Finite Group Databasebeta

← C3×#9519 #10628126

Subgroups — C3×#9519 #10628126

Order 768.

644 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3004C211yesmaximal, characteristic
2C3×#2976C211yesmaximal, characteristic
3C3×#1787C211yesmaximal, characteristic, complemented
4C3×#1784C211yesmaximal, characteristic, complemented
5C3×#1513C211yesmaximal, characteristic
6C3×#1305C211yesmaximal, characteristic, complemented
7C3×#1293C211yesmaximal, characteristic, complemented
8#9519C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×#137—21no—
11C3×#525C2211yescharacteristic
12C3×#451—21no—
13C3×#451C412yescomplemented
14C3×#448—21no—
15C3×#448C412yescomplemented
16C3×#436—21no—
17C3×#435—21no—
18C3×#435—21no—
19C3×C8×D4—21no—
20C3×#433—21no—
21C3×#419—21no—
22C2×C3×#103—21no—
23C2×C3×#103—21no—
24C2×C3×#101—21no—
25C2×C3×#101—21no—
26C3×#408—21no—
27C3×#407C2211yescharacteristic
28C3×#407C2211yescharacteristic
29C3×#407C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C3×#328—21no—
33C3×#327—21no—
34#3004C2×C311yescharacteristic
35#2976C2×C311yescharacteristic
36#1787C2×C311yescharacteristic, complemented
37#1784C2×C311yescharacteristic, complemented
38#1513C2×C311yescharacteristic
39#1305C2×C311yescharacteristic, complemented
40#1293C2×C311yescharacteristic, complemented
41C2×C3×#41D411yescharacteristic
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8—21no—
47C22×C3×Q8—42no—
48C22×C3×Q8—42no—
49C22×C3×Q8C2×C412yes—
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C3×#137—42no—
53C3×#137—42no—
54C2×C3×Q16—22no—
55C2×C3×SD16—22no—
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16D411yescharacteristic
60C2×C3×M16D411yescharacteristic
61C22×C3×C8—21no—
62C22×C3×C8D411yescharacteristic
63C22×C3×C8D411yescharacteristic
64C3×#123—21no—
65C3×C22⋊Q8C2×C411yescharacteristic
66C3×C22⋊Q8D411yescharacteristic
67C3×#121C2×C411yescharacteristic
68C3×#121D411yescharacteristic
69C3×#120—21no—
70C3×C4×D4—21no—
71C3×C4×D4—21no—
72C3×#117D411yescharacteristic
73C2×C3×C4⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×#103—22no—
78C3×#103—22no—
79C3×#103—42no—
80C3×#103—42no—
81C3×#101—42no—
82C3×#101—42no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3×C22⋊C8—21no—
87C3×C22⋊C8—21no—
88C3×C22⋊C8—21no—
89C3×C22⋊C8—21no—
90C3×C22⋊C8—21no—
91C3×C22⋊C8—21no—
92C3×C22⋊C8C2×C412yes—
93C3×#97—21no—
94C3×C4×C8—21no—
95C3×#95—21no—
96C23×Q8C22×C311yescharacteristic
97C2×#137—21no—
98#525C22×C311yescharacteristic
99#451—21no—
100#451C3×C412yescomplemented
101#448—21no—
102#448C3×C412yescomplemented
103#436—21no—
104#435—21no—
105#435—21no—
106C8×D4—21no—
107#433—21no—
108#419—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#101—21no—
112C2×#101—21no—
113#408—21no—
114#407C22×C311yescharacteristic
115#407C22×C311yescharacteristic
116#407C22×C311yescharacteristic
117C2×C22⋊C8C22×C311yescharacteristic
118#380C22×C311yescharacteristic
119#328—21no—
120#327—21no—
121C24×C3#4111yescharacteristic
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—44no—
133C2×C3×Q8—41no—
134C2×C3×Q8—41no—
135C2×C3×Q8—41no—
136C2×C3×Q8—41no—
137C2×C3×Q8—41no—
138C2×C3×Q8—41no—
139C2×C3×Q8—44no—
140C2×C3×Q8—44no—
141C2×C3×Q8—41no—
142C2×C3×Q8—41no—
143C2×C3×Q8C22⋊C411yescharacteristic
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4C22⋊C411yescharacteristic
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—44no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4—41no—
156C22×C3×C4C22×C411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4#4111yescharacteristic
162C3×Q16—44no—
163C3×SD16—44no—
164C3×M16—41no—
165C3×M16—41no—
166C3×M16—41no—
167C3×M16—41no—
168C3×M16—41no—
169C3×M16—41no—
170C3×M16—81no—
171C2×C3×C8—22no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—41no—
177C2×C3×C8—41no—
178C2×C3×C8—41no—
179C2×C3×C8—81no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—41no—
183C3×C4⋊C4—41no—
184C3×C4⋊C4C22⋊C411yescharacteristic
185C3×C4⋊C4C22⋊C411yescharacteristic
186C3×C22⋊C4—21no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—21no—
190C3×C22⋊C4—21no—
191C3×C22⋊C4—81no—
192C3×C42—21no—
193C2×#41C3×D411yescharacteristic
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8C2×C3×C412yes—
202C23×C4—21no—
203C23×C4C23×C311yescharacteristic
204#137—42no—
205#137—42no—
206C2×Q16—22no—
207C2×SD16—22no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16C3×D411yescharacteristic
