Finite Group Databasebeta

← #10592686⋊C3 #13142625

Subgroups — #10592686⋊C3 #13142625

Order 1728.

911 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C212yesmaximal
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592686C311yesmaximal, characteristic, complemented
7C3×#4987—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—22no—
14C2×C4×(C3⋊D3)⋊C3—22no—
15C2×C4×(C3⋊D3)⋊C3C414yescomplemented
16C4⋊C4×H(3,𝔽3)—22no—
17#3752⋊C3—24no—
18C4⋊((C3⋊Dic3)⋊C3)—22no—
19C22×C4×C3⋊D3C2×C311yescharacteristic
20C22×C3×C4×D3—31no—
21C2×C32×C4⋊C4—21no—
22C2×C32×C4⋊C4D311yescharacteristic, complemented
23C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
24C2×#3752C2×C312yes—
25C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
26C2×C3×(C2×D3)⋊C4—32no—
27C2×C3×C4⋊Dic3—31no—
28#63658C2×C312yescomplemented
29C3×C3⋊#95—32no—
30C23×H(3,𝔽3)C2311yescharacteristic
31C22×(C3⋊D3)⋊C3—21no—
32C22×(C3⋊D3)⋊C3—21no—
33C22×(C3⋊D3)⋊C3—21no—
34C22×(C3⋊D3)⋊C3—21no—
35C22×(C3⋊D3)⋊C3C2×C412yes—
36C22×(C3⋊D3)⋊C3D412yes—
37C22×(C3⋊D3)⋊C3Q812yes—
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)—22no—
41C2×C4×H(3,𝔽3)C2×C412yes—
42C2×C4×H(3,𝔽3)D412yes—
43C2×C4×H(3,𝔽3)D412yes—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—22no—
47C2×(C3⋊Dic3)⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2×C412yes—
50C4×(C3⋊D3)⋊C3—28no—
51C3×#380—91no—
52#4987—31no—
53#4987—91no—
54C23×C3⋊D3C22×C311yescharacteristic
55C23×C3×D3—31no—
56C22×C32×C4—21no—
57C22×C32×C4—22no—
58C22×C32×C4C2×D312yes—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C22×C312yes—
61C22×C32×C4C22×C311yescharacteristic
62C22×C3⋊Dic3C22×C312yes—
63C22×C3⋊Dic3C22×C311yescharacteristic
64C2×C4×C3⋊D3—22no—
65C2×C4×C3⋊D3—22no—
66C2×C4×C3⋊D3C3×C414yescomplemented
67C22×C3×Dic3—31no—
68C22×C3×Dic3—32no—
69C2×C3×C4×D3—34no—
70C2×C3×C4×D3—62no—
71C2×C3×C4×D3—62no—
72C32×C4⋊C4—24no—
73C32×C4⋊C4—22no—
74C32×C4⋊C4—22no—
75#3752—24no—
76C4⋊(C3⋊Dic3)—22no—
77C3×(C2×D3)⋊C4—64no—
78C3×C4⋊Dic3—62no—
79C22×H(3,𝔽3)C22×C411yescharacteristic
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C22×H(3,𝔽3)C2×Q811yescharacteristic
84C22×H(3,𝔽3)#4111yescharacteristic
85C22×H(3,𝔽3)#4111yescharacteristic
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3—22no—
89C2×(C3⋊D3)⋊C3—22no—
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3C4⋊C414yescomplemented
93C4×H(3,𝔽3)—42no—
94C4×H(3,𝔽3)C22⋊C414yescomplemented
95(C3⋊Dic3)⋊C3—22no—
96(C3⋊Dic3)⋊C3—42no—
97C23×C3×C4—91no—
98C22×C4×D3—31no—
99C22×C4×D3—91no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—61no—
103C2×C3×C4⋊C4—91no—
104C2×C3×C4⋊C4C3×D311yescharacteristic
105C2×C3×C22⋊C4—92no—
106C2×(C2×D3)⋊C4—32no—
107C2×(C2×D3)⋊C4—92no—
108C2×C4⋊Dic3—31no—
109C2×C4⋊Dic3—91no—
110C3×#95—92no—
111C3⋊#95—32no—
112C3⋊#95—92no—
113C23×C32—21no—
114C23×C32C22×D311yescharacteristic
115C23×C32C23×C311yescharacteristic
116C22×C3⋊D3—21no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3C2×C3×C412yes—
121C22×C3⋊D3C3×D412yes—
122C22×C3⋊D3C3×Q812yes—
123C22×C3×D3—32no—
124C22×C3×D3—32no—
125C22×C3×D3—32no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C2×C32×C4—24no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—42no—
141C2×C32×C4—42no—
142C2×C32×C4C4×D312yes—
143C2×C32×C4D1212yes—
144C2×C32×C4#6712yes—
145C2×C32×C4C2×C3×C412yes—
146C2×C32×C4C3×D412yes—
147C2×C32×C4C3×D412yes—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—22no—
153C2×C3⋊Dic3C2×C3×C412yes—
154C4×C3⋊D3—28no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C3×C4×D3—68no—
162#380—91noSylow
163C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
164C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
165C2×H(3,𝔽3)#11711yescharacteristic
