Finite Group Databasebeta

← C3×#10938 #10629545

Subgroups — C3×#10938 #10629545

Order 768.

1234 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#99C212yesmaximal
2C3×#2308C211yesmaximal, characteristic
3C3×#2273C211yesmaximal, characteristic, complemented
4C2×C3×#342C211yesmaximal, characteristic, complemented
5C2×C3×#342C212yesmaximal, complemented
6#10938C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4C2212yes—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2212yes—
12C2×C3×#99—24no—
13C2×C3×#99—24no—
14C2×C3×#99C418yescomplemented
15C3×#386—22no—
16C3×#386—22no—
17C3×#381C412yescomplemented
18C3×#379C412yescomplemented
19C3×C4×C22⋊C4—22no—
20C3×C4×C22⋊C4—24no—
21C3×#342—22no—
22C3×#342—24no—
23C22×#99C2×C312yes—
24#2308C2×C311yescharacteristic
25#2273C2×C311yescharacteristic, complemented
26C2×#342C2×C312yescomplemented
27C2×#342C2×C311yescharacteristic, complemented
28C25×C3C2×C411yescharacteristic
29C25×C3C2311yescharacteristic
30C22×C3×D4—21no—
31C22×C3×D4—21no—
32C22×C3×D4—22no—
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4C2×C414yes—
38C23×C3×C4—22no—
39C23×C3×C4—21no—
40C23×C3×C4—22no—
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4D412yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4C2×C412yes—
58C2×C3×C22⋊C4C2×C414yes—
59C2×C3×C22⋊C4C2×C412yes—
60C2×C3×C22⋊C4C2×C414yes—
61C2×C3×C22⋊C4D412yes—
62C2×C3×C22⋊C4D414yes—
63C2×C3×C22⋊C4D414yes—
64C2×C3×C42—24no—
65C3×#99—216no—
66C3×#95—24no—
67C3×#95—24no—
68C23×D4C22×C311yescharacteristic
69C22×C22⋊C4C22×C311yescharacteristic
70C22×C22⋊C4C22×C312yes—
71C22×C22⋊C4C22×C311yescharacteristic
72C22×C22⋊C4C22×C312yes—
73C2×#99—24no—
74C2×#99—24no—
75C2×#99C3×C418yescomplemented
76#386—22no—
77#386—22no—
78#381C3×C412yescomplemented
79#379C3×C412yescomplemented
80C4×C22⋊C4—22no—
81C4×C22⋊C4—24no—
82#342—22no—
83#342—24no—
84C24×C3—21no—
85C24×C3—21no—
86C24×C3—22no—
87C24×C3—22no—
88C24×C3—82no—
89C24×C3—82no—
90C24×C3C4212yescomplemented
91C24×C3C22×C411yescharacteristic
92C24×C3C22×C412yes—
93C24×C3C2×D411yescharacteristic
94C24×C3C2×D412yes—
95C24×C3C2×D411yescharacteristic
96C24×C3C2×D412yes—
97C24×C3#4112yes—
98C24×C3#4112yes—
99C24×C3#4112yes—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—24no—
103C2×C3×D4—24no—
104C2×C3×D4—24no—
105C2×C3×D4—24no—
106C2×C3×D4—41no—
107C2×C3×D4—42no—
108C2×C3×D4—42no—
109C2×C3×D4—42no—
110C2×C3×D4—41no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—41no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—24no—
122C22×C3×C4—24no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—82no—
132C22×C3×C4—84no—
133C22×C3×C4—82no—
134C22×C3×C4—84no—
135C22×C3×C4C4212yescomplemented
136C22×C3×C4C22⋊C414yes—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—24no—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—84no—
149C3×C22⋊C4C22⋊C418yescomplemented
150C3×C42—84no—
151C25C2×C3×C411yescharacteristic
152C25C23×C311yescharacteristic
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4C2×C3×C414yes—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4C2×C3×C411yescharacteristic
165C23×C4C3×D412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4C2×C3×C412yes—
181C2×C22⋊C4C2×C3×C414yes—
182C2×C22⋊C4C2×C3×C412yes—
183C2×C22⋊C4C2×C3×C414yes—
184C2×C22⋊C4C3×D412yes—
185C2×C22⋊C4C3×D414yes—
186C2×C22⋊C4C3×D414yes—
187C2×C42—24no—
188#99—216no—
189#95—24no—
190#95—24no—
191C23×C3—22no—
192C23×C3—24no—
193C23×C3—42no—
194C23×C3—44no—
195C23×C3—42no—
196C23×C3—44no—
197C23×C3—42no—
198C23×C3—44no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—41no—
202C23×C3—84no—
203C23×C3—82no—
204C23×C3—82no—
205C23×C3—82no—
206C23×C3—82no—
207C23×C3—82no—
208C23×C3—82no—
209C23×C3—84no—
210C23×C3C2×C4211yescharacteristic
211C23×C3C2×C22⋊C411yescharacteristic
212C23×C3C2×C22⋊C412yes—
213C23×C3#11711yescharacteristic
214C23×C3#11712yes—
215C23×C3C4×D412yes—
216C23×C3C4×D412yes—
217C23×C3C4×D412yes—
218C23×C3C4×D412yes—
219C23×C3C4×D412yes—
220C23×C3C4×D412yes—
221C23×C3#12011yescharacteristic
222C23×C3#12112yes—
223C23×C3#12112yes—
224C23×C3#12112yes—
225C23×C3#12311yescharacteristic
226C23×C3#12312yes—
227C23×C3#12412yes—
228C23×C3#12411yescharacteristic
229C23×C3#12612yes—
230C23×C3#12711yescharacteristic
