Finite Group Databasebeta

← #10595982⋊C3 #13152442

Subgroups — #10595982⋊C3 #13152442

Order 1728.

890 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C211yesmaximal, characteristic
3C4×#638⋊C3C212yesmaximal
4#64133⋊C3C211yesmaximal, characteristic, complemented
5#64133⋊C3C212yesmaximal, complemented
6#64106⋊C3C211yesmaximal, characteristic, complemented
7#64105⋊C3C212yesmaximal, complemented
8C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
9#64098⋊C3C212yesmaximal, complemented
10#64097⋊C3C211yesmaximal, characteristic, complemented
11#64096⋊C3C211yesmaximal, characteristic, complemented
12#10595982C311yesmaximal, characteristic, complemented
13C3×#5511—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2212yes—
16C2×#638⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20C2×#634⋊C3C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C4⋊C4×H(3,𝔽3)C2212yes—
24#3757⋊C3C2211yescharacteristic
25#3757⋊C3C2212yes—
26#3752⋊C3C2211yescharacteristic
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34#3750⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C2×C32×C4⋊C4—21no—
38C2×C32×C4⋊C4D311yescharacteristic, complemented
39C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
40C4×#638C2×C311yescharacteristic
41C4×#638C2×C312yes—
42#64133C2×C312yescomplemented
43#64133C2×C311yescharacteristic, complemented
44#64106C2×C311yescharacteristic, complemented
45#64105C2×C312yescomplemented
46C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
47#64098C2×C312yescomplemented
48#64097C2×C311yescharacteristic, complemented
49#64096C2×C311yescharacteristic, complemented
50C3×C4×#67—31no—
51C3×C4×#67—32no—
52C3×#924—32no—
53C3×#924—31no—
54C3×#897—31no—
55C3×#896—32no—
56C3×D3×C4⋊C4—31no—
57C3×#889—32no—
58C3×#888—31no—
59C3×#887—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)—42no—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68#638⋊C3Q814yescomplemented
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—42no—
76#634⋊C3—22no—
77#634⋊C3—42no—
78C3×#554—91no—
79#5511—31no—
80#5511—91no—
81C22×C32×C4—21no—
82C22×C32×C4—22no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C2×D312yes—
85C22×C32×C4C22×C311yescharacteristic
86C22×C32×C4C22×C312yes—
87C2×#638C22×C311yescharacteristic
88C2×C4×C3⋊D3C22×C311yescharacteristic
89C2×C4×C3⋊D3C22×C312yes—
90C2×#634C22×C311yescharacteristic
91C2×#634C22×C312yes—
92C2×C3×#67—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—32no—
95C2×C3×Dic6—31no—
96C2×C3×Dic6—32no—
97C32×C4⋊C4—21no—
98C32×C4⋊C4—21no—
99C32×C4⋊C4—22no—
100C32×C4⋊C4C2×D311yescharacteristic
101C32×C4⋊C4C2×D311yescharacteristic
102C32×C4⋊C4C2×D312yes—
103C32×C4⋊C4C22×C311yescharacteristic
104C32×C4⋊C4C22×C311yescharacteristic
105C32×C4⋊C4C22×C312yes—
106#3757C22×C312yes—
107#3757C22×C311yescharacteristic
108#3752C22×C311yescharacteristic
109#3752C22×C312yes—
110C4⋊(C3⋊Dic3)C22×C311yescharacteristic
111C4⋊(C3⋊Dic3)C22×C312yes—
112#3750C22×C312yes—
113#3750C22×C312yes—
114#3750C22×C311yescharacteristic
115#3750C22×C312yes—
116#3750C22×C312yes—
117C4×C3⋊Dic3C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C312yes—
119C3×#217—32no—
120C3×#217—31no—
121C3×(C2×D3)⋊C4—31no—
122C3×(C2×D3)⋊C4—32no—
