Finite Group Databasebeta

← #10592868⋊C3 #13143372

Subgroups — #10592868⋊C3 #13143372

Order 1728.

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

normal subgroups only

This group has 726 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×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64155⋊C3C211yesmaximal, characteristic, complemented
4C2×#3755⋊C3C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63647⋊C3C211yesmaximal, characteristic, complemented
8#10592868C311yesmaximal, characteristic, complemented
9C3×#5169—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
12C2×D4×H(3,𝔽3)C2211yescharacteristic
13C2×#638⋊C3—21no—
14D4×(C3⋊D3)⋊C3—22no—
15D4×(C3⋊D3)⋊C3—22no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18SD16×H(3,𝔽3)—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20#3755⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
23#3750⋊C3—21no—
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
25C2×D4×C3⋊D3C2×C311yescharacteristic
26C2×C3×D3×D4—31no—
27C2×C32×SD16—21no—
28C2×C32×SD16D311yescharacteristic, complemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64155C2×C311yescharacteristic, complemented
31C2×#3755C2×C311yescharacteristic, complemented
32C3×#946—31no—
33C2×C3×#215—31no—
34#63659C2×C311yescharacteristic, complemented
35#63648C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37C3×#832—31no—
38C3×#821—31no—
39C3×(C2×D3)⋊C8—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—42no—
44C22×(C3⋊D3)⋊C3—42no—
45C22×(C3⋊D3)⋊C3D411yescharacteristic
46Q8×H(3,𝔽3)—21no—
47Q8×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)—21no—
49D4×H(3,𝔽3)D412yes—
50C2×C4×H(3,𝔽3)—21no—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52#638⋊C3—24no—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3D411yescharacteristic
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)D412yes—
57C4×(C3⋊D3)⋊C3—21no—
58C4×(C3⋊D3)⋊C3—21no—
59C8×H(3,𝔽3)—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61C3×#450—91no—
62#5169—31no—
63#5169—91no—
64C23×C3⋊D3—21no—
65C23×C3×D3—61no—
66C2×C32×Q8—21no—
67C2×C32×Q8C2×D311yescharacteristic, complemented
68C2×C32×Q8C22×C311yescharacteristic, complemented
69C2×C32×D4—21no—
70C2×C32×D4C2×D311yescharacteristic
71C2×C32×D4C22×C311yescharacteristic
72C2×#638—21no—
73D4×C3⋊D3—22no—
74D4×C3⋊D3—22no—
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C2×C4×C3⋊D3C22×C311yescharacteristic
77C2×C3×#67—61no—
78C3×D3×D4—62no—
79C3×D3×D4—62no—
80C2×C3×D12—31no—
81C2×C3×C4×D3—31no—
82C32×SD16—24no—
83C32×SD16—22no—
84C32×SD16—22no—
85C2×C32×C8—21no—
86C2×C32×C8C2×D311yescharacteristic, complemented
87C2×C32×C8C22×C311yescharacteristic, complemented
88#3755—22no—
89#3752—21no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
91#3750—21no—
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
93C3×#215—62no—
94C3×(C2×D3)⋊C4—61no—
95C3×C4⋊Dic3—31no—
96C3×Dic3⋊C4—61no—
97C2×C3×C3⋊C8—31no—
98C22×H(3,𝔽3)—22no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)—41no—
101C22×H(3,𝔽3)C2×D411yescharacteristic
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—48no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3SD1612yescomplemented
109C4×H(3,𝔽3)—41no—
110C4×H(3,𝔽3)C2×D411yescharacteristic
111C4×H(3,𝔽3)C2×D411yescharacteristic
112(C3⋊Dic3)⋊C3—21no—
113(C3⋊Dic3)⋊C3—41no—
114C22×C3×D4—91no—
115C2×D3×D4—31no—
116C2×D3×D4—91no—
117C2×C3×SD16—31no—
118C2×C3×SD16—31no—
119C2×C3×SD16—61no—
120C2×C3×SD16—91no—
121C2×C3×SD16C3×D311yescharacteristic
122C3×C22⋊Q8—91no—
123#946—31no—
124#946—91no—
125C2×#215—31no—
126C2×#215—91no—
127C3×#102—91no—
128C3×#102—91no—
129C3×C22⋊C8—91no—
130#832—31no—
131#832—91no—
132#821—31no—
133#821—91no—
134(C2×D3)⋊C8—31no—
135(C2×D3)⋊C8—91no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—41no—
139C22×C3⋊D3—24no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3—42no—
142C22×C3⋊D3—42no—
143C22×C3⋊D3C3×D411yescharacteristic
144C22×C3×D3—31no—
145C22×C3×D3—64no—
146C22×C3×D3—61no—
147C22×C3×D3—122no—
148C22×C3×D3—122no—
149C32×Q8—22no—
