Finite Group Databasebeta

← #10596204⋊C3 #13153329

Subgroups — #10596204⋊C3 #13153329

Order 1728.

824 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64156⋊C3C212yesmaximal, complemented
3#64155⋊C3C212yesmaximal, complemented
4#64107⋊C3C212yesmaximal, complemented
5#64106⋊C3C211yesmaximal, characteristic, complemented
6#64104⋊C3C211yesmaximal, characteristic
7#64103⋊C3C212yesmaximal, complemented
8#64102⋊C3C211yesmaximal, characteristic, complemented
9#64096⋊C3C211yesmaximal, characteristic, complemented
10#64084⋊C3C211yesmaximal, characteristic, complemented
11#64081⋊C3C211yesmaximal, characteristic, complemented
12#10596204C311yesmaximal, characteristic, complemented
13C3×#5733—31nomaximal
14C2×Q8×H(3,𝔽3)C2212yes—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C42×H(3,𝔽3)C2211yescharacteristic
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2211yescharacteristic
28#3752⋊C3C2211yescharacteristic
29#3752⋊C3C2212yes—
30#3752⋊C3C2212yes—
31C4⋊((C3⋊Dic3)⋊C3)C2212yes—
32#3750⋊C3C2211yescharacteristic
33#3750⋊C3C2211yescharacteristic
34#3750⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37C32×C4⋊Q8—21no—
38C32×C4⋊Q8D311yescharacteristic, complemented
39C32×C4⋊Q8C2×C311yescharacteristic, complemented
40#64156C2×C312yescomplemented
41#64155C2×C312yescomplemented
42#64107C2×C312yescomplemented
43#64106C2×C311yescharacteristic, complemented
44#64104C2×C311yescharacteristic
45#64103C2×C312yescomplemented
46#64102C2×C311yescharacteristic, complemented
47#64096C2×C311yescharacteristic, complemented
48#64084C2×C311yescharacteristic, complemented
49#64081C2×C311yescharacteristic, complemented
50C3×#947—32no—
51C3×#946—32no—
52C3×#898—32no—
53C3×#897—31no—
54C3×#895—31no—
55C3×#894—32no—
56C3×#893—31no—
57C3×#887—31no—
58C3×#875—31no—
59C3×#872—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62Q8×H(3,𝔽3)—42no—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4⋊((C3⋊D3)⋊C3)—42no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—42no—
76#634⋊C3—22no—
77C3×#542—91no—
78#5733—31no—
79#5733—91no—
80C2×C32×Q8—22no—
81C2×C32×Q8C2×D312yes—
82C2×C32×Q8C22×C312yes—
83C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
84C2×C4⋊(C3⋊D3)C22×C312yes—
85C2×C4×C3⋊D3C22×C311yescharacteristic
86C2×C4×C3⋊D3C22×C312yes—
87C2×#634C22×C311yescharacteristic
88C2×C3×D12—31no—
89C2×C3×D12—32no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—32no—
92C2×C3×Dic6—31no—
93C32×C4⋊C4—21no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4—22no—
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C2×D311yescharacteristic
98C32×C4⋊C4C2×D312yes—
99C32×C4⋊C4C22×C311yescharacteristic
100C32×C4⋊C4C22×C311yescharacteristic
101C32×C4⋊C4C22×C312yes—
102C32×C42—21no—
103C32×C42C2×D311yescharacteristic
104C32×C42C22×C311yescharacteristic
105#3752C22×C311yescharacteristic
106#3752C22×C311yescharacteristic
107#3752C22×C312yes—
108#3752C22×C312yes—
109#3752C22×C312yes—
110#3752C22×C312yes—
111#3752C22×C312yes—
112C4⋊(C3⋊Dic3)C22×C312yes—
113#3750C22×C311yescharacteristic
114#3750C22×C311yescharacteristic