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—31no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×C4×Dic3—31no—
131C3×C4×Dic3—32no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)C2×Q812yes—
135C22×H(3,𝔽3)C2411yescharacteristic
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3C2×Q812yes—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—22no—
141C4×H(3,𝔽3)—22no—
142C4×H(3,𝔽3)#4112yes—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3C2×Q812yes—
148C3×C4⋊Q8—91no—
149C3×#125—92no—
150C3×C22⋊Q8—92no—
151C3×C22⋊Q8—91no—
152C3×C22⋊Q8—92no—
153C3×C22⋊Q8—91no—
154C3×C4×Q8—91no—
155C3×C4×D4—92no—
156C3×C4×D4—91no—
157C2×C3×C4⋊C4—31no—
158C2×C3×C4⋊C4—31no—
159C2×C3×C4⋊C4—61no—
160C2×C3×C4⋊C4—91no—
161C2×C3×C4⋊C4C3×D311yescharacteristic
162C4×#67—31no—
163C4×#67—32no—
164C4×#67—91no—
165C4×#67—92no—
166#924—32no—
167#924—31no—
168#924—92no—
169#924—91no—
170#897—31no—
171#897—91no—
172#896—32no—
173#896—92no—
174D3×C4⋊C4—31no—
175D3×C4⋊C4—91no—
176#889—32no—
177#889—92no—
178#888—31no—
179#888—91no—
180#887—31no—
181#887—91no—
182C23×C32—21no—
183C23×C32C22×D311yescharacteristic
184C23×C32C23×C311yescharacteristic
185C22×C3⋊D3C23×C311yescharacteristic
186C22×C3×D3—31no—
187C2×C32×C4—24no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—22no—
193C2×C32×C4—22no—
194C2×C32×C4—42no—
195C2×C32×C4—42no—
196C2×C32×C4—44no—
197C2×C32×C4C22×D312yes—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D312yes—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C23×C312yes—
202C2×C32×C4C23×C311yescharacteristic
203C2×C32×C4C23×C312yes—
204C2×C32×C4C23×C311yescharacteristic
205#638C3×Q814yescomplemented
206C2×C3⋊Dic3C23×C312yes—
207C2×C3⋊Dic3C23×C311yescharacteristic
208C2×C3⋊Dic3C23×C311yescharacteristic
209C2×C3⋊Dic3C23×C312yes—
210C2×C3⋊Dic3C23×C311yescharacteristic
211C4×C3⋊D3—22no—
212C4×C3⋊D3—42no—
213#634—22no—
214#634—42no—
215C3×#67—34no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—32no—
221C3×C4×D3—62no—
222C3×C4×D3—122no—
223C3×Dic6—62no—
224C3×Dic6—122no—
225#554—91noSylow
226C2×H(3,𝔽3)—22no—
227C2×H(3,𝔽3)C22×Q811yescharacteristic
228C2×H(3,𝔽3)C2×#4111yescharacteristic
229C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
230(C3⋊D3)⋊C3—22no—
231C2×C3×Q8—91no—
232C2×C3×Q8—92no—
233C2×C3×D4—91no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—32no—
237C22×C3×C4—32no—
238C22×C3×C4—61no—
239C22×C3×C4—62no—
240C22×C3×C4—91no—
241C22×C3×C4—92no—
242C22×C3×C4C2×C3×D311yescharacteristic
243C22×C3×C4C2×C3×D312yes—
244C2×#67—31no—
245C2×#67—91no—
246C2×C4×D3—31no—
247C2×C4×D3—32no—
248C2×C4×D3—91no—
249C2×C4×D3—92no—
250C2×Dic6—31no—
251C2×Dic6—32no—
252C2×Dic6—91no—
253C2×Dic6—92no—
254C3×C4⋊C4—31no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—32no—
257C3×C4⋊C4—31no—
258C3×C4⋊C4—32no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—61no—
261C3×C4⋊C4—62no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—91no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4—92no—
268C3×C4⋊C4—91no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4C2×C3×D311yescharacteristic
271C3×C4⋊C4C2×C3×D311yescharacteristic
272C3×C4⋊C4C2×C3×D312yes—
273C3×C22⋊C4—91no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—92no—
276C3×C22⋊C4—91no—
