Finite Group Databasebeta

← C3×#10593402 #13166763

Subgroups — C3×#10593402 #13166763

Order 1728.

764 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3797C211yesmaximal, characteristic, complemented
2C2×C3×D32⋊C4C212yesmaximal
3C3×#63921C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic, complemented
5C3×#63737C212yesmaximal, complemented
6#10593402C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic
8C22×C3×#157C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6—21no—
11C2×C3×#627—21no—
12C2×C3×#624C2211yescharacteristic
13C2×C3×#624C2212yes—
14C3×#3797—24no—
15C3×D32⋊C4—24no—
16C32×(C2×D3)⋊C4—21no—
17C3×(C2×D3)⋊Dic3—21no—
18C2×C3×D3×Q8—61no—
19C2×#3797C2×C311yescharacteristic, complemented
20C2×D32⋊C4C2×C312yes—
21C3×D3×C22⋊C4—61no—
22#63921C2×C311yescharacteristic
23#63781C2×C311yescharacteristic, complemented
24#63737C2×C312yescomplemented
25C22×C3×C3⋊D3C2311yescharacteristic
26C22×C32×D3—21no—
27C2×C3×D32—22no—
28C2×C3×D32—22no—
29C2×C3×D32—21no—
30C2×C3×D32Q812yescomplemented
31C2×C3×#157—22no—
32C2×C3×#157—22no—
33C2×C3×#157—22no—
34C2×C33×C4D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—21no—
39C32×Dic6—24no—
40C3×#627—24no—
41C3×#624—22no—
42C3×#624—22no—
43C3×#624D414yes—
44C3×#393—91nomaximal
45C23×C3×D3—61no—
46C22×D32C22×C311yescharacteristic
47C22×#157C22×C312yes—
48C2×C32×Q8—61no—
49C2×C4×C3⋊D3C22×C311yescharacteristic
50C3×D3×Q8—68no—
51C2×C3×C4×D3—62no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—61no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—41no—
58C2×C3×Dic6—62no—
59C2×#627—21no—
60C2×#624C22×C311yescharacteristic
61C2×#624C22×C312yes—
62#3797—24no—
63D32⋊C4—24no—
64C32×C22⋊C4—61no—
65C3×#217—61no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—41no—
69C3×(C2×D3)⋊C4—61no—
70(C2×D3)⋊Dic3—21no—
71C22×C33C2×D411yescharacteristic
72C2×C3×C3⋊D3C2×D412yes—
73C2×C3×C3⋊D3C2×Q811yescharacteristic
74C2×C3×C3⋊D3#4112yes—
75C2×C3×C3⋊D3#4111yescharacteristic
76C2×C32×D3—22no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C3×D32—24no—
80C3×D32—42no—
81C3×#157—42no—
82C33×C4—21no—
83C3×C3⋊Dic3—21no—
84C32×Dic3—24no—
85C32×Dic3—41no—
86C22×C3×Q8—91no—
87C2×D3×Q8—61no—
88C2×C3×C22⋊C4—91no—
89C2×C3×C22⋊C4—92no—
90D3×C22⋊C4—61no—
91C3×#95—91no—
92C3×#95—92no—
93C23×C32—61no—
94C22×C3⋊D3C23×C311yescharacteristic
95C22×C3×D3—21no—
96C22×C3×D3—21no—
97C22×C3×D3—41no—
98C22×C3×D3—62no—
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C22×C3×D3—62no—
102C22×C3×D3—122no—
103C22×C3×D3—122no—
104C2×D32—21no—
105C2×D32—22no—
106C2×D32—22no—
107C2×D32C3×Q812yescomplemented
108C2×#157—22no—
109C2×#157—22no—
110C2×#157—22no—
111C32×Q8—64no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—62no—
117C2×C32×C4—61no—
118C2×C32×C4C3×D411yescharacteristic
119C2×C3⋊Dic3C3×D411yescharacteristic
120C4×C3⋊D3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—42no—