115#3750C22×C312yes—
116C4×C3⋊Dic3C22×C311yescharacteristic
117C4×C3⋊Dic3C22×C312yes—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—31no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—31no—
125C3×C4⋊Dic3—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—31no—
128C3×Dic3⋊C4—31no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—32no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3—22no—
134C2×(C3⋊D3)⋊C3—22no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3—22no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)—22no—
140C4×H(3,𝔽3)—22no—
141C4×H(3,𝔽3)—21no—
142C4×H(3,𝔽3)#4112yes—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—21no—
147C3×C4⋊Q8—31no—
148C3×C4⋊Q8—31no—
149C3×C4⋊Q8—61no—
150C3×C4⋊Q8C3×D311yescharacteristic
151C3×#126—92no—
152C3×#124—91no—
153C3×#124—92no—
154C3×#123—92no—
155C3×C22⋊Q8—91no—
156C3×C22⋊Q8—92no—
157C3×#121—91no—
158C3×C4×Q8—91no—
159C3×C4×D4—91no—
160C3×#117—91no—
161#947—32no—
162#947—92no—
163#946—32no—
164#946—92no—
165#898—32no—
166#898—92no—
167#897—31no—
168#897—91no—
169#895—31no—
170#895—91no—
171#894—32no—
172#894—92no—
173#893—31no—
174#893—91no—
175#887—31no—
176#887—91no—
177#875—31no—
178#875—91no—
179#872—31no—
180#872—91no—
181C22×C3⋊D3C23×C311yescharacteristic
182C22×C3⋊D3C23×C312yes—
183C22×C3×D3—32no—
184C22×C3×D3—31no—
185C32×Q8—42no—
186C32×Q8—42no—
187C32×Q8—44no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—22no—
192C2×C32×C4—22no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C23×C312yes—
199C2×C32×C4C23×C311yescharacteristic
200C2×C32×C4C23×C311yescharacteristic
201C2×C32×C4C23×C312yes—
202C2×C32×C4C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C311yescharacteristic
204C2×C3⋊Dic3C23×C311yescharacteristic
205C2×C3⋊Dic3C23×C312yes—
206C2×C3⋊Dic3C23×C311yescharacteristic
207C4⋊(C3⋊D3)—22no—
208C4⋊(C3⋊D3)—42no—
209C4×C3⋊D3—22no—
210C4×C3⋊D3—42no—
211#634—22no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—31no—
216C3×D12—62no—
217C3×D12—122no—
218C3×C4×D3—62no—
219C3×C4×D3—122no—
220C3×Dic6—62no—
221#542—91noSylow
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
224C2×H(3,𝔽3)2−1+411yescharacteristic
225(C3⋊D3)⋊C3—41no—
226(C3⋊D3)⋊C3—42no—
227C2×C3×Q8—32no—
228C2×C3×Q8—32no—
229C2×C3×Q8—62no—
230C2×C3×Q8—91no—
231C2×C3×Q8C2×C3×D312yes—
232C2×C3×D4—91no—
233C2×C3×D4—92no—
234C22×C3×C4—91no—
235C22×C3×C4—92no—
236C2×D12—31no—
237C2×D12—32no—
238C2×D12—91no—
239C2×D12—92no—
240C2×C4×D3—31no—
241C2×C4×D3—32no—
242C2×C4×D3—91no—
243C2×C4×D3—92no—
244C2×Dic6—31no—
245C2×Dic6—91no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—32no—
249C3×C4⋊C4—31no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—32no—
252C3×C4⋊C4—61no—
253C3×C4⋊C4—61no—
254C3×C4⋊C4—62no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4—91no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—92no—
259C3×C4⋊C4C2×C3×D311yescharacteristic
260C3×C4⋊C4C2×C3×D311yescharacteristic
261C3×C4⋊C4C2×C3×D312yes—
262C3×C22⋊C4—91no—
263C3×C22⋊C4—91no—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C42—31no—
