Finite Group Databasebeta

← C3×#29308 #10647915

Subgroups — C3×#29308 #10647915

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2628C211yesmaximal, characteristic
2C3×#2316C211yesmaximal, characteristic
3C3×#2085C211yesmaximal, characteristic, complemented
4C3×#2085C211yesmaximal, characteristic, complemented
5C3×#2058C211yesmaximal, characteristic, complemented
6C3×#2057C211yesmaximal, characteristic, complemented
7C3×#1944C211yesmaximal, characteristic, complemented
8C3×#1943C211yesmaximal, characteristic, complemented
9C3×#1942C211yesmaximal, characteristic, complemented
10C3×#1941C211yesmaximal, characteristic, complemented
11C3×#1854C211yesmaximal, characteristic
12C3×#1854C211yesmaximal, characteristic
13C3×#1853C211yesmaximal, characteristic
14C3×#1852C211yesmaximal, characteristic
15C3×#1750C211yesmaximal, characteristic
16#29308C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×#97C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33C3×#400—21no—
34C3×#399—21no—
35C3×#395—21no—
36C3×#393—21no—
37C3×#389—21no—
38C3×#389—21no—
39C3×#389C2211yescharacteristic
40C3×#389C2211yescharacteristic
41C3×#389C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#382C2211yescharacteristic
46C3×#380—21no—
47C3×#380—21no—
48C3×#380C2211yescharacteristic
49C3×#380C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2628C2×C311yescharacteristic
61#2316C2×C311yescharacteristic
62#2085C2×C311yescharacteristic, complemented
63#2085C2×C311yescharacteristic, complemented
64#2058C2×C311yescharacteristic, complemented
65#2057C2×C311yescharacteristic, complemented
66#1944C2×C311yescharacteristic, complemented
67#1943C2×C311yescharacteristic, complemented
68#1942C2×C311yescharacteristic, complemented
69#1941C2×C311yescharacteristic, complemented
70#1854C2×C311yescharacteristic
71#1854C2×C311yescharacteristic
72#1853C2×C311yescharacteristic
73#1852C2×C311yescharacteristic
74#1750C2×C311yescharacteristic
75C22×C3×Q8—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—41no—
83C3×C4⋊Q8—41no—
84C3×#117—41no—
85C3×#117—41no—
86C3×#117—41no—
87C3×#117—41no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4D411yescharacteristic
112C2×C3×C22⋊C4D411yescharacteristic
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2311yescharacteristic
116C3×#107—22no—
117C3×#107—22no—
118C3×#107—41no—
119C3×#107—41no—
120C3×#106—22no—
121C3×#106—22no—
122C3×#106—41no—
123C3×#106—41no—
124C3×C22⋊C8Q814yescomplemented
125C3×C22⋊C8Q814yescomplemented
126C3×#97—41no—
127C3×C4×C8—41no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C2×C4⋊Q8C22×C311yescharacteristic
139C2×#117C22×C311yescharacteristic
140C2×#117C22×C311yescharacteristic
141C22×C4⋊C4C22×C311yescharacteristic
142C2×#107C22×C311yescharacteristic
143C2×#107C22×C311yescharacteristic
144C2×#107C22×C311yescharacteristic
145C2×#107C22×C311yescharacteristic
146C2×#106C22×C311yescharacteristic
147C2×#106C22×C311yescharacteristic
148C2×#106C22×C311yescharacteristic
149C2×#106C22×C311yescharacteristic
150C2×C22⋊C8C22×C311yescharacteristic
151C2×C22⋊C8C22×C311yescharacteristic
152C2×#97C22×C311yescharacteristic
153C2×C4×C8C22×C311yescharacteristic
154#400—21no—
155#399—21no—
156#395—21no—
157#393—21no—
158#389—21no—
159#389—21no—
160#389C22×C311yescharacteristic
161#389C22×C311yescharacteristic
162#389C22×C311yescharacteristic
163#389C22×C311yescharacteristic
164#389C22×C311yescharacteristic
165#389C22×C311yescharacteristic
166#382C22×C311yescharacteristic
167#380—21no—
168#380—21no—
169#380C22×C311yescharacteristic
170#380C22×C311yescharacteristic
171#340C22×C311yescharacteristic
172#340C22×C311yescharacteristic
173#340C22×C311yescharacteristic
174#340C22×C311yescharacteristic
175#340C22×C311yescharacteristic
176#340C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#336C22×C311yescharacteristic
180#336C22×C311yescharacteristic
181C24×C3C2×D411yescharacteristic
182C2×C3×Q8—42no—
183C2×C3×Q8—81no—
184C2×C3×Q8—81no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—24no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—81no—
199C22×C3×C4—81no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×Q812yes—
206C22×C3×C4C2×Q812yes—
207C22×C3×C4C2411yescharacteristic
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8C2×Q812yes—
217C2×C3×C8C2×Q812yes—
218C2×C3×C8C2×Q812yes—
219C2×C3×C8C2×Q812yes—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—81no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4—81no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4#4112yes—
245C3×C4⋊C4#4112yes—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—81no—
