Finite Group Databasebeta

← C3×#73657 #11807109

Subgroups — C3×#73657 #11807109

Order 1152.

1089 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#888C211yesmaximal, characteristic
2C32×C4×C4⋊C4C211yesmaximal, characteristic
3C3×#4984C211yesmaximal, characteristic
4C3×Dic3×C4⋊C4C211yesmaximal, characteristic
5C3×Dic3×C4⋊C4C211yesmaximal, characteristic
6C3×C4×Dic3⋊C4C212yesmaximal
7C3×C4×Dic3⋊C4C211yesmaximal, characteristic
8C3×#4929C212yesmaximal
9C3×#4928C211yesmaximal, characteristic
10C3×#4647C212yesmaximal
11C3×#4646C212yesmaximal
12C3×#2369—31nomaximal
13#73657C311yesmaximal, characteristic, direct product factor
14C22×C3×Dic6C2212yes—
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C42C2211yescharacteristic
18C2×C32×C42C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4⋊Dic3C2212yes—
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×Dic3⋊C4C2211yescharacteristic
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×Dic3⋊C4C2212yes—
25C2×C3×Dic3⋊C4C2211yescharacteristic
26C2×C3×C4×Dic3C2212yes—
27C2×C3×C4×Dic3C2211yescharacteristic
28C2×C3×C4×Dic3C2212yes—
29C2×C3×C4×Dic3C2211yescharacteristic
30C3×#888C418yescomplemented
31C32×#95C2212yes—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2211yescharacteristic
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2211yescharacteristic
38C2×C3×C4⋊Q8—31no—
39C2×#888C2×C311yescharacteristic
40C3×#391—32no—
41C3×#391—32no—
42C3×#389—31no—
43C3×#389—31no—
44C3×#389—32no—
45C3×C4×C4⋊C4—21no—
46C3×C4×C4⋊C4—31no—
47C3×C4×C4⋊C4—31no—
48C3×C4×C4⋊C4—32no—
49C3×C4×C4⋊C4—31no—
50C3×C4×C4⋊C4D311yescharacteristic
51C3×C4×C4⋊C4C2×C311yescharacteristic
52#4984C2×C311yescharacteristic
53Dic3×C4⋊C4C2×C311yescharacteristic
54Dic3×C4⋊C4C2×C311yescharacteristic
55C4×Dic3⋊C4C2×C311yescharacteristic
56C4×Dic3⋊C4C2×C312yes—
57#4929C2×C312yes—
58#4928C2×C311yescharacteristic
59#4647C2×C312yes—
60#4646C2×C312yes—
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2312yes—
63C22×C32×C4C2312yes—
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C3×Dic3C2311yescharacteristic
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C2×C3×Dic6—42no—
73C2×C3×Dic6—42no—
74C2×C3×Dic6C2×C418yes—
75C32×C4⋊C4C2×C414yes—
76C32×C4⋊C4Q814yescomplemented
77C32×C42—22no—
78C32×C42—42no—
79C3×C4⋊Dic3—22no—
80C3×C4⋊Dic3—42no—
81C3×C4⋊Dic3C2×C414yes—
82C3×Dic3⋊C4—22no—
83C3×Dic3⋊C4—42no—
84C3×Dic3⋊C4C2×C418yes—
85C3×Dic3⋊C4Q814yescomplemented
86C3×C4×Dic3—22no—
87C3×C4×Dic3—24no—
88C3×C4×Dic3—42no—
89C3×C4×Dic3C2×C414yes—
90#2369—31noSylow
91C22×C3×Q8—32no—
92C22×Dic6C22×C312yes—
93C3×C4⋊Q8—38no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—31no—
97C2×C3×C4⋊C4—32no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—32no—
102C2×C3×C4⋊C4—32no—
103C2×C3×C4⋊C4C2×D311yescharacteristic
104C2×C3×C4⋊C4C2×D311yescharacteristic
105C2×C3×C4⋊C4C22×C311yescharacteristic
106C2×C3×C4⋊C4C22×C311yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42—22no—
109C2×C3×C42—31no—
110C2×C3×C42—32no—
