Finite Group Databasebeta

← #13138061

Subgroups — #13138061

Order 1728.

609 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 111 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⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic
2C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
4#11720532C211yesmaximal, characteristic, complemented
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590601—31nomaximal
9D3×#810—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×((C3⋊D3)⋊C3)⋊C4—21no—
15C2×#6196—21no—
16((C3⋊D3)⋊C3)⋊Q8—22no—
17((C3⋊D3)⋊C3)⋊Q8C414yescomplemented
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21#3750⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23((C3⋊D3)⋊C3)⋊C8—22no—
24C2×#3797—31no—
25C2×D3×Dic6—31no—
26C4⋊C4×C3⋊D3D311yescharacteristic
27C3×D3×C4⋊C4—31no—
28C2×#3710—31no—
29C2×D3×C3⋊C8—31no—
30#63637D311yescharacteristic, complemented
31C3×#810—31no—
32C3×#810—31no—
33C3×#810—31no—
34C3×#810—61no—
35#63567—31no—
36#63566—31no—
37#63566—31no—
38#63565—31no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43#6229—21no—
44#6229C2×C412yescomplemented
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4×(C3⋊D3)⋊C3—42no—
49C4×(C3⋊D3)⋊C3C2×C412yes—
50C4×(C3⋊D3)⋊C3D412yes—
51#634⋊C3—21no—
52#634⋊C3C2×C412yescomplemented
53((C3⋊D3)⋊C3)⋊C4—24no—
54#6196—24no—
55#6180—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57C2×#810—91no—
58D3×#103—91no—
59D3×#103—91no—
60C2×C4×C3⋊D3—21no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×#627—61no—
70C2×#627—61no—
71C2×#624—61no—
72C2×D3×Dic3—61no—
73#3797—34no—
74#3797—62no—
75D3×Dic6—34no—
76D3×Dic6—62no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—21no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C3×C4⋊Dic3—31no—
84C3×Dic3⋊C4—61no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—61no—
89#3710—62no—
90D3×C3⋊C8—62no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3C22⋊C412yes—
93C2×(C3⋊D3)⋊C3SD1612yes—
94C2×(C3⋊D3)⋊C3Q1612yes—
95C4×H(3,𝔽3)—41no—
96C4×H(3,𝔽3)C22×C411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98#1084—22no—
99(C3⋊Dic3)⋊C3—22no—
100(C3⋊Dic3)⋊C3—41no—
101(C3⋊Dic3)⋊C3C22⋊C412yes—
102C2×D3×Q8—91no—
103C2×D3×Q8—91no—
104C22×Dic6—91no—
105C2×C3×C4⋊C4—91no—
106C22×C3⋊C8—91no—
107C2×D3×C8—91no—
108C2×D3×C8—91no—
109D3×C4⋊C4—31no—
110D3×C4⋊C4—91no—
111C3×#103—91no—
112C3×#103—91no—
113#835—91no—
114#835—91no—
115#816—91no—
116#816—91no—
117#810—31no—
118#810—31no—
119#810—31no—
120#810—61no—
121#810—91no—
122#810—91no—
123#810—91no—
124C22×C3⋊D3#6711yescharacteristic
125C22×C3×D3—31no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3#6711yescharacteristic
135C4×C3⋊D3—42no—
136C4×C3⋊D3C4×D312yes—
137C4×C3⋊D3D1212yes—
138#634—21no—
139#634C4×D312yescomplemented
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—121no—
146C3×C4×D3—32no—
147C3×C4×D3—32no—
148C3×C4×D3—122no—
149C3×Dic6—32no—
150C3×Dic6—32no—
151C3×Dic6—32no—
152C3×Dic6—62no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156C3×Dic6—121no—
157#627—64no—
158#627—64no—
159#624—64no—
160D3×Dic3—64no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165C3×C3⋊C8—121no—
166C2×#103—271noSylow
167C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
168C2×H(3,𝔽3)C2×SD1611yescharacteristic
169C2×H(3,𝔽3)C2×Q1611yescharacteristic
170(C3⋊D3)⋊C3#10314yescomplemented
171C2×C3×Q8—91no—
172C2×C3×Q8—91no—
173C22×C3×C4—91no—
174C22×C3×C4—181no—
175C22×Dic3—181no—
176D3×Q8—94no—
177D3×Q8—94no—
178D3×Q8—182no—
179D3×Q8—182no—
180C2×C4×D3—31no—
181C2×C4×D3—61no—
182C2×C4×D3—91no—
183C2×C4×D3—181no—
184C2×C4×D3—181no—
185C2×C4×D3—181no—
186C2×Dic6—31no—
187C2×Dic6—31no—
188C2×Dic6—31no—
189C2×Dic6—61no—
190C2×Dic6—91no—
191C2×Dic6—94no—
192C2×Dic6—182no—
193C2×Dic6—181no—
194C2×Dic6—181no—
195C2×Dic6—181no—
196C2×C3×C8—91no—
197C2×C3×C8—91no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—91no—
202C3×C4⋊C4—181no—
203C3×C4⋊C4D3211yescharacteristic
204C4⋊Dic3—31no—
205C4⋊Dic3—91no—
206Dic3⋊C4—61no—
207Dic3⋊C4—181no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—61no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—182no—
214D3×C8—182no—
215D3×C8—182no—
216C22×C32—21no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×#6711yescharacteristic
219C2×C3⋊D3(C2×D3)⋊C412yes—
220C2×C3⋊D3#21412yes—
221C2×C3⋊D3#21612yes—
