Finite Group Databasebeta

← #13140905

Subgroups — #13140905

Order 1728.

609 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 123 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×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10591552—31nomaximal
9D3×#835—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3—22no—
12Q8×(C3⋊D3)⋊C3C414yescomplemented
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#634⋊C3—21no—
16C2×#634⋊C3C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4—21no—
18C4⋊#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88448—21no—
23((C3⋊D3)⋊C3)⋊C8—22no—
24C2×Q8×C3⋊D3D311yescharacteristic
25C2×C3×D3×Q8—31no—
26#63900—31no—
27D3×C4⋊Dic3—31no—
28C2×#3710—31no—
29C2×D3×C3⋊C8—31no—
30#63662D311yescharacteristic, complemented
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—61no—
35#63567—31no—
36#63567—31no—
37#63566—31no—
38#63565—31no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)C2×C412yescomplemented
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4×(C3⋊D3)⋊C3—24no—
49C4×(C3⋊D3)⋊C3C2×C412yes—
50C4×(C3⋊D3)⋊C3D412yes—
51#634⋊C3—21no—
52#634⋊C3—24no—
53#634⋊C3C2×C412yescomplemented
54((C3⋊D3)⋊C3)⋊C4—42no—
55#6180—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57C2×#835—91no—
58D3×#103—91no—
59D3×#103—91no—
60C2×C32×Q8—21no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C2×D311yescharacteristic
63Q8×C3⋊D3—22no—
64Q8×C3⋊D3Dic314yescomplemented
65C2×C4×C3⋊D3—21no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634—21no—
68C2×#634C2×D311yescharacteristic
69C3×D3×Q8—34no—
70C3×D3×Q8—62no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75C2×#624—61no—
76C2×D3×Dic3—61no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—61no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—61no—
87#3724—61no—
88Dic3⋊Dic3—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)—22no—
96C4×H(3,𝔽3)C22×C411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98#1084—41no—
99(C3⋊Dic3)⋊C3—22no—
100(C3⋊Dic3)⋊C3—41no—
101(C3⋊Dic3)⋊C3C22⋊C412yes—
102C22×C3×Q8—91no—
103C2×D3×Q8—31no—
104C2×D3×Q8—91no—
105C2×C4⋊Dic3—91no—
106C22×C3⋊C8—91no—
107C2×D3×C8—91no—
108C2×D3×C8—91no—
109D3×C4⋊C4—91no—
110D3×C4⋊C4—91no—
111C3×#103—91no—
112C3×#103—91no—
113#835—31no—
114#835—31no—
115#835—31no—
116#835—61no—
117#835—91no—
118#835—91no—
119#835—91no—
120#816—91no—
121#816—91no—
122#810—91no—
123#810—91no—
124C22×C3⋊D3#6711yescharacteristic
125C22×C3×D3—31no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—22no—
129C32×Q8—41no—
130C32×Q8C4×D312yescomplemented
131C32×Q8C2×Dic312yescomplemented
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3#6711yescharacteristic
141C4×C3⋊D3—24no—
142C4×C3⋊D3C2×Dic312yes—
143C4×C3⋊D3#6712yes—
144#634—21no—
145#634—24no—
146#634C2×Dic312yescomplemented
147C2×C3×Dic3—31no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—121no—
153C3×C4×D3—32no—
154C3×C4×D3—32no—
155C3×C4×D3—64no—
156C3×Dic6—32no—
157C3×Dic6—64no—
158C3×Dic6—61no—
159#624—122no—
160D3×Dic3—122no—
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—31no—
172C2×C3×Q8—31no—
173C2×C3×Q8—61no—
174C2×C3×Q8—91no—
175C2×C3×Q8—94no—
176C2×C3×Q8—181no—
177C2×C3×Q8—182no—
178C2×C3×Q8D3211yescharacteristic
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181C22×Dic3—181no—
182D3×Q8—34no—
183D3×Q8—62no—
184D3×Q8—94no—
185D3×Q8—182no—
186C2×C4×D3—31no—
187C2×C4×D3—61no—
188C2×C4×D3—91no—
189C2×C4×D3—181no—
190C2×C4×D3—181no—
191C2×C4×D3—181no—
192C2×Dic6—31no—
193C2×Dic6—61no—
194C2×Dic6—91no—
195C2×Dic6—181no—
196C2×C3×C8—91no—
197C2×C3×C8—91no—
198C3×C4⋊C4—91no—
199C3×C4⋊C4—91no—
200C4⋊Dic3—31no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—61no—
204C4⋊Dic3—91no—
205C4⋊Dic3—181no—
206Dic3⋊C4—181no—
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#21512yes—
220C2×C3⋊D3#21612yes—