126C2×C3×Dic3—41no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131C3×C4×D3—64no—
132C3×C4×D3—122no—
133C3×C4×D3—122no—
134C3×Dic6—24no—
135C3×Dic6—24no—
136C3×Dic6—44no—
137C3×Dic6—68no—
138#627—24no—
139#624—22no—
140#624—22no—
141#624C3×D414yes—
142#393—91noSylow
143C2×C33#12011yescharacteristic
144C2×C33C22⋊Q811yescharacteristic
145C2×C33#12411yescharacteristic
146C3×C3⋊D3C22⋊Q812yes—
147C3×C3⋊D3#12412yes—
148C32×D3—42no—
149C24×C3—91no—
150C23×D3—61no—
151C2×C3×Q8—61no—
152C2×C3×Q8—61no—
153C2×C3×Q8—121no—
154C2×C3×Q8—181no—
155C2×C3×Q8—184no—
156C22×C3×C4—92no—
157C22×C3×C4—91no—
158C22×C3×C4—91no—
159C22×C3×C4—92no—
160D3×Q8—68no—
161C2×C4×D3—62no—
162C2×C4×D3—61no—
163C2×C4×D3—61no—
164C2×C4×D3—61no—
165C2×Dic6—21no—
166C2×Dic6—62no—
167C3×C22⋊C4—61no—
168C3×C22⋊C4—61no—
169C3×C22⋊C4—121no—
170C3×C22⋊C4—181no—
171C3×C22⋊C4—184no—
172#217—61no—
173(C2×D3)⋊C4—21no—
174(C2×D3)⋊C4—61no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—62no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32C2×C3×D411yescharacteristic
183C2×C3⋊D3C2×C3×D412yes—
184C2×C3⋊D3C2×C3×Q811yescharacteristic
185C2×C3⋊D3C3×#4112yes—
186C2×C3⋊D3C3×#4111yescharacteristic
187C2×C3×D3—22no—
188C2×C3×D3—22no—
189C2×C3×D3—42no—
190C2×C3×D3—41no—
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C2×C3×D3—41no—
194C2×C3×D3—62no—
195C2×C3×D3—62no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—62no—
199C2×C3×D3—61no—
200C2×C3×D3—62no—
201C2×C3×D3—61no—
202C2×C3×D3—81no—
203C2×C3×D3—81no—
204C2×C3×D3—124no—
205C2×C3×D3—124no—
206D32—24no—
207D32—42no—
208#157—42no—
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—41no—
212C32×C4—42no—
213C32×C4—64no—
214C32×C4—81no—
215C32×C4—121no—
216C3⋊Dic3—21no—
217C3×Dic3—24no—
218C3×Dic3—24no—
219C3×Dic3—44no—
220C3×Dic3—41no—
221C3×Dic3—41no—
222C3×Dic3—62no—
223C3×Dic3—81no—
224C3×Dic3—121no—
225C22×Q8—91no—
226C2×C22⋊C4—91no—
227C2×C22⋊C4—92no—
228#95—91no—
229#95—92no—
230C33#39311yescharacteristic, complemented, Sylow
231C23×C3—61no—
232C23×C3—61no—
233C23×C3—92no—
234C23×C3—91no—
235C23×C3—121no—
236C23×C3—181no—
237C23×C3—182no—
238C23×C3—182no—
239C22×D3—21no—
240C22×D3—62no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—62no—
244C22×D3—122no—
245C22×D3—122no—
246C3×Q8—64no—
247C3×Q8—64no—
248C3×Q8—124no—
249C3×Q8—184no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—94no—
259C2×C3×C4—91no—
260C2×C3×C4—122no—
261C2×C3×C4—121no—
262C2×C3×C4—182no—
263C2×C3×C4—182no—
264C2×C3×C4—182no—
265C2×C3×C4—182no—
266C2×C3×C4—182no—
267C2×C3×C4—182no—
268C2×C3×C4GO+(4,𝔽2)11yescharacteristic
269C2×Dic3—22no—
270C2×Dic3—21no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C4×D3—64no—
274C4×D3—64no—
275C4×D3—64no—
276C4×D3—122no—
277C4×D3—122no—
278Dic6—24no—
279Dic6—68no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—122no—