277C3×C42—92no—
278C3×C42—91no—
279#217—32no—
280#217—31no—
281#217—92no—
282#217—91no—
283(C2×D3)⋊C4—31no—
284(C2×D3)⋊C4—32no—
285(C2×D3)⋊C4—91no—
286(C2×D3)⋊C4—92no—
287C4⋊Dic3—31no—
288C4⋊Dic3—32no—
289C4⋊Dic3—91no—
290C4⋊Dic3—92no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—31no—
294Dic3⋊C4—32no—
295Dic3⋊C4—32no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—91no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301C4×Dic3—31no—
302C4×Dic3—32no—
303C4×Dic3—91no—
304C4×Dic3—92no—
305C22×C32—22no—
306C22×C32—22no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—22no—
313C22×C32D3×Q812yes—
314C22×C32C2×C3×Q812yes—
315C22×C32C23×D311yescharacteristic
316C22×C32C24×C311yescharacteristic
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3C2×C3×Q812yes—
320C2×C3×D3—32no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—21no—
328C32×C4—22no—
329C32×C4—22no—
330C32×C4—22no—
331C32×C4—42no—
332C32×C4—42no—
333C32×C4#23512yes—
334C32×C4C3×#4112yes—
335C3⋊Dic3—22no—
336C3⋊Dic3—21no—
337C3⋊Dic3—21no—
338C3⋊Dic3—22no—
339C3⋊Dic3C2×C3×Q812yes—
340C3×Dic3—32no—
341C3×Dic3—62no—
342C3×Dic3—61no—
343C3×Dic3—61no—
344C3×Dic3—62no—
345C4⋊Q8—91no—
346#125—92no—
347C22⋊Q8—92no—
348C22⋊Q8—91no—
349C22⋊Q8—92no—
350C22⋊Q8—91no—
351C4×Q8—91no—
352C4×D4—92no—
353C4×D4—91no—
354C2×C4⋊C4—91no—
355C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
356H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
357C23×C3—31no—
358C23×C3—31no—
359C23×C3—61no—
360C23×C3—91no—
361C23×C3C22×C3×D311yescharacteristic
362C22×D3—31no—
363C22×D3—91no—
364C3×Q8—182no—
365C3×Q8—362no—
366C3×D4—94no—
367C2×C3×C4—22no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—32no—
371C2×C3×C4—31no—
372C2×C3×C4—32no—
373C2×C3×C4—32no—
374C2×C3×C4—31no—
375C2×C3×C4—32no—
376C2×C3×C4—42no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—64no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—62no—
383C2×C3×C4—62no—
384C2×C3×C4—91no—
385C2×C3×C4—92no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—92no—
389C2×C3×C4—122no—
390C2×C3×C4—122no—
391C2×C3×C4—124no—
392C2×C3×C4—182no—
393C2×C3×C4—362no—
394C2×C3×C4C22×C3×D312yes—
395C2×C3×C4C22×C3×D311yescharacteristic
396C2×C3×C4C22×C3×D312yes—
397C2×C3×C4C22×C3×D311yescharacteristic
398#67—34no—
399#67—94no—
400C2×Dic3—31no—
401C2×Dic3—32no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—32no—
405C2×Dic3—91no—
406C2×Dic3—92no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—92no—
410C4×D3—62no—
411C4×D3—122no—
412C4×D3—182no—
413C4×D3—362no—
414Dic6—62no—
415Dic6—122no—
416Dic6—182no—
417Dic6—362no—
418C2×C32—24no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32C2×#23511yescharacteristic
425C2×C32#100411yescharacteristic
426C2×C32C2×D3×Q811yescharacteristic
427C2×C32C22×C3×Q811yescharacteristic
428C2×C32C2×C3×#4111yescharacteristic
429C2×C32C3×H(5,𝔽2)11yescharacteristic
430C3⋊D3—22no—
431C3×D3—62no—
432C2×Q8—91no—
433C2×Q8—92no—
434C2×D4—91no—
435C22×C4—91no—
436C22×C4—92no—
437C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
438C22×C4C2×(C3⋊D3)⋊C312yes—
439C4⋊C4—92no—
440C4⋊C4—91no—
441C4⋊C4—91no—
442C4⋊C4—92no—
443C4⋊C4—92no—
444C4⋊C4—92no—
445C4⋊C4—92no—
446C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