111C2×C3×C42—31no—
112C2×C3×C42—32no—
113C2×C3×C42C2×D311yescharacteristic
114C2×C3×C42C2×D312yes—
115C2×C3×C42C22×C311yescharacteristic
116C2×C3×C42C22×C312yes—
117C2×C4⋊Dic3C22×C311yescharacteristic
118C2×C4⋊Dic3C22×C312yes—
119C2×C4⋊Dic3C22×C311yescharacteristic
120C2×Dic3⋊C4C22×C311yescharacteristic
121C2×Dic3⋊C4C22×C312yes—
122C2×Dic3⋊C4C22×C312yes—
123C2×Dic3⋊C4C22×C311yescharacteristic
124C2×C4×Dic3C22×C312yes—
125C2×C4×Dic3C22×C311yescharacteristic
126C2×C4×Dic3C22×C312yes—
127C2×C4×Dic3C22×C311yescharacteristic
128#888C3×C418yescomplemented
129C3×#95—22no—
130C3×#95—32no—
131C3×#95—32no—
132C3×#95—32no—
133C3×#95—31no—
134C3×#95—32no—
135C3×#95—31no—
136C3×#95C2×D312yes—
137C3×#95C22×C312yes—
138C3⋊#95C22×C311yescharacteristic
139C3⋊#95C22×C312yes—
140C3⋊#95C22×C312yes—
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C312yes—
143C3⋊#95C22×C311yescharacteristic
144C23×C32C2411yescharacteristic
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4C22×C414yes—
155C2×C32×C4C22×C412yes—
156C2×C32×C4C2×Q814yes—
157C2×C32×C4C2×Q812yes—
158C2×C32×C4C2×Q812yes—
159C2×C32×C4#4112yes—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3C22×C412yes—
173C2×C3×Dic3C22×C414yes—
174C2×C3×Dic3C22×C412yes—
175C2×C3×Dic3C2×Q812yes—
176C2×C3×Dic3C2×Q812yes—
177C2×C3×Dic3#4112yes—
178C3×Dic6—48no—
179C2×C4⋊Q8—31no—
180#391—32no—
181#391—32no—
182#389—31no—
183#389—31no—
184#389—32no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—32no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4C3×D311yescharacteristic
190C2×C3×Q8—38no—
191C2×C3×Q8—122no—
192C2×C3×Q8—122no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—31no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—32no—
202C22×C3×C4—31no—
203C22×C3×C4—32no—
204C22×C3×C4C22×D311yescharacteristic
205C22×C3×C4C22×D312yes—
206C22×C3×C4C22×D312yes—
207C22×C3×C4C22×D311yescharacteristic
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C23×C311yescharacteristic
210C22×C3×C4C23×C312yes—
211C22×C3×C4C23×C311yescharacteristic
212C22×C3×C4C23×C312yes—
213C22×C3×C4C23×C311yescharacteristic
214C22×Dic3C23×C311yescharacteristic
215C22×Dic3C23×C312yes—
216C22×Dic3C23×C311yescharacteristic
217C22×Dic3C23×C312yes—
218C22×Dic3C23×C311yescharacteristic
219C22×Dic3C23×C311yescharacteristic
220C2×Dic6—42no—
221C2×Dic6—42no—
222C2×Dic6C2×C3×C418yes—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—38no—
226C3×C4⋊C4—34no—
227C3×C4⋊C4—34no—
228C3×C4⋊C4—62no—
229C3×C4⋊C4—62no—
230C3×C4⋊C4—122no—
231C3×C4⋊C4—122no—
232C3×C4⋊C4Dic614yescomplemented
233C3×C4⋊C4C4×D314yes—
234C3×C4⋊C4C2×C3×C414yes—
235C3×C4⋊C4C3×Q814yescomplemented
236C3×C42—24no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—34no—
240C3×C42—42no—
241C3×C42—42no—
242C3×C42—44no—
243C3×C42—64no—
244C3×C42—62no—
245C3×C42—122no—
246C4⋊Dic3—22no—
247C4⋊Dic3—42no—
248C4⋊Dic3C2×C3×C414yes—
249Dic3⋊C4—22no—
250Dic3⋊C4—42no—
251Dic3⋊C4C2×C3×C418yes—
252Dic3⋊C4C3×Q814yescomplemented
253C4×Dic3—22no—
254C4×Dic3—24no—