270C3×C42—31no—
271C3×C42—61no—
272C3×C42—91no—
273C3×C42—92no—
274C3×C42C2×C3×D311yescharacteristic
275(C2×D3)⋊C4—31no—
276(C2×D3)⋊C4—31no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—91no—
283(C2×D3)⋊C4—91no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289C4⋊Dic3—32no—
290C4⋊Dic3—92no—
291Dic3⋊C4—31no—
292Dic3⋊C4—31no—
293Dic3⋊C4—32no—
294Dic3⋊C4—91no—
295Dic3⋊C4—91no—
296Dic3⋊C4—92no—
297C4×Dic3—31no—
298C4×Dic3—32no—
299C4×Dic3—91no—
300C4×Dic3—92no—
301C22×C32—21no—
302C22×C32C23×D311yescharacteristic
303C22×C32C24×C311yescharacteristic
304C2×C3⋊D3—21no—
305C2×C3⋊D3—22no—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—21no—
308C2×C3⋊D3—21no—
309C2×C3⋊D3—22no—
310C2×C3×D3—62no—
311C2×C3×D3—61no—
312C2×C3×D3—61no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—61no—
316C32×C4—22no—
317C32×C4—22no—
318C32×C4—22no—
319C32×C4—22no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—22no—
327C32×C4—42no—
328C32×C4—42no—
329C32×C4#23912yes—
330C32×C4C3×#4112yes—
331C3⋊Dic3—22no—
332C3⋊Dic3—21no—
333C3⋊Dic3—21no—
334C3⋊Dic3—21no—
335C3×Dic3—61no—
336C3×Dic3—62no—
337C3×Dic3—61no—
338C3×Dic3—61no—
339C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
340#126—92no—
341#124—92no—
342#124—91no—
343#123—92no—
344C22⋊Q8—92no—
345C22⋊Q8—91no—
346#121—91no—
347C4×Q8—91no—
348C4×D4—91no—
349#117—91no—
350H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
351C23×C3—91no—
352C23×C3—92no—
353C22×D3—32no—
354C22×D3—31no—
355C22×D3—91no—
356C22×D3—92no—
357C3×Q8—42no—
358C3×Q8—122no—
359C3×Q8—122no—
360C3×Q8—124no—
361C3×Q8—182no—
362C3×D4—182no—
363C3×D4—362no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—32no—
369C2×C3×C4—32no—
370C2×C3×C4—32no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—32no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—91no—
380C2×C3×C4—92no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—182no—
384C2×C3×C4—362no—
385C2×C3×C4C22×C3×D312yes—
386C2×C3×C4C22×C3×D311yescharacteristic
387C2×C3×C4C22×C3×D311yescharacteristic
388C2×C3×C4C22×C3×D312yes—
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—32no—
393C2×Dic3—31no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—92no—
397C2×Dic3—91no—
398D12—62no—
399D12—122no—
400D12—182no—
401D12—362no—
402C4×D3—62no—
403C4×D3—122no—
404C4×D3—182no—
405C4×D3—362no—
406Dic6—62no—
407Dic6—182no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32#100411yescharacteristic
412C2×C32C2×#23911yescharacteristic
413C2×C32#100711yescharacteristic
414C2×C32C2×C3×#4111yescharacteristic
415C2×C32C3×H(5,𝔽2)11yescharacteristic
416C2×C32C3×2−1+411yescharacteristic
417C3⋊D3—41no—
418C3⋊D3—42no—
419C3×D3—122no—
420C3×D3—121no—
421C2×Q8—91no—
422C2×Q8C2×(C3⋊D3)⋊C312yes—
423C2×D4—91no—
424C2×D4—92no—
425C22×C4—91no—
426C22×C4—92no—
427C4⋊C4—91no—
428C4⋊C4—91no—
429C4⋊C4—92no—
430C4⋊C4—92no—
431C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
432C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
433C4⋊C4C2×(C3⋊D3)⋊C312yes—
434C22⋊C4—91no—
435C22⋊C4—91no—
436C22⋊C4—92no—
437C22⋊C4—92no—