250C3×C22⋊C4—81no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C22×Q8—21no—
257C23×C4—21no—
258C23×C4C23×C311yescharacteristic
259C22×C8C23×C311yescharacteristic
260C22×C8C23×C311yescharacteristic
261C22×C8C23×C311yescharacteristic
262C22×C8C23×C311yescharacteristic
263C4⋊Q8—41no—
264C4⋊Q8—41no—
265#117—41no—
266#117—41no—
267#117—41no—
268#117—41no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C3×D411yescharacteristic
280C2×C4⋊C4C3×D411yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C4⋊C4C23×C311yescharacteristic
283C2×C4⋊C4C23×C311yescharacteristic
284C2×C4⋊C4C23×C311yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4C3×D411yescharacteristic
293C2×C22⋊C4C3×D411yescharacteristic
294C2×C42—21no—
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#107—22no—
298#107—22no—
299#107—41no—
300#107—41no—
301#106—22no—
302#106—22no—
303#106—41no—
304#106—41no—
305C22⋊C8C3×Q814yescomplemented
306C22⋊C8C3×Q814yescomplemented
307#97—41no—
308C4×C8—41no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3C4⋊Q812yes—
326C23×C3C22×D411yescharacteristic
327C3×Q8—84no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—84no—
371C2×C3×C4C4⋊Q812yes—
372C2×C3×C4C22×Q811yescharacteristic
373C2×C3×C4C22×Q811yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C3×C8—22no—
379C3×C8—22no—
380C3×C8—22no—
381C3×C8—22no—
382C24C2×C3×D411yescharacteristic
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×Q812yes—
407C22×C4C2×C3×Q812yes—
408C22×C4C24×C311yescharacteristic
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C3×Q812yes—
418C2×C8C2×C3×Q812yes—
419C2×C8C2×C3×Q812yes—
420C2×C8C2×C3×Q812yes—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4C3×#4112yes—
446C4⋊C4C3×#4112yes—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—42no—
450C22⋊C4—81no—
451C22⋊C4—81no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—42no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3#52511yescharacteristic
465C22×C3C2×C4⋊Q811yescharacteristic
466C22×C3#53411yescharacteristic
467C22×C3#57611yescharacteristic
468C22×C3#57711yescharacteristic
469C22×C3#57711yescharacteristic
470C22×C3#57811yescharacteristic
471C3×C4—22no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4#55411yescharacteristic
485C3×C4#55411yescharacteristic
486C3×C4#55411yescharacteristic
487C3×C4#55411yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23C3×C4⋊Q812yes—
495C23C22×C3×D411yescharacteristic
496Q8—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4C3×C4⋊Q812yes—
541C2×C4C22×C3×Q811yescharacteristic
542C2×C4C22×C3×Q811yescharacteristic
543C2×C4C2×C3×#4111yescharacteristic
544C2×C4C2×C3×#4111yescharacteristic
545C2×C4C3×H(5,𝔽2)11yescharacteristic
546C2×C4C3×H(5,𝔽2)11yescharacteristic
547C8—22no—
548C8—22no—
549C8—22no—
550C8—22no—
551C2×C3—42no—
552C2×C3#258311yescharacteristic
553C2×C3#311811yescharacteristic
554C2×C3#312111yescharacteristic
555C2×C3#316311yescharacteristic
556C2×C3#316411yescharacteristic
557C2×C3#319811yescharacteristic
558C2×C3#320111yescharacteristic
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C3×#52511yescharacteristic
567C22C2×C3×C4⋊Q811yescharacteristic
568C22C3×#53411yescharacteristic
569C22C3×#57611yescharacteristic
570C22C3×#57711yescharacteristic
571C22C3×#57711yescharacteristic
572C22C3×#57811yescharacteristic
573C4—22no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4C3×#55411yescharacteristic
587C4C3×#55411yescharacteristic
588C4C3×#55411yescharacteristic
589C4C3×#55411yescharacteristic
590C3#2930811yescharacteristic, direct product factor, Sylow
591C2—42no—
592C2C3×#258311yescharacteristic
593C2C3×#311811yescharacteristic
594C2C3×#312111yescharacteristic
595C2C3×#316311yescharacteristic
596C2C3×#316411yescharacteristic
597C2C3×#319811yescharacteristic
598C2C3×#320111yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2085C2semidirect
4. C3×#2085C2semidirect
5. C3×#2058C2semidirect
6. C3×#2057C2semidirect
7. C3×#1944C2semidirect
8. C3×#1943C2semidirect
9. C3×#1942C2semidirect
10. C3×#1941C2semidirect
16. #29308C3direct (N × H)
62. #2085C2×C3semidirect
63. #2085C2×C3semidirect
64. #2058C2×C3semidirect
65. #2057C2×C3semidirect
66. #1944C2×C3semidirect
67. #1943C2×C3semidirect
68. #1942C2×C3semidirect
69. #1941C2×C3semidirect
124. C3×C22⋊C8Q8semidirect
125. C3×C22⋊C8Q8semidirect
305. C22⋊C8C3×Q8semidirect
306. C22⋊C8C3×Q8semidirect
590. C3#29308direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26282
C3×#23162
C3×#20852
C3×#20582
C3×#20572
C3×#19442
C3×#19432
C3×#19422
C3×#19412
C3×#18542
C3×#18532
C3×#18522
C3×#17502
#293083