255C4×Dic3—42no—
256C4×Dic3C2×C3×C414yes—
257C22×C32C23×C411yescharacteristic
258C22×C32C22×Q811yescharacteristic
259C22×C32C22×Q811yescharacteristic
260C22×C32C2×#4111yescharacteristic
261C22×C32C2×#4111yescharacteristic
262C22×C32H(5,𝔽2)11yescharacteristic
263C22×C32H(5,𝔽2)11yescharacteristic
264C32×C4—22no—
265C32×C4—24no—
266C32×C4—24no—
267C32×C4—41no—
268C32×C4C4×Q814yes—
269C3×Dic3—22no—
270C3×Dic3—24no—
271C3×Dic3—22no—
272C3×Dic3—41no—
273C3×Dic3—42no—
274C3×Dic3C4×Q814yes—
275C22×Q8—32no—
276C4⋊Q8—38no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4C2×C3×D311yescharacteristic
285C2×C4⋊C4C2×C3×D311yescharacteristic
286C2×C42—31no—
287C2×C42—32no—
288C2×C42—31no—
289C2×C42—32no—
290C2×C42C2×C3×D311yescharacteristic
291C2×C42C2×C3×D312yes—
292#95—32no—
293#95—32no—
294#95—32no—
295#95—31no—
296#95—32no—
297#95—31no—
298#95C2×C3×D312yes—
299C23×C3—21no—
300C23×C3C23×D311yescharacteristic
301C23×C3C24×C311yescharacteristic
302C3×Q8—128no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—32no—
334C2×C3×C4—34no—
335C2×C3×C4—32no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—62no—
345C2×C3×C4—61no—
346C2×C3×C4—62no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—61no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4C2×Dic614yes—
355C2×C3×C4C2×Dic612yes—
356C2×C3×C4C2×C4×D314yes—
357C2×C3×C4C2×C4×D312yes—
358C2×C3×C4#23512yes—
359C2×C3×C4D3×Q812yes—
360C2×C3×C4C22×C3×C414yes—
361C2×C3×C4C22×C3×C412yes—
362C2×C3×C4C2×C3×Q814yes—
363C2×C3×C4C2×C3×Q812yes—
364C2×C3×C4C2×C3×Q812yes—
365C2×C3×C4C3×#4112yes—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—21no—
378C2×Dic3C22×C3×C412yes—
379C2×Dic3C22×C3×C414yes—
380C2×Dic3C22×C3×C412yes—
381C2×Dic3C2×C3×Q812yes—
382C2×Dic3C2×C3×Q812yes—
383C2×Dic3C3×#4112yes—
384Dic6—48no—
385C2×C32C2×C4×Q811yescharacteristic
386C2×C32C2×C4×Q811yescharacteristic
387C2×C32#51811yescharacteristic
388C2×C32#55411yescharacteristic
389C2×C32#55411yescharacteristic
390C2×C32#55511yescharacteristic
391C2×C32Q8211yescharacteristic
392C2×Q8—38no—
393C2×Q8—122no—
394C2×Q8—122no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—32no—
399C22×C4—31no—
400C22×C4—32no—
401C22×C4C22×C3×D311yescharacteristic
402C22×C4C22×C3×D312yes—
403C22×C4C22×C3×D312yes—
404C22×C4C22×C3×D311yescharacteristic
405C22×C4C22×C3×D311yescharacteristic
406C4⋊C4—38no—
407C4⋊C4—34no—
408C4⋊C4—34no—
409C4⋊C4—62no—
410C4⋊C4—62no—
411C4⋊C4—122no—
412C4⋊C4—122no—
413C4⋊C4C3×Dic614yescomplemented
414C4⋊C4C3×C4×D314yes—
415C42—22no—
416C42—34no—
417C42—42no—
418C42—64no—
419C42—62no—
420C42—122no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3C22×Dic611yescharacteristic
429C22×C3C22×C4×D311yescharacteristic
430C22×C3C2×#23511yescharacteristic
431C22×C3#100411yescharacteristic
432C22×C3C2×D3×Q811yescharacteristic
433C22×C3D3×#4111yescharacteristic
434C22×C3#100911yescharacteristic
435C22×C3C23×C3×C411yescharacteristic
436C22×C3C22×C3×Q811yescharacteristic
437C22×C3C22×C3×Q811yescharacteristic