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)#12111yescharacteristic
168C2×H(3,𝔽3)C22⋊Q811yescharacteristic
169C2×H(3,𝔽3)C22⋊Q811yescharacteristic
170(C3⋊D3)⋊C3—24no—
171C24×C3—91no—
172C23×D3—31no—
173C23×D3—91no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—32no—
177C22×C3×C4—32no—
178C22×C3×C4—61no—
179C22×C3×C4—62no—
180C22×C3×C4—94no—
181C22×C3×C4—91no—
182C22×C3×C4—92no—
183C22×C3×C4—182no—
184C22×C3×C4—182no—
185C22×C3×C4C2×C3×D312yes—
186C22×C3×C4C2×C3×D311yescharacteristic
187C22×Dic3—31no—
188C22×Dic3—32no—
189C22×Dic3—91no—
190C22×Dic3—92no—
191C2×C4×D3—34no—
192C2×C4×D3—62no—
193C2×C4×D3—62no—
194C2×C4×D3—94no—
195C2×C4×D3—182no—
196C2×C4×D3—182no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—62no—
199C3×C4⋊C4—62no—
200C3×C4⋊C4—64no—
201C3×C4⋊C4—182no—
202C3×C22⋊C4—184no—
203(C2×D3)⋊C4—64no—
204(C2×D3)⋊C4—184no—
205C4⋊Dic3—62no—
206C4⋊Dic3—182no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32C2×C4×D311yescharacteristic
215C22×C32C2×D1211yescharacteristic
216C22×C32D3×D411yescharacteristic
217C22×C32#23711yescharacteristic
218C22×C32D3×Q811yescharacteristic
219C22×C32#23911yescharacteristic
220C22×C32C2×#6711yescharacteristic
221C22×C32C22×C3×C411yescharacteristic
222C22×C32C2×C3×D411yescharacteristic
223C22×C32C2×C3×D411yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C22×C32C2×C3×Q811yescharacteristic
226C22×C32C3×#4111yescharacteristic
227C22×C32C3×#4111yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—22no—
230C2×C3⋊D3—22no—
231C2×C3⋊D3—22no—
232C2×C3⋊D3—22no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3C3×C4⋊C414yescomplemented
235C2×C3×D3—34no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C32×C4—24no—
243C32×C4—44no—
244C32×C4—42no—
245C32×C4—42no—
246C32×C4(C2×D3)⋊C414yescomplemented
247C32×C4C3×C22⋊C414yescomplemented
248C3⋊Dic3—22no—
249C3⋊Dic3—42no—
250C3×Dic3—62no—
251C3×Dic3—122no—
252C23×C4—91no—
253C2×C4⋊C4—91no—
254C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
255C2×C22⋊C4—92no—
256#95—92no—
257H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
258C23×C3—31no—
259C23×C3—31no—
260C23×C3—61no—
261C23×C3—92no—
262C23×C3—92no—
263C23×C3—92no—
264C23×C3—181no—
265C23×C3—181no—
266C23×C3—181no—
267C23×C3—181no—
268C23×C3C22×C3×D311yescharacteristic
269C22×D3—32no—
270C22×D3—32no—
271C22×D3—32no—
272C22×D3—61no—
273C22×D3—61no—
274C22×D3—61no—
275C22×D3—61no—
276C22×D3—92no—
277C22×D3—92no—
278C22×D3—92no—
279C22×D3—181no—
280C22×D3—181no—
281C22×D3—181no—
282C22×D3—181no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—32no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
289C2×C3×C4—32no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—64no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—92no—
303C2×C3×C4—122no—
304C2×C3×C4—122no—
305C2×C3×C4—188no—
306C2×C3×C4—181no—
307C2×C3×C4—181no—
308C2×C3×C4—182no—
309C2×C3×C4—182no—
310C2×C3×C4—182no—
311C2×C3×C4C3×C4×D312yes—
312C2×C3×C4C3×D1212yes—
313C2×C3×C4C3×#6712yes—
314C2×Dic3—32no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—92no—
321C2×Dic3—181no—
322C2×Dic3—181no—
323C2×Dic3—182no—
324C2×Dic3—182no—
325C2×Dic3—182no—
326C4×D3—68no—
327C4×D3—188no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32D3×C4⋊C411yescharacteristic
336C2×C32#89211yescharacteristic
337C2×C32#89511yescharacteristic
338C2×C32#89711yescharacteristic
339C2×C32C2×(C2×D3)⋊C411yescharacteristic
340C2×C32#93811yescharacteristic
341C2×C32#94611yescharacteristic
342C2×C32C2×C3×C22⋊C411yescharacteristic
343C2×C32C2×C3×C4⋊C411yescharacteristic
344C2×C32C3×#11711yescharacteristic
345C2×C32C3×#12111yescharacteristic
346C2×C32C3×#12111yescharacteristic
347C2×C32C3×C22⋊Q811yescharacteristic
348C2×C32C3×C22⋊Q811yescharacteristic
349C3⋊D3—24no—
