Finite Group Databasebeta

← #10592472⋊C3 #13141790

Subgroups — #10592472⋊C3 #13141790

Order 1728.

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

normal subgroups only

This group has 743 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, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592472C311yesmaximal, characteristic, complemented
9C3×#4773—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)—21no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
21#3753⋊C3—22no—
22#3752⋊C3—21no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
24#3745⋊C3—22no—
25C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D3×D4—31no—
27C2×#3753C2×C311yescharacteristic, complemented
28C2×#3745C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic, complemented
30C2×C3×#213—31no—
31C2×C3×D24—31no—
32C3×#895—31no—
33C32×#102—21no—
34C32×#102D311yescharacteristic, complemented
35C32×#102C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63636C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#809—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—21no—
44C22×(C3⋊D3)⋊C3—42no—
45C22×(C3⋊D3)⋊C3—42no—
46C22×(C3⋊D3)⋊C3D411yescharacteristic
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)D412yescomplemented
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51#638⋊C3—24no—
52C2×(C3⋊Dic3)⋊C3D411yescharacteristic
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)—42no—
57C4⋊((C3⋊D3)⋊C3)D412yescomplemented
58C4×(C3⋊D3)⋊C3—21no—
59C4×(C3⋊D3)⋊C3—21no—
60C8×H(3,𝔽3)—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62C3×#447—91no—
63#4773—31no—
64#4773—91no—
65C23×C3⋊D3—21no—
66C23×C3×D3—61no—
67C2×C32×D4—21no—
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C22×C311yescharacteristic
70C2×#638—21no—
71D4×C3⋊D3—22no—
72D4×C3⋊D3—22no—
73C2×C4⋊(C3⋊D3)—21no—
74C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
76C2×C4×C3⋊D3C22×C311yescharacteristic
77C2×C3×#67—61no—
78C3×D3×D4—62no—
79C3×D3×D4—62no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×C4×D3—31no—
84C2×C32×C8—21no—
85C2×C32×C8C2×D311yescharacteristic, complemented
86C2×C32×C8C22×C311yescharacteristic, complemented
87C32×C4⋊C4—21no—
88C32×C4⋊C4C2×D311yescharacteristic, complemented
89C32×C4⋊C4C22×C311yescharacteristic, complemented
90#3753—22no—
91#3752—21no—
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
93#3745—22no—
94C3×#213—62no—
95C3×(C2×D3)⋊C4—61no—
96C2×C3×C3⋊C8—31no—
97C3×D24—62no—
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)⋊C3—41no—
109C2×(C3⋊D3)⋊C3—41no—
110C2×(C3⋊D3)⋊C3—41no—
111C2×(C3⋊D3)⋊C3D812yescomplemented
112C4×H(3,𝔽3)—41no—
113C4×H(3,𝔽3)C2×D411yescharacteristic
114C4×H(3,𝔽3)C2×D411yescharacteristic
115(C3⋊Dic3)⋊C3—21no—
116C22×C3×D4—91no—
117C2×D3×D4—31no—
118C2×D3×D4—91no—
119C2×C3×D8—91no—
120C2×C3×D8—91no—
121C3×#121—91no—
122C2×#213—31no—
123C2×#213—91no—
124C2×D24—31no—
125C2×D24—91no—
126#895—31no—
127#895—91no—
128C3×#102—31no—
129C3×#102—31no—
130C3×#102—61no—
131C3×#102—91no—
132C3×#102C3×D311yescharacteristic
133C3×C22⋊C8—91no—
134(C2×D3)⋊C8—31no—
135(C2×D3)⋊C8—91no—
136#809—31no—
137#809—91no—
138C23×C32—21no—
139C23×C32—21no—
140C23×C32—41no—
141C22×C3⋊D3—24no—
142C22×C3⋊D3—21no—
143C22×C3⋊D3—21no—
144C22×C3⋊D3—42no—
145C22×C3⋊D3—42no—
146C22×C3⋊D3C3×D411yescharacteristic
147C22×C3×D3—31no—
148C22×C3×D3—64no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—122no—
152C22×C3×D3—122no—
153C32×D4—21no—
154C32×D4—21no—
155C32×D4—22no—
156C32×D4—41no—
157C32×D4D1212yescomplemented
158C32×D4C3×D412yescomplemented
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C23×C311yescharacteristic
165#638—24no—
166C2×C3⋊Dic3C3×D411yescharacteristic
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)—42no—
171C4⋊(C3⋊D3)C3×D412yescomplemented
172C4×C3⋊D3—21no—
173C4×C3⋊D3—21no—
174C3×#67—64no—
175C2×C3×Dic3—31no—
176C3×D12—32no—
177C3×D12—61no—
178C3×D12—61no—
179C3×D12—61no—
180C3×D12—122no—
181C3×C4×D3—61no—
182C3×C4×D3—61no—
183C32×C8—21no—
184C32×C8—21no—