447C4⋊C4C2×(C3⋊D3)⋊C312yes—
448C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
449C22⋊C4—92no—
450C22⋊C4—91no—
451C22⋊C4—92no—
452C22⋊C4—91no—
453C42—92no—
454C42—91no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—32no—
460C22×C3—32no—
461C22×C3—61no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—61no—
468C22×C3—62no—
469C22×C3—92no—
470C22×C3—181no—
471C22×C3—181no—
472C22×C3C3×D3×Q812yes—
473C22×C3C23×C3×D311yescharacteristic
474C2×D3—32no—
475C2×D3—61no—
476C2×D3—61no—
477C2×D3—92no—
478C2×D3—181no—
479C2×D3—181no—
480C3×C4—21no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—62no—
492C3×C4—62no—
493C3×C4—92no—
494C3×C4—122no—
495C3×C4—122no—
496C3×C4—181no—
497C3×C4—182no—
498C3×C4—182no—
499C3×C4—181no—
500C3×C4C3×#23512yes—
501Dic3—32no—
502Dic3—62no—
503Dic3—61no—
504Dic3—61no—
505Dic3—62no—
506Dic3—92no—
507Dic3—182no—
508Dic3—181no—
509Dic3—182no—
510Dic3—181no—
511C32—21no—
512C32#551111yescharacteristic, complemented
513C32C3×#55411yescharacteristic, complemented
514C23—91no—
515C23C22×(C3⋊D3)⋊C311yescharacteristic
516Q8—182no—
517Q8—362no—
518D4—94no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—91no—
522C2×C4—92no—
523C2×C4—91no—
524C2×C4—91no—
525C2×C4—92no—
526C2×C4—182no—
527C2×C4—362no—
528C2×C4C22×(C3⋊D3)⋊C312yes—
529C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
530C2×C4C22×(C3⋊D3)⋊C312yes—
531C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
532C2×C3—22no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—62no—
540C2×C3—62no—
541C2×C3—64no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—182no—
546C2×C3C2×C3×#23511yescharacteristic
547C2×C3C3×#100411yescharacteristic
548C2×C3C2×C3×D3×Q811yescharacteristic
549D3—62no—
550D3—182no—
551C22—21no—
552C22—21no—
553C22—92no—
554C22—181no—
555C22—181no—
556C22Q8×(C3⋊D3)⋊C312yes—
557C22C23×(C3⋊D3)⋊C311yescharacteristic
558C4—21no—
559C4—22no—
560C4—22no—
561C4—92no—
562C4—181no—
563C4—182no—
564C4—182no—
565C4—181no—
566C4#3828⋊C312yes—
567C3—31no—
568C3—31no—
569C3—61no—
570C3C3×#551111yescharacteristic
571C2—22no—
572C2—182no—
573C2C2×#3828⋊C311yescharacteristic
574C2#64361⋊C311yescharacteristic
575C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC2×#638⋊C3C22
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. #64133⋊C3C2semidirect
5. #64133⋊C3C2semidirect
6. #64106⋊C3C2semidirect
7. #64105⋊C3C2semidirect
8. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
9. #64098⋊C3C2semidirect
10. #64097⋊C3C2semidirect
11. #64096⋊C3C2semidirect
12. #10595982C3semidirect
38. C2×C32×C4⋊C4D3semidirect
39. C2×C32×C4⋊C4C2×C3semidirect
42. #64133C2×C3semidirect
43. #64133C2×C3semidirect
44. #64106C2×C3semidirect
45. #64105C2×C3semidirect
46. C4⋊C4×C3⋊D3C2×C3semidirect
47. #64098C2×C3semidirect
48. #64097C2×C3semidirect
49. #64096C2×C3semidirect
68. #638⋊C3Q8semidirect
205. #638C3×Q8semidirect
355. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
356. H(3,𝔽3)#554semidirect
512. C32#5511semidirect
513. C32C3×#554semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C4×#638⋊C32
#64133⋊C32
#64106⋊C32
#64105⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64098⋊C32
#64097⋊C32
#64096⋊C32
#105959823
C3×#55113