222C2×C3×D3—32no—
223C2×C3×D3—32no—
224C2×C3×D3—32no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—81no—
230C32×C4C2×C4×D311yescharacteristic
231C32×C4C2×C4×D311yescharacteristic
232C32×C4C2×D1211yescharacteristic
233C32×C4D3×D411yescharacteristic
234C3⋊Dic3—22no—
235C3⋊Dic3—41no—
236C3⋊Dic3(C2×D3)⋊C412yes—
237C3×Dic3—32no—
238C3×Dic3—62no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—122no—
242C3×Dic3—121no—
243C22×Q8—271no—
244C22×C8—271no—
245C2×C4⋊C4—91no—
246#103—91no—
247#103—91no—
248#103—271no—
249#103—271no—
250H(3,𝔽3)C2×#10311yescharacteristic, complemented, Sylow
251C23×C3—91no—
252C22×D3—31no—
253C22×D3—91no—
254C3×Q8—92no—
255C3×Q8—92no—
256C3×Q8—181no—
257C3×Q8—181no—
258C2×C3×C4—21no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—92no—
265C2×C3×C4—92no—
266C2×C3×C4—91no—
267C2×C3×C4—121no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—362no—
272C2×C3×C4C2×D3211yescharacteristic
273C2×Dic3—31no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—91no—
279C2×Dic3—121no—
280C2×Dic3—181no—
281C2×Dic3—181no—
282C2×Dic3—184no—
283C4×D3—32no—
284C4×D3—32no—
285C4×D3—92no—
286C4×D3—92no—
287C4×D3—122no—
288C4×D3—184no—
289C4×D3—184no—
290C4×D3—362no—
291Dic6—32no—
292Dic6—32no—
293Dic6—32no—
294Dic6—62no—
295Dic6—61no—
296Dic6—61no—
297Dic6—61no—
298Dic6—92no—
299Dic6—121no—
300Dic6—181no—
301Dic6—184no—
302Dic6—184no—
303Dic6—184no—
304C3×C8—181no—
305C3×C8—181no—
306C3⋊C8—61no—
307C3⋊C8—61no—
308C3⋊C8—61no—
309C3⋊C8—121no—
310C3⋊C8—181no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32D3×C22⋊C411yescharacteristic
315C2×C32D3×SD1611yescharacteristic
316C2×C32D3×Q1611yescharacteristic
317C2×C32C2×(C2×D3)⋊C411yescharacteristic
318C2×C32C2×#21411yescharacteristic
319C2×C32C2×#21611yescharacteristic
320C3⋊D3#81014yescomplemented
321C3×D3—34no—
322C2×Q8—91no—
323C2×Q8—91no—
324C2×Q8—274no—
325C2×Q8—541no—
326C2×Q8—542no—
327C22×C4—91no—
328C22×C4—181no—
329C22×C4—541no—
330C2×C8—91no—
331C2×C8—91no—
332C2×C8—542no—
333C4⋊C4—91no—
334C4⋊C4—181no—
335C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—61no—
339C22×C3—92no—
340C22×C3—92no—
341C22×C3—92no—
342C22×C3D3×#6711yescharacteristic
343C2×D3—32no—
344C2×D3—32no—
345C2×D3—32no—
346C2×D3—92no—
347C2×D3—92no—
348C2×D3—92no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—31no—
353C3×C4—41no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—92no—
357C3×C4—121no—
358C3×C4—121no—
359C3×C4—182no—
360C3×C4—182no—
361C3×C4—241no—
362C3×C4—361no—
363C3×C4C4×D3211yescharacteristic
364C3×C4D3×D1211yescharacteristic
365Dic3—32no—
366Dic3—62no—
367Dic3—62no—
368Dic3—62no—
369Dic3—92no—
370Dic3—122no—
371Dic3—121no—
372Dic3—182no—
373Dic3—361no—
374C32—21no—
375C32D3×#10311yescharacteristic, complemented
376C32C2×#81011yescharacteristic, complemented
377C23—91no—
378Q8—92no—
379Q8—92no—
380Q8—181no—
381Q8—181no—
382Q8—544no—
383C2×C4—21no—
384C2×C4—91no—
385C2×C4—92no—
386C2×C4—92no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—362no—
391C2×C4—544no—
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C8—181no—
394C8—181no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—94no—
405C2×C3D3×(C2×D3)⋊C411yescharacteristic
406C2×C3D3×#21411yescharacteristic
407C2×C3D3×#21611yescharacteristic
408D3—34no—
409D3—94no—
410C22—92no—
411C22—92no—
412C22—92no—
413C22#8867611yescharacteristic
414C4—41no—
415C4—92no—
416C4—182no—
417C4—182no—
418C4—361no—
419C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
420C4#8865411yescharacteristic
421C3—31no—
422C3—31no—
423C3—61no—
424C3D3×#81011yescharacteristic
425C2—94no—
426C2#1172116011yescharacteristic
427C2#1172119811yescharacteristic
428C2#1172123611yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#103
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11720532C2semidirect
5. #63637⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720313C2semidirect
17. ((C3⋊D3)⋊C3)⋊Q8C4semidirect
30. #63637D3semidirect
44. #6229C2×C4semidirect
52. #634⋊C3C2×C4semidirect
139. #634C4×D3semidirect
170. (C3⋊D3)⋊C3#103semidirect
250. H(3,𝔽3)C2×#103semidirect
320. C3⋊D3#810semidirect
375. C32D3×#103semidirect
376. C32C2×#810semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
C4⋊C4×(C3⋊D3)⋊C32
C2×((C3⋊D3)⋊C3)⋊C82
#117205322
#63637⋊C32
#117203262
#117203132
#105906013
D3×#8103