185C32×C8—41no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—61no—
188#447—91noSylow
189C2×H(3,𝔽3)—42no—
190C2×H(3,𝔽3)#12011yescharacteristic
191C2×H(3,𝔽3)C2×D811yescharacteristic
192C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
193(C3⋊D3)⋊C3—22no—
194(C3⋊D3)⋊C3—42no—
195(C3⋊D3)⋊C3—81no—
196C24×C3—181no—
197C23×D3—61no—
198C23×D3—181no—
199C2×C3×D4—31no—
200C2×C3×D4—31no—
201C2×C3×D4—61no—
202C2×C3×D4—91no—
203C2×C3×D4—91no—
204C2×C3×D4—181no—
205C2×C3×D4—181no—
206C2×C3×D4—182no—
207C2×C3×D4—182no—
208C2×C3×D4C2×C3×D311yescharacteristic
209C22×C3×C4—91no—
210C2×#67—61no—
211C2×#67—181no—
212D3×D4—62no—
213D3×D4—62no—
214D3×D4—182no—
215D3×D4—182no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—61no—
219C2×D12—91no—
220C2×D12—91no—
221C2×D12—181no—
222C2×C4×D3—31no—
223C2×C4×D3—91no—
224C3×D8—182no—
225C3×D8—182no—
226C2×C3×C8—31no—
227C2×C3×C8—31no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C2×C3×C8C2×C3×D311yescharacteristic
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4C2×C3×D311yescharacteristic
235C3×C22⋊C4—181no—
236#213—62no—
237#213—182no—
238(C2×D3)⋊C4—61no—
239(C2×D3)⋊C4—181no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—91no—
242D24—62no—
243D24—182no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—42no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—41no—
251C22×C32—41no—
252C22×C32—81no—
253C22×C32—81no—
254C22×C32D3×D411yescharacteristic
255C22×C32C2×C3×D411yescharacteristic
256C2×C3⋊D3—21no—
257C2×C3⋊D3—21no—
258C2×C3⋊D3—22no—
259C2×C3⋊D3—48no—
260C2×C3⋊D3—41no—
261C2×C3⋊D3—41no—
262C2×C3⋊D3—41no—
263C2×C3⋊D3—41no—
264C2×C3⋊D3—41no—
265C2×C3⋊D3C3×D812yescomplemented
266C2×C3×D3—32no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—62no—
270C2×C3×D3—128no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—81no—
281C32×C4C2×D1211yescharacteristic
282C32×C4D3×D411yescharacteristic
283C32×C4C2×C3×D411yescharacteristic
284C32×C4C2×C3×D411yescharacteristic
285C3⋊Dic3—21no—
286C3×Dic3—61no—
287C22×D4—91no—
288C2×D8—91no—
289C2×D8—91no—
290#121—91no—
291#102—91no—
292#102(C3⋊D3)⋊C311yescharacteristic, complemented
293C22⋊C8—91no—
294H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
295C23×C3—21no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—91no—
299C23×C3—121no—
300C23×C3—181no—
301C23×C3—181no—
302C23×C3—184no—
303C23×C3—362no—
304C23×C3—362no—
305C22×D3—31no—
306C22×D3—64no—
307C22×D3—61no—
308C22×D3—61no—
309C22×D3—91no—
310C22×D3—122no—
311C22×D3—122no—
312C22×D3—181no—
313C22×D3—184no—
314C22×D3—181no—
315C22×D3—362no—
316C22×D3—362no—
317C3×D4—21no—
318C3×D4—32no—
319C3×D4—32no—
320C3×D4—61no—
321C3×D4—61no—
322C3×D4—62no—
323C3×D4—92no—
324C3×D4—121no—
325C3×D4—181no—
326C3×D4—181no—
327C3×D4—181no—
328C3×D4—184no—
329C3×D4—362no—
330C3×D4C3×D1212yes—
331C2×C3×C4—21no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—91no—
338C2×C3×C4—121no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4C22×C3×D311yescharacteristic
342#67—64no—
343#67—184no—
344C2×Dic3—31no—
345C2×Dic3—91no—
346D12—32no—
347D12—61no—
348D12—61no—
349D12—61no—
350D12—92no—
351D12—122no—
352D12—181no—
353D12—181no—
354D12—181no—
355D12—362no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—181no—
359C4×D3—181no—
360C3×C8—21no—
361C3×C8—61no—
362C3×C8—61no—
363C3×C8—121no—
364C3×C8—181no—
365C3⋊C8—61no—
366C3⋊C8—181no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—42no—
371C2×C32—42no—
372C2×C32—82no—
373C2×C32#88211yescharacteristic
374C2×C32D3×D811yescharacteristic
375C2×C32#91411yescharacteristic
376C2×C32C3×#12011yescharacteristic
377C2×C32C2×C3×D811yescharacteristic
378C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
379C3⋊D3—22no—
380C3⋊D3—42no—
381C3⋊D3—81no—
382C3×D3—62no—
383C3×D3—122no—
384C3×D3—241no—
385C24—181no—
386C2×D4—91no—
387C2×D4—91no—
388C2×D4—181no—
389C2×D4—181no—
390C2×D4—182no—
391C2×D4—182no—
392C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
393C22×C4—91no—
394D8—182no—
395D8—182no—