293C2×C32C3×#12011yescharacteristic
294C2×C32C3×C22⋊Q811yescharacteristic
295C2×C32C3×#12411yescharacteristic
296C3⋊D3C3×C22⋊Q812yes—
297C3⋊D3C3×#12412yes—
298C3×D3—42no—
299C3×D3—42no—
300C3×D3—62no—
301C3×D3—62no—
302C3×D3—62no—
303C3×D3—62no—
304C3×D3—82no—
305C24—91no—
306C2×Q8—61no—
307C2×Q8—181no—
308C2×Q8—184no—
309C22×C4—92no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4—92no—
313C22⋊C4—61no—
314C22⋊C4—181no—
315C22⋊C4—184no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—91no—
323C22×C3—91no—
324C22×C3—92no—
325C22×C3—92no—
326C22×C3—122no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—121no—
331C22×C3—184no—
332C22×C3—241no—
333C22×C3—241no—
334C22×C3—362no—
335C22×C3C2×GO+(4,𝔽2)11yescharacteristic
336C2×D3—22no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—62no—
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—62no—
347C2×D3—124no—
348C2×D3—124no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—41no—
352C3×C4—42no—
353C3×C4—64no—
354C3×C4—64no—
355C3×C4—81no—
356C3×C4—124no—
357C3×C4—121no—
358C3×C4—121no—
359C3×C4—181no—
360C3×C4—241no—
361C3×C4—362no—
362Dic3—24no—
363Dic3—41no—
364Dic3—62no—
365Dic3—121no—
366C32—21no—
367C32—21no—
368C32—41no—
369C32—41no—
370C32C3×#39311yescharacteristic, complemented
371C23—61no—
372C23—92no—
373C23—91no—
374C23—181no—
375C23—182no—
376C23—182no—
377Q8—64no—
378Q8—184no—
379C2×C4—62no—
380C2×C4—61no—
381C2×C4—94no—
382C2×C4—91no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4C3×GO+(4,𝔽2)11yescharacteristic
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3—92no—
403C2×C3—92no—
404C2×C3—122no—
405C2×C3—122no—
406C2×C3—242no—
407C2×C3D32⋊Q811yescharacteristic
408C2×C3#6422111yescharacteristic
409C2×C3#6424111yescharacteristic
410D3—42no—
411D3—62no—
412D3—62no—
413D3—62no—
414D3—62no—
415C22—62no—
416C22—92no—
417C22—91no—
418C22—91no—
419C22—92no—
420C22—121no—
421C22—121no—
422C22—184no—
423C22—362no—
424C22C2×C3×GO+(4,𝔽2)11yescharacteristic
425C4—21no—
426C4—64no—
427C4—121no—
428C4—181no—
429C4—362no—
430C3—21no—
431C3—21no—
432C3—41no—
433C3—41no—
434C3#1059340211yescharacteristic, direct product factor
435C2—92no—
436C2—92no—
437C2—122no—
438C2C3×D32⋊Q811yescharacteristic
439C2C3×#6422111yescharacteristic
440C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×D32C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#393
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3797C2semidirect
4. C3×#63781C2semidirect
5. C3×#63737C2semidirect
6. #10593402C3direct (N × H)
19. C2×#3797C2×C3semidirect
23. #63781C2×C3semidirect
24. #63737C2×C3semidirect
30. C2×C3×D32Q8semidirect
107. C2×D32C3×Q8semidirect
230. C33#393semidirect
370. C32C3×#393semidirect
434. C3#10593402direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×C3×D32⋊C42
C3×#639212
C3×#637812
C3×#637372
#105934023
C3×#3939