221C2×C3⋊D3#21712yes—
222C2×C3×D3—32no—
223C2×C3×D3—32no—
224C2×C3×D3—32no—
225C32×C4—22no—
226C32×C4—22no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—42no—
230C32×C4C2×C4×D311yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4C22×Dic311yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C3⋊Dic3—22no—
235C3⋊Dic3—41no—
236C3⋊Dic3#21712yes—
237C3×Dic3—32no—
238C3×Dic3—62no—
239C3×Dic3—121no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—241no—
243C22×Q8—91no—
244C22×C8—271no—
245C2×C4⋊C4—271no—
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—21no—
255C3×Q8—32no—
256C3×Q8—32no—
257C3×Q8—61no—
258C3×Q8—61no—
259C3×Q8—62no—
260C3×Q8—92no—
261C3×Q8—121no—
262C3×Q8—181no—
263C3×Q8—184no—
264C3×Q8D3×Dic312yes—
265C2×C3×C4—21no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—91no—
274C2×C3×C4—121no—
275C2×C3×C4—184no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C2×D3211yescharacteristic
280C2×Dic3—31no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—91no—
286C2×Dic3—121no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289C2×Dic3—362no—
290C4×D3—32no—
291C4×D3—32no—
292C4×D3—64no—
293C4×D3—92no—
294C4×D3—92no—
295C4×D3—184no—
296C4×D3—362no—
297C4×D3—362no—
298Dic6—32no—
299Dic6—64no—
300Dic6—61no—
301Dic6—92no—
302Dic6—184no—
303Dic6—181no—
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×#21511yescharacteristic
318C2×C32C2×#21611yescharacteristic
319C2×C32C2×#21711yescharacteristic
320C3⋊D3#83514yescomplemented
321C3×D3—34no—
322C2×Q8—94no—
323C2×Q8—91no—
324C2×Q8—181no—
325C2×Q8—182no—
326C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
327C22×C4—91no—
328C22×C4—181no—
329C22×C4—541no—
330C2×C8—91no—
331C2×C8—91no—
332C2×C8—542no—
333C4⋊C4—91no—
334C4⋊C4—91no—
335C4⋊C4—541no—
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—22no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—31no—
353C3×C4—31no—
354C3×C4—62no—
355C3×C4—62no—
356C3×C4—61no—
357C3×C4—61no—
358C3×C4—92no—
359C3×C4—122no—
360C3×C4—182no—
361C3×C4—361no—
362C3×C4—361no—
363C3×C4C2×D3×Dic311yescharacteristic
364C3×C4D3×#6711yescharacteristic
365Dic3—32no—
366Dic3—62no—
367Dic3—92no—
368Dic3—121no—
369Dic3—121no—
370Dic3—121no—
371Dic3—182no—
372Dic3—241no—
373Dic3—361no—
374C32—21no—
375C32D3×#10311yescharacteristic, complemented
376C32C2×#83511yescharacteristic, complemented
377C23—91no—
378Q8—21no—
379Q8—92no—
380Q8—184no—
381Q8—181no—
382Q8((C3⋊D3)⋊C3)⋊C412yescomplemented
383C2×C4—21no—
384C2×C4—91no—
385C2×C4—92no—
386C2×C4—92no—
387C2×C4—184no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4—1082no—
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×#21511yescharacteristic
406C2×C3D3×#21611yescharacteristic
407C2×C3D3×#21711yescharacteristic
408D3—34no—
409D3—94no—
410C22—92no—
411C22—92no—
412C22—92no—
413C22#8867611yescharacteristic
414C4—22no—
415C4—92no—
416C4—182no—
417C4—361no—
418C4—361no—
419C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
420C4#8867611yescharacteristic
421C3—31no—
422C3—31no—
423C3—61no—
424C3D3×#83511yescharacteristic
425C2—94no—
426C2#1172123211yescharacteristic
427C2#1172123611yescharacteristic
428C2#1172131011yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #11720583C2semidirect
5. #63662⋊C3C2semidirect
6. #11720313C2semidirect
7. #11720309C2semidirect
12. Q8×(C3⋊D3)⋊C3C4semidirect
30. #63662D3semidirect
41. Q8×H(3,𝔽3)C2×C4semidirect
53. #634⋊C3C2×C4semidirect
64. Q8×C3⋊D3Dic3semidirect
130. C32×Q8C4×D3semidirect
131. C32×Q8C2×Dic3semidirect
146. #634C2×Dic3semidirect
170. (C3⋊D3)⋊C3#103semidirect
250. H(3,𝔽3)C2×#103semidirect
320. C3⋊D3#835semidirect
375. C32D3×#103semidirect
376. C32C2×#835semidirect
382. Q8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
#117211292
C2×((C3⋊D3)⋊C3)⋊C82
#117205832
#63662⋊C32
#117203132
#117203092
#105915523
D3×#8353