438C22×C3C2×C3×#4111yescharacteristic
439C22×C3C2×C3×#4111yescharacteristic
440C22×C3C3×H(5,𝔽2)11yescharacteristic
441C22×C3C3×H(5,𝔽2)11yescharacteristic
442C3×C4—24no—
443C3×C4—22no—
444C3×C4—24no—
445C3×C4—24no—
446C3×C4—24no—
447C3×C4—22no—
448C3×C4—24no—
449C3×C4—24no—
450C3×C4—34no—
451C3×C4—44no—
452C3×C4—44no—
453C3×C4—42no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—64no—
459C3×C4—121no—
460C3×C4—122no—
461C3×C4C4×Dic614yes—
462C3×C4C3×C4×Q814yes—
463Dic3—22no—
464Dic3—24no—
465Dic3—22no—
466Dic3—41no—
467Dic3—42no—
468Dic3C3×C4×Q814yes—
469C32#236911yescharacteristic, complemented, Sylow
470C23C23×C3×D311yescharacteristic
471Q8—128no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—34no—
486C2×C4—32no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—62no—
490C2×C4—61no—
491C2×C4—62no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—61no—
499C2×C4C2×C3×Dic614yes—
500C2×C4C2×C3×Dic612yes—
501C2×C4C2×C3×C4×D314yes—
502C2×C4C2×C3×C4×D312yes—
503C2×C4C3×#23512yes—
504C2×C4C3×D3×Q812yes—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3C2×C4×Dic611yescharacteristic
513C2×C3#551111yescharacteristic
514C2×C3#553211yescharacteristic
515C2×C3#553911yescharacteristic
516C2×C3#554511yescharacteristic
517C2×C3Q8×Dic611yescharacteristic
518C2×C3C4×D3×Q811yescharacteristic
519C2×C3C2×C3×C4×Q811yescharacteristic
520C2×C3C2×C3×C4×Q811yescharacteristic
521C2×C3C3×#51811yescharacteristic
522C2×C3C3×#55411yescharacteristic
523C2×C3C3×#55411yescharacteristic
524C2×C3C3×#55511yescharacteristic
525C2×C3C3×Q8211yescharacteristic
526C22C22×C3×Dic611yescharacteristic
527C22C22×C3×C4×D311yescharacteristic
528C22C2×C3×#23511yescharacteristic
529C22C3×#100411yescharacteristic
530C22C2×C3×D3×Q811yescharacteristic
531C22C3×D3×#4111yescharacteristic
532C22C3×#100911yescharacteristic
533C4—22no—
534C4—24no—
535C4—24no—
536C4—34no—
537C4—41no—
538C4—62no—
539C4—62no—
540C4—64no—
541C4—121no—
542C4—122no—
543C4C3×C4×Dic614yes—
544C3—21no—
545C3#7365711yescharacteristic, direct product factor
546C3C3×#236911yescharacteristic, complemented
547C2C2×C3×C4×Dic611yescharacteristic
548C2C3×#551111yescharacteristic
549C2C3×#553211yescharacteristic
550C2C3×#553911yescharacteristic
551C2C3×#554511yescharacteristic
552C2C3×Q8×Dic611yescharacteristic
553C2C3×C4×D3×Q811yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×C4×C4⋊C4C2
socle, prime radicalC23×C32C24
cosocleC23×C3C24×C3
hypercenterC3×C4×C4⋊C4D3
hypocenterC3C3×#2369
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
13. #73657C3direct (N × H)
30. C3×#888C4semidirect
76. C32×C4⋊C4Q8semidirect
85. C3×Dic3⋊C4Q8semidirect
128. #888C3×C4semidirect
232. C3×C4⋊C4Dic6semidirect
235. C3×C4⋊C4C3×Q8semidirect
252. Dic3⋊C4C3×Q8semidirect
413. C4⋊C4C3×Dic6semidirect
469. C32#2369semidirect
545. C3#73657direct (N × H)
546. C3C3×#2369semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8882
C32×C4×C4⋊C42
C3×#49842
C3×Dic3×C4⋊C42
C3×C4×Dic3⋊C42
C3×#49292
C3×#49282
C3×#46472
C3×#46462
C3×#23693
#736573