150C32×Q8—21no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—21no—
154C32×Q8—41no—
155C32×D4—21no—
156C32×D4—21no—
157C32×D4—22no—
158C32×D4—41no—
159C32×D4#6712yes—
160C32×D4C3×D412yes—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C23×C311yescharacteristic
167#638—24no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3C3×D411yescharacteristic
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)C3×D412yes—
172C4×C3⋊D3—21no—
173C4×C3⋊D3—21no—
174C3×#67—64no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—61no—
177C3×D12—32no—
178C3×D12—61no—
179C3×C4×D3—61no—
180C3×C4×D3—61no—
181C32×C8—22no—
182C32×C8—21no—
183C32×C8—21no—
184C3⋊(C3⋊C8)—21no—
185C3×C3⋊C8—61no—
186#450—91noSylow
187C2×H(3,𝔽3)—42no—
188C2×H(3,𝔽3)#12011yescharacteristic
189C2×H(3,𝔽3)C2×SD1611yescharacteristic
190C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
191(C3⋊D3)⋊C3—22no—
192(C3⋊D3)⋊C3—42no—
193C24×C3—181no—
194C23×D3—61no—
195C23×D3—181no—
196C2×C3×Q8—31no—
197C2×C3×Q8—31no—
198C2×C3×Q8—61no—
199C2×C3×Q8C2×C3×D311yescharacteristic
200C2×C3×D4—31no—
201C2×C3×D4—31no—
202C2×C3×D4—61no—
203C2×C3×D4—91no—
204C2×C3×D4—181no—
205C2×C3×D4—182no—
206C2×C3×D4—182no—
207C2×C3×D4C2×C3×D311yescharacteristic
208C22×C3×C4—91no—
209C2×#67—61no—
210C2×#67—181no—
211D3×D4—62no—
212D3×D4—62no—
213D3×D4—182no—
214D3×D4—182no—
215C2×D12—31no—
216C2×D12—91no—
217C2×C4×D3—31no—
218C2×C4×D3—91no—
219C3×SD16—22no—
220C3×SD16—62no—
221C3×SD16—62no—
222C3×SD16—64no—
223C3×SD16—182no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—91no—
228C2×C3×C8C2×C3×D311yescharacteristic
229C3×C4⋊C4—91no—
230C3×C4⋊C4—181no—
231C3×C22⋊C4—181no—
232#215—62no—
233#215—182no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—181no—
236C4⋊Dic3—31no—
237C4⋊Dic3—91no—
238Dic3⋊C4—61no—
239Dic3⋊C4—181no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—91no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—42no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—81no—
251C22×C32—81no—
252C22×C32D3×D411yescharacteristic
253C22×C32C2×C3×D411yescharacteristic
254C2×C3⋊D3—21no—
255C2×C3⋊D3—21no—
256C2×C3⋊D3—22no—
257C2×C3⋊D3—48no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—41no—
260C2×C3⋊D3C3×SD1612yescomplemented
261C2×C3×D3—32no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—62no—
265C2×C3×D3—128no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—42no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4D3×D411yescharacteristic
274C32×C4C2×#6711yescharacteristic
275C32×C4C2×C3×D411yescharacteristic
276C32×C4C2×C3×D411yescharacteristic
277C3⋊Dic3—21no—
278C3⋊Dic3—41no—
279C3×Dic3—61no—
280C3×Dic3—121no—
281C22×D4—91no—
282C2×SD16—91no—
283C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
284C22⋊Q8—91no—
285#102—91no—
286#102—91no—
287C22⋊C8—91no—
288H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—61no—
291C23×C3—61no—
292C23×C3—91no—
293C23×C3—121no—
294C23×C3—181no—
295C23×C3—184no—
296C23×C3—362no—
297C23×C3—362no—
298C22×D3—31no—
299C22×D3—64no—
300C22×D3—61no—
301C22×D3—91no—
302C22×D3—122no—
303C22×D3—122no—
304C22×D3—181no—
305C22×D3—184no—
306C22×D3—362no—
307C22×D3—362no—
308C3×Q8—21no—
309C3×Q8—21no—
310C3×Q8—61no—
311C3×Q8—61no—
312C3×Q8—61no—
313C3×Q8—61no—
314C3×Q8—62no—
315C3×Q8—121no—
316C3×D4—21no—
317C3×D4—32no—
318C3×D4—32no—
319C3×D4—61no—
320C3×D4—61no—
321C3×D4—62no—
322C3×D4—92no—
323C3×D4—121no—
324C3×D4—181no—
325C3×D4—184no—
326C3×D4C3×#6712yes—
327C2×C3×C4—21no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—91no—
334C2×C3×C4—121no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4C22×C3×D311yescharacteristic
339#67—64no—
340#67—184no—
341C2×Dic3—31no—
342C2×Dic3—61no—
343C2×Dic3—91no—
344C2×Dic3—181no—
345D12—32no—
346D12—61no—
347D12—92no—
348D12—181no—
349C4×D3—61no—
350C4×D3—61no—
351C4×D3—181no—
352C4×D3—181no—
353C3×C8—21no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—62no—
357C3×C8—181no—
358C3⋊C8—61no—
359C3⋊C8—181no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—42no—
364C2×C32—42no—
365C2×C32—82no—