212C2×M16C3×D411yescharacteristic
213C22×C8—21no—
214C22×C8C3×D411yescharacteristic
215C22×C8C3×D411yescharacteristic
216#123—21no—
217C22⋊Q8C2×C3×C411yescharacteristic
218C22⋊Q8C3×D411yescharacteristic
219#121C2×C3×C411yescharacteristic
220#121C3×D411yescharacteristic
221#120—21no—
222C4×D4—21no—
223C4×D4—21no—
224#117C3×D411yescharacteristic
225C2×C4⋊C4—21no—
226C2×C22⋊C4—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229#103—22no—
230#103—22no—
231#103—42no—
232#103—42no—
233#101—42no—
234#101—42no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8C2×C3×C412yes—
245#97—21no—
246C4×C8—21no—
247#95—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3C4×D411yescharacteristic
254C23×C3#12111yescharacteristic
255C23×C3#12311yescharacteristic
256C3×Q8—24no—
257C3×Q8—24no—
258C3×Q8—41no—
259C3×Q8—42no—
260C3×Q8—42no—
261C3×Q8—42no—
262C3×Q8—42no—
263C3×Q8—42no—
264C3×Q8—42no—
265C3×D4—41no—
266C3×D4—41no—
267C3×D4—41no—
268C3×D4—81no—
269C3×D4—81no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—81no—
292C2×C3×C4—81no—
293C2×C3×C4—81no—
294C2×C3×C4C2×C22⋊C411yescharacteristic
295C2×C3×C4C4×D411yescharacteristic
296C2×C3×C4#12011yescharacteristic
297C2×C3×C4#12111yescharacteristic
298C3×C8—42no—
299C3×C8—41no—
300C3×C8—41no—
301C3×C8—41no—
302C3×C8—41no—
303C3×C8—81no—
304C24C3×#4111yescharacteristic
305#41—41no—
306#41—41no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—44no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8C3×C22⋊C411yescharacteristic
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4C3×C22⋊C411yescharacteristic
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—44no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C22×C3×C411yescharacteristic
340C22×C4C2×C3×D411yescharacteristic
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C3×#4111yescharacteristic
345Q16—44no—
346SD16—44no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—81no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—81no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C3×C22⋊C411yescharacteristic
368C4⋊C4C3×C22⋊C411yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—81no—
375C42—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3#38611yescharacteristic
390C22×C3#47111yescharacteristic
391C22×C3#47311yescharacteristic
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—44no—
396C3×C4—41no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—161no—
400C3×C4B(4,𝔽2)11yescharacteristic
401C3×C4B(4,𝔽2)11yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—41no—
406C23—41no—
407C23C3×C4×D411yescharacteristic
408C23C3×#12111yescharacteristic
409C23C3×#12311yescharacteristic
410Q8—24no—
411Q8—24no—
412Q8—41no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—81no—
423D4—81no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4C2×C3×C22⋊C411yescharacteristic
449C2×C4C3×C4×D411yescharacteristic
450C2×C4C3×#12011yescharacteristic
451C2×C4C3×#12111yescharacteristic
452C8—42no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—41no—
457C8—81no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—81no—
463C2×C3#188911yescharacteristic
464C2×C3#190211yescharacteristic
465C2×C3#190811yescharacteristic
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22C3×#38611yescharacteristic
480C22C3×#47111yescharacteristic
481C22C3×#47311yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—44no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—161no—
490C4C3×B(4,𝔽2)11yescharacteristic
491C4C3×B(4,𝔽2)11yescharacteristic
492C3#951911yescharacteristic, direct product factor, Sylow
493C2—21no—
494C2—21no—
495C2—41no—
496C2—41no—
497C2—81no—
498C2C3×#188911yescharacteristic
499C2C3×#190211yescharacteristic
500C2C3×#190811yescharacteristic
subgroupG/H
center, socleC22×C3#386
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC3×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1787C2semidirect
4. C3×#1784C2semidirect
6. C3×#1305C2semidirect
7. C3×#1293C2semidirect
8. #9519C3direct (N × H)
13. C3×#451C4semidirect
15. C3×#448C4semidirect
36. #1787C2×C3semidirect
37. #1784C2×C3semidirect
39. #1305C2×C3semidirect
40. #1293C2×C3semidirect
100. #451C3×C4semidirect
102. #448C3×C4semidirect
492. C3#9519direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30042
C3×#29762
C3×#17872
C3×#17842
C3×#15132
C3×#13052
C3×#12932
#95193