231C3×D4—44no—
232C3×D4—44no—
233C3×D4—44no—
234C3×D4—44no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—48no—
240C2×C3×C4—44no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—44no—
245C2×C3×C4—44no—
246C2×C3×C4—42no—
247C2×C3×C4—44no—
248C2×C3×C4—82no—
249C2×C3×C4—84no—
250C2×C3×C4—84no—
251C2×C3×C4—84no—
252C2×C3×C4—82no—
253C2×C3×C4—82no—
254C2×C3×C4—84no—
255C2×C3×C4—84no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—82no—
261C24—82no—
262C24C3×C4212yescomplemented
263C24C22×C3×C411yescharacteristic
264C24C22×C3×C412yes—
265C24C2×C3×D411yescharacteristic
266C24C2×C3×D412yes—
267C24C2×C3×D411yescharacteristic
268C24C2×C3×D412yes—
269C24C3×#4112yes—
270C24C3×#4112yes—
271C24C3×#4112yes—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—41no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—84no—
305C22×C4—82no—
306C22×C4—84no—
307C22×C4C3×C4212yescomplemented
308C22×C4C3×C22⋊C414yes—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—84no—
321C22⋊C4C3×C22⋊C418yescomplemented
322C42—84no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—24no—
326C22×C3—22no—
327C22×C3—24no—
328C22×C3—42no—
329C22×C3—42no—
330C22×C3—42no—
331C22×C3—41no—
332C22×C3—42no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—44no—
345C22×C3—42no—
346C22×C3—84no—
347C22×C3—82no—
348C22×C3—84no—
349C22×C3—82no—
350C22×C3C4×C22⋊C411yescharacteristic
351C22×C3C4×C22⋊C412yes—
352C22×C3#38612yes—
353C22×C3#38611yescharacteristic
354C22×C3#38812yes—
355C22×C3#38812yes—
356C22×C3#38812yes—
357C22×C3#39011yescharacteristic
358C22×C3#39012yes—
359C3×C4—48no—
360C3×C4—44no—
361C3×C4—42no—
362C3×C4—82no—
363C3×C4—84no—
364C23—22no—
365C23—24no—
366C23—42no—
367C23—44no—
368C23—42no—
369C23—44no—
370C23—42no—
371C23—44no—
372C23—41no—
373C23—42no—
374C23—41no—
375C23—84no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—84no—
383C23C2×C3×C4211yescharacteristic
384C23C2×C3×C22⋊C411yescharacteristic
385C23C2×C3×C22⋊C412yes—
386C23C3×#11711yescharacteristic
387C23C3×#11712yes—
388C23C3×C4×D412yes—
389C23C3×C4×D412yes—
390C23C3×C4×D412yes—
391C23C3×C4×D412yes—
392C23C3×C4×D412yes—
393C23C3×C4×D412yes—
394C23C3×#12011yescharacteristic
395C23C3×#12112yes—
396C23C3×#12112yes—
397C23C3×#12112yes—
398C23C3×#12311yescharacteristic
399C23C3×#12312yes—
400C23C3×#12412yes—
401C23C3×#12411yescharacteristic
402C23C3×#12612yes—
403C23C3×#12711yescharacteristic
404D4—44no—
405D4—44no—
406D4—44no—
407D4—44no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—48no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C3—22no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—22no—
435C2×C3—22no—
436C2×C3—22no—
437C2×C3—82no—
438C2×C3#143711yescharacteristic
439C2×C3#191312yes—
440C22—22no—
441C22—22no—
442C22—24no—
443C22—22no—
444C22—24no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—42no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—44no—
462C22—42no—
463C22—84no—
464C22—82no—
465C22—84no—
466C22—82no—
467C22C3×C4×C22⋊C411yescharacteristic
468C22C3×C4×C22⋊C412yes—
469C22C3×#38612yes—
470C22C3×#38611yescharacteristic
471C22C3×#38812yes—
472C22C3×#38812yes—
473C22C3×#38812yes—
474C22C3×#39011yescharacteristic
475C22C3×#39012yes—
476C4—48no—
477C4—44no—
478C4—42no—
479C4—82no—
480C4—84no—
481C3#1093811yescharacteristic, direct product factor, Sylow
482C2—22no—
483C2—21no—
484C2—22no—
485C2—21no—
486C2—22no—
487C2—22no—
488C2—22no—
489C2—22no—
490C2—82no—
491C2C3×#143711yescharacteristic
492C2C3×#191312yes—
subgroupG/H
center, socleC22×C3#386
commutatorC23C2×C3×C42
Frattini, cosocleC25C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2273C2semidirect
4. C2×C3×#342C2semidirect
5. C2×C3×#342C2semidirect
6. #10938C3direct (N × H)
14. C2×C3×#99C4semidirect
17. C3×#381C4semidirect
18. C3×#379C4semidirect
25. #2273C2×C3semidirect
26. C2×#342C2×C3semidirect
27. C2×#342C2×C3semidirect
75. C2×#99C3×C4semidirect
78. #381C3×C4semidirect
79. #379C3×C4semidirect
90. C24×C3C42semidirect
135. C22×C3×C4C42semidirect
149. C3×C22⋊C4C22⋊C4semidirect
262. C24C3×C42semidirect
307. C22×C4C3×C42semidirect
321. C22⋊C4C3×C22⋊C4semidirect
481. C3#10938direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#992
C3×#23082
C3×#22732
C2×C3×#3422
#109383