366C2×C32D3×SD1611yescharacteristic
367C2×C32#91411yescharacteristic
368C2×C32#93711yescharacteristic
369C2×C32C3×#12011yescharacteristic
370C2×C32C2×C3×SD1611yescharacteristic
371C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
372C3⋊D3—22no—
373C3⋊D3—42no—
374C3×D3—62no—
375C3×D3—122no—
376C24—181no—
377C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
378C2×D4—91no—
379C2×D4—181no—
380C2×D4—182no—
381C2×D4—182no—
382C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
383C22×C4—91no—
384SD16—22no—
385SD16—182no—
386C2×C8—91no—
387C2×C8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
388C4⋊C4—91no—
389C4⋊C4—181no—
390C22⋊C4—181no—
391C22×C3—22no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—62no—
397C22×C3—62no—
398C22×C3—61no—
399C22×C3—92no—
400C22×C3—122no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3—181no—
406C22×C3—181no—
407C22×C3—182no—
408C22×C3—241no—
409C22×C3—241no—
410C22×C3—361no—
411C22×C3—361no—
412C22×C3—368no—
413C22×C3C3×D3×D411yescharacteristic
414C2×D3—32no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—62no—
418C2×D3—92no—
419C2×D3—128no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—181no—
423C2×D3—181no—
424C2×D3—182no—
425C2×D3—368no—
426C2×D3—361no—
427C2×D3—361no—
428C3×C4—31no—
429C3×C4—31no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—41no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—122no—
438C3×C4—181no—
439C3×C4—361no—
440C3×C4C3×D3×D411yescharacteristic
441C3×C4C2×C3×#6711yescharacteristic
442Dic3—61no—
443Dic3—121no—
444Dic3—181no—
445Dic3—361no—
446C32—21no—
447C32#516911yescharacteristic, complemented
448C32C3×#45011yescharacteristic, complemented
449C23—21no—
450C23—91no—
451C23—184no—
452C23—181no—
453C23—362no—
454C23—362no—
455Q8—21no—
456Q8—21no—
457D4—21no—
458D4—92no—
459D4—184no—
460D4—181no—
461D4#638⋊C312yes—
462C2×C4—21no—
463C2×C4—91no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
468C8—21no—
469C8—181no—
470C2×C3—31no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—42no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—122no—
481C2×C3—122no—
482C2×C3—182no—
483C2×C3—242no—
484C2×C3—362no—
485C2×C3C3×D3×SD1611yescharacteristic
486C2×C3C3×#91411yescharacteristic
487C2×C3C3×#93711yescharacteristic
488D3—62no—
489D3—122no—
490D3—182no—
491D3—362no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—92no—
496C22—181no—
497C22—181no—
498C22—182no—
499C22—368no—
500C22—361no—
501C22—361no—
502C22D4×(C3⋊D3)⋊C311yescharacteristic
503C4—41no—
504C4—181no—
505C4—361no—
506C4D4×(C3⋊D3)⋊C311yescharacteristic
507C4C2×#638⋊C311yescharacteristic
508C3—31no—
509C3—31no—
510C3—61no—
511C3C3×#516911yescharacteristic
512C2—42no—
513C2—182no—
514C2—362no—
515C2SD16×(C3⋊D3)⋊C311yescharacteristic
516C2#64123⋊C311yescharacteristic
517C2#64146⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×(C3⋊D3)⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#450
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
2. C2×SD16×H(3,𝔽3)C2semidirect
3. #64155⋊C3C2semidirect
4. C2×#3755⋊C3C2semidirect
5. #63659⋊C3C2semidirect
6. #63648⋊C3C2semidirect
7. #63647⋊C3C2semidirect
8. #10592868C3semidirect
11. C2×Q8×H(3,𝔽3)C22semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
22. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
24. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
30. #64155C2×C3semidirect
31. C2×#3755C2×C3semidirect
34. #63659C2×C3semidirect
35. #63648C2×C3semidirect
36. #63647C2×C3semidirect
67. C2×C32×Q8C2×D3semidirect
68. C2×C32×Q8C22×C3semidirect
86. C2×C32×C8C2×D3semidirect
87. C2×C32×C8C22×C3semidirect
90. C4⋊(C3⋊Dic3)C22×C3semidirect
92. C2×C3⋊(C3⋊C8)C22×C3semidirect
108. C2×(C3⋊D3)⋊C3SD16semidirect
260. C2×C3⋊D3C3×SD16semidirect
283. C2×SD16(C3⋊D3)⋊C3semidirect
288. H(3,𝔽3)#450semidirect
377. C2×Q8C2×(C3⋊D3)⋊C3semidirect
387. C2×C8C2×(C3⋊D3)⋊C3semidirect
447. C32#5169semidirect
448. C32C3×#450semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
C2×SD16×H(3,𝔽3)2
#64155⋊C32
C2×#3755⋊C32
#63659⋊C32
#63648⋊C32
#63647⋊C32
#105928683
C3×#51693