396C2×C8—91no—
397C2×C8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
398C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
399C22⋊C4—181no—
400C22×C3—22no—
401C22×C3—31no—
402C22×C3—31no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—61no—
408C22×C3—92no—
409C22×C3—122no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—121no—
413C22×C3—121no—
414C22×C3—181no—
415C22×C3—181no—
416C22×C3—182no—
417C22×C3—241no—
418C22×C3—241no—
419C22×C3—361no—
420C22×C3—361no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3—361no—
424C22×C3—368no—
425C22×C3C3×D3×D411yescharacteristic
426C2×D3—32no—
427C2×D3—61no—
428C2×D3—61no—
429C2×D3—62no—
430C2×D3—92no—
431C2×D3—128no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—181no—
438C2×D3—181no—
439C2×D3—182no—
440C2×D3—368no—
441C2×D3—361no—
442C2×D3—361no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—361no—
446C3×C4—31no—
447C3×C4—31no—
448C3×C4—31no—
449C3×C4—31no—
450C3×C4—41no—
451C3×C4—61no—
452C3×C4—61no—
453C3×C4—121no—
454C3×C4—121no—
455C3×C4—181no—
456C3×C4—241no—
457C3×C4C2×C3×D1211yescharacteristic
458C3×C4C3×D3×D411yescharacteristic
459Dic3—61no—
460Dic3—181no—
461C32—21no—
462C32#477311yescharacteristic, complemented
463C32C3×#44711yescharacteristic, complemented
464C23—21no—
465C23—91no—
466C23—184no—
467C23—181no—
468C23—181no—
469C23—362no—
470C23—362no—
471D4—21no—
472D4—92no—
473D4—184no—
474D4—181no—
475D4—181no—
476D4—181no—
477D4—362no—
478D4C4⋊((C3⋊D3)⋊C3)12yescomplemented
479C2×C4—21no—
480C2×C4—91no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
484C8—21no—
485C8—181no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—42no—
493C2×C3—61no—
494C2×C3—61no—
495C2×C3—61no—
496C2×C3—122no—
497C2×C3—122no—
498C2×C3—182no—
499C2×C3—242no—
500C2×C3—362no—
501C2×C3—721no—
502C2×C3C3×#88211yescharacteristic
503C2×C3C3×D3×D811yescharacteristic
504C2×C3C3×#91411yescharacteristic
505D3—62no—
506D3—122no—
507D3—182no—
508D3—241no—
509D3—362no—
510D3—721no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—92no—
515C22—181no—
516C22—181no—
517C22—182no—
518C22—368no—
519C22—361no—
520C22—361no—
521C22—361no—
522C22—361no—
523C22—361no—
524C22D4×(C3⋊D3)⋊C311yescharacteristic
525C4—41no—
526C4—181no—
527C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
528C4D4×(C3⋊D3)⋊C311yescharacteristic
529C3—31no—
530C3—31no—
531C3—61no—
532C3C3×#477311yescharacteristic
533C2—42no—
534C2—182no—
535C2—362no—
536C2—721no—
537C2#64091⋊C311yescharacteristic
538C2D8×(C3⋊D3)⋊C311yescharacteristic
539C2#64123⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#447
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. C2×D4×(C3⋊D3)⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. #64104⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63647⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592472C3semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
20. C4⋊C4×H(3,𝔽3)C22semidirect
23. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
25. C2×D4×C3⋊D3C2×C3semidirect
27. C2×#3753C2×C3semidirect
28. C2×#3745C2×C3semidirect
29. #64104C2×C3semidirect
34. C32×#102D3semidirect
35. C32×#102C2×C3semidirect
36. #63647C2×C3semidirect
37. #63636C2×C3semidirect
48. D4×H(3,𝔽3)D4semidirect
57. C4⋊((C3⋊D3)⋊C3)D4semidirect
75. C2×C4⋊(C3⋊D3)C22×C3semidirect
85. C2×C32×C8C2×D3semidirect
86. C2×C32×C8C22×C3semidirect
88. C32×C4⋊C4C2×D3semidirect
89. C32×C4⋊C4C22×C3semidirect
92. C2×C3⋊(C3⋊C8)C22×C3semidirect
111. C2×(C3⋊D3)⋊C3D8semidirect
157. C32×D4D12semidirect
158. C32×D4C3×D4semidirect
171. C4⋊(C3⋊D3)C3×D4semidirect
265. C2×C3⋊D3C3×D8semidirect
292. #102(C3⋊D3)⋊C3semidirect
294. H(3,𝔽3)#447semidirect
397. C2×C8C2×(C3⋊D3)⋊C3semidirect
398. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
462. C32#4773semidirect
463. C32C3×#447semidirect
478. D4C4⋊((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×D4×(C3⋊D3)⋊C32
C2×#3753⋊C32
C2×#3745⋊C32
#64104⋊C32
H(3,𝔽3)×#1022
#63647⋊C32
#63636⋊C32
#105924723
C3×#47733