350C3×D3—64no—
351C24—91no—
352C22×C4—91no—
353C22×C4—94no—
354C22×C4—92no—
355C22×C4—182no—
356C22×C4—182no—
357C22×C4C2×(C3⋊D3)⋊C312yes—
358C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
359C4⋊C4—22no—
360C4⋊C4—182no—
361C22⋊C4—184no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—61no—
377C22×C3—61no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—94no—
384C22×C3—182no—
385C22×C3—182no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3—182no—
389C22×C3—182no—
390C22×C3C2×C3×C4×D311yescharacteristic
391C22×C3C2×C3×D1211yescharacteristic
392C22×C3C3×D3×D411yescharacteristic
393C22×C3C3×#23711yescharacteristic
394C22×C3C3×D3×Q811yescharacteristic
395C22×C3C3×#23911yescharacteristic
396C22×C3C2×C3×#6711yescharacteristic
397C2×D3—34no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—62no—
403C2×D3—62no—
404C2×D3—94no—
405C2×D3—182no—
406C2×D3—182no—
407C2×D3—182no—
408C2×D3—182no—
409C2×D3—182no—
410C2×D3—182no—
411C3×C4—34no—
412C3×C4—34no—
413C3×C4—42no—
414C3×C4—64no—
415C3×C4—122no—
416C3×C4—122no—
417C3×C4—124no—
418C3×C4—182no—
419C3×C4—362no—
420C3×C4C3×(C2×D3)⋊C414yes—
421Dic3—62no—
422Dic3—122no—
423Dic3—182no—
424Dic3—362no—
425C32—21no—
426C32#498711yescharacteristic, complemented
427C32C3×#38011yescharacteristic, complemented
428C23—92no—
429C23—92no—
430C23—92no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23—181no—
435C23C22×(C3⋊D3)⋊C311yescharacteristic
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—92no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—188no—
443C2×C4—182no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4C4×(C3⋊D3)⋊C312yes—
447C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
448C2×C4#638⋊C312yes—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—184no—
471C2×C3C3×D3×C4⋊C411yescharacteristic
472C2×C3C3×#89211yescharacteristic
473C2×C3C3×#89511yescharacteristic
474C2×C3C3×#89711yescharacteristic
475C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
476C2×C3C3×#93811yescharacteristic
477C2×C3C3×#94611yescharacteristic
478D3—64no—
479D3—184no—
480C22—94no—
481C22—182no—
482C22—182no—
483C22—182no—
484C22—182no—
485C22—182no—
486C22—182no—
487C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
488C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
489C22D4×(C3⋊D3)⋊C311yescharacteristic
490C22#3830⋊C311yescharacteristic
491C22Q8×(C3⋊D3)⋊C311yescharacteristic
492C22#8872211yescharacteristic
493C22C2×#638⋊C311yescharacteristic
494C4—42no—
495C4—182no—
496C4—362no—
497C4#3752⋊C314yescomplemented
498C3—31no—
499C3—31no—
500C3—61no—
501C3C3×#498711yescharacteristic
502C2—184no—
503C2C4⋊C4×(C3⋊D3)⋊C311yescharacteristic
504C2#64101⋊C311yescharacteristic
505C2#64104⋊C311yescharacteristic
506C2#64106⋊C311yescharacteristic
507C2C2×#3752⋊C311yescharacteristic
508C2#64147⋊C311yescharacteristic
509C2#64155⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC23×C32C23×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#380
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #63658⋊C3C2semidirect
6. #10592686C3semidirect
15. C2×C4×(C3⋊D3)⋊C3C4semidirect
22. C2×C32×C4⋊C4D3semidirect
23. C2×C32×C4⋊C4C2×C3semidirect
25. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
28. #63658C2×C3semidirect
66. C2×C4×C3⋊D3C3×C4semidirect
92. C2×(C3⋊D3)⋊C3C4⋊C4semidirect
94. C4×H(3,𝔽3)C22⋊C4semidirect
234. C2×C3⋊D3C3×C4⋊C4semidirect
246. C32×C4(C2×D3)⋊C4semidirect
247. C32×C4C3×C22⋊C4semidirect
254. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
257. H(3,𝔽3)#380semidirect
426. C32#4987semidirect
427. C32C3×#380semidirect
497. C4#3752⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×C4⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63658⋊C32
#105926863
C3×#49873