—181no—
379C22×C3—181no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3—361no—
383C22×C3—361no—
384C22×C3—361no—
385C22×C3C3×D3×D411yescharacteristic
386C2×D3—62no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—121no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—182no—
396C2×D3—181no—
397C2×D3—181no—
398C2×D3—181no—
399C2×D3—361no—
400C2×D3—361no—
401C2×D3—361no—
402C2×D3—361no—
403C2×D3—361no—
404C3×C4—22no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—31no—
408C3×C4—31no—
409C3×C4—41no—
410C3×C4—62no—
411C3×C4—62no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—122no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—181no—
418C3×C4—241no—
419C3×C4C2×C3×D1211yescharacteristic
420C3×C4C3×D3×D411yescharacteristic
421Dic3—61no—
422Dic3—181no—
423C32—21no—
424C32#481011yescharacteristic, complemented
425C32C3×#44911yescharacteristic, complemented
426C23—91no—
427C23—181no—
428C23—181no—
429Q8—21no—
430Q8C4⋊((C3⋊D3)⋊C3)12yescomplemented
431D4—92no—
432D4—184no—
433D4—181no—
434D4—181no—
435D4—181no—
436D4—362no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—91no—
440C2×C4—184no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
444C8—21no—
445C8—181no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—61no—
453C2×C3—61no—
454C2×C3—61no—
455C2×C3—361no—
456C2×C3—362no—
457C2×C3—721no—
458C2×C3C3×#88211yescharacteristic
459C2×C3C3×#91411yescharacteristic
460C2×C3C3×#91911yescharacteristic
461D3—121no—
462D3—122no—
463D3—241no—
464D3—361no—
465D3—362no—
466D3—721no—
467C22—181no—
468C22—181no—
469C22—182no—
470C22—181no—
471C22—361no—
472C22—361no—
473C22—361no—
474C22—361no—
475C22—361no—
476C22D4×(C3⋊D3)⋊C311yescharacteristic
477C4—22no—
478C4—41no—
479C4—181no—
480C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
481C4D4×(C3⋊D3)⋊C311yescharacteristic
482C3—31no—
483C3—31no—
484C3—61no—
485C3C3×#481011yescharacteristic
486C2—361no—
487C2—362no—
488C2—721no—
489C2#64091⋊C311yescharacteristic
490C2#64123⋊C311yescharacteristic
491C2#64128⋊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
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#449
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×#88722C2semidirect
2. C2×#3755⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. #64104⋊C3C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #63647⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592509C3semidirect
25. C2×#3832C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. C2×#3745C2×C3semidirect
29. #64104C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
36. #63647C2×C3semidirect
37. #63636C2×C3semidirect
44. Q8×H(3,𝔽3)D4semidirect
54. C4⋊((C3⋊D3)⋊C3)D4semidirect
143. C32×Q8D12semidirect
144. C32×Q8C3×D4semidirect
160. C4⋊(C3⋊D3)C3×D4semidirect
273. #103(C3⋊D3)⋊C3semidirect
276. H(3,𝔽3)#449semidirect
424. C32#4810semidirect
425. C32C3×#449semidirect
430. 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×#887222
C2×#3755⋊C32
C2×#3745⋊C32
#64104⋊C32
H(3,𝔽3)×#1032
#63647⋊C32
#63636⋊C32
#105925093
C3×#48103