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—92no—
441C42—91no—
442C42—92no—
443C42C2×(C3⋊D3)⋊C311yescharacteristic
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—61no—
447C22×C3—181no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—181no—
451C22×C3—181no—
452C22×C3—182no—
453C22×C3C23×C3×D311yescharacteristic
454C2×D3—62no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—61no—
460C2×D3—181no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—181no—
464C2×D3—181no—
465C2×D3—182no—
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—32no—
471C3×C4—32no—
472C3×C4—62no—
473C3×C4—62no—
474C3×C4—62no—
475C3×C4—62no—
476C3×C4—61no—
477C3×C4—61no—
478C3×C4—61no—
479C3×C4—61no—
480C3×C4—62no—
481C3×C4—62no—
482C3×C4—62no—
483C3×C4—122no—
484C3×C4—122no—
485C3×C4—181no—
486C3×C4—181no—
487C3×C4—182no—
488C3×C4—181no—
489C3×C4C3×#23912yes—
490Dic3—61no—
491Dic3—62no—
492Dic3—61no—
493Dic3—61no—
494Dic3—182no—
495Dic3—181no—
496Dic3—181no—
497Dic3—181no—
498C32—21no—
499C32#573311yescharacteristic, complemented
500C32C3×#54211yescharacteristic, complemented
501C23—92no—
502C23—91no—
503Q8—42no—
504Q8—182no—
505D4—182no—
506D4—362no—
507C2×C4—91no—
508C2×C4—92no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—182no—
512C2×C4—362no—
513C2×C4C22×(C3⋊D3)⋊C312yes—
514C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
515C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
516C2×C4C22×(C3⋊D3)⋊C312yes—
517C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—61no—
527C2×C3—361no—
528C2×C3—362no—
529C2×C3C3×#100411yescharacteristic
530C2×C3C2×C3×#23911yescharacteristic
531C2×C3C3×#100711yescharacteristic
532D3—122no—
533D3—121no—
534D3—361no—
535D3—362no—
536C22—182no—
537C22—181no—
538C22—181no—
539C22—182no—
540C22—182no—
541C22—181no—
542C22C23×(C3⋊D3)⋊C311yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—181no—
548C4—181no—
549C4—182no—
550C4—181no—
551C4#8872212yes—
552C3—31no—
553C3—31no—
554C3—61no—
555C3C3×#573311yescharacteristic
556C2—362no—
557C2—361no—
558C2#64361⋊C311yescharacteristic
559C2C2×#8872211yescharacteristic
560C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radical#64104⋊C3C2
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
2. #64156⋊C3C2semidirect
3. #64155⋊C3C2semidirect
4. #64107⋊C3C2semidirect
5. #64106⋊C3C2semidirect
7. #64103⋊C3C2semidirect
8. #64102⋊C3C2semidirect
9. #64096⋊C3C2semidirect
10. #64084⋊C3C2semidirect
11. #64081⋊C3C2semidirect
12. #10596204C3semidirect
38. C32×C4⋊Q8D3semidirect
39. C32×C4⋊Q8C2×C3semidirect
40. #64156C2×C3semidirect
41. #64155C2×C3semidirect
42. #64107C2×C3semidirect
43. #64106C2×C3semidirect
45. #64103C2×C3semidirect
46. #64102C2×C3semidirect
47. #64096C2×C3semidirect
48. #64084C2×C3semidirect
49. #64081C2×C3semidirect
339. C4⋊Q8(C3⋊D3)⋊C3semidirect
350. H(3,𝔽3)#542semidirect
499. C32#5733semidirect
500. C32C3×#542semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#64156⋊C32
#64155⋊C32
#64107⋊C32
#64106⋊C32
#64104⋊C32
#64103⋊C32
#64102⋊C32
#64096⋊C32
#64084⋊C32
#64081⋊C32
#105962043
C3×#57333