Finite Group Databasebeta

← #10592466⋊C3 #13141782

Subgroups — #10592466⋊C3 #13141782

Order 1728.

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

normal subgroups only

This group has 442 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
1#64145⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic
3C2×#3746⋊C3C211yesmaximal, characteristic, complemented
4#64096⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592466C311yesmaximal, characteristic, complemented
9C3×#4767—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3—21no—
16#3757⋊C3—21no—
17#3754⋊C3—22no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3746⋊C3—22no—
23#64145C2×C311yescharacteristic
24C2×#3754C2×C311yescharacteristic
25C2×#3746C2×C311yescharacteristic, complemented
26#64096C2×C311yescharacteristic, complemented
27C3×#936—31no—
28C2×C3×#214—31no—
29C2×C3×Dic12—31no—
30C3×#887—31no—
31C32×#102—21no—
32C32×#102D311yescharacteristic
33C32×#102C2×C311yescharacteristic
34#63641C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—31no—
37C3×#810—31no—
38C23×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47#634⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51C8×H(3,𝔽3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53C3×#464—91no—
54#4767—31no—
55#4767—91no—
56C2×C32×D4—21no—
57C2×C32×D4C2×D311yescharacteristic
58C2×C32×D4C22×C311yescharacteristic
59C2×#634C22×C311yescharacteristic
60C2×#634C22×C311yescharacteristic
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C22×C311yescharacteristic
69#3757—21no—
70#3757—21no—
71#3754—22no—
72#3750—21no—
73C4×C3⋊Dic3—21no—
74C4×C3⋊Dic3C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76#3746—22no—
77C3×#217—61no—
78C3×#217—61no—
79C3×#214—62no—
80C3×Dic3⋊C4—61no—
81C3×C4×Dic3—31no—
82C3×C4×Dic3—61no—
83C2×C3×C3⋊C8—31no—
84C3×Dic12—62no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92(C3⋊Dic3)⋊C3—22no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96C2×C3×Q16—91no—
97C2×C3×SD16—91no—
98C3×#124—91no—
99C3×C4×Q8—91no—
100#936—31no—
101#936—91no—
102C2×#214—31no—
103C2×#214—91no—
104C2×Dic12—31no—
105C2×Dic12—91no—
106#887—31no—
107#887—91no—
108C3×C4⋊C8—91no—
109C3×#103—91no—
110C3×#102—31no—
111C3×#102—31no—
112C3×#102—61no—
113C3×#102C3×D311yescharacteristic
114Dic3⋊C8—31no—
115Dic3⋊C8—91no—
116#810—31no—
117#810—91no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—41no—
121C32×D4—22no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—21no—
126C32×D4—41no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C23×C311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C3×D411yescharacteristic
136C2×C3⋊Dic3C3×D411yescharacteristic
137#634—21no—
138#634—21no—
139#634—21no—
140#634C3×D412yes—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C3×Dic6—32no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—41no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—61no—
154#464—91noSylow
155C2×H(3,𝔽3)—81no—
156C2×H(3,𝔽3)#12111yescharacteristic
157C2×H(3,𝔽3)#13511yescharacteristic
158C2×H(3,𝔽3)#13711yescharacteristic
159C2×C3×Q8—91no—
160C2×C3×Q8—91no—
161C2×C3×D4—31no—
162C2×C3×D4—31no—
163C2×C3×D4—61no—
164C2×C3×D4C2×C3×D311yescharacteristic
165C2×Dic6—31no—
166C2×Dic6—31no—
167C2×Dic6—91no—
168C2×Dic6—91no—
169C3×Q16—182no—
170C3×SD16—182no—
171C2×C3×C8—31no—
172C2×C3×C8—31no—
173C2×C3×C8—61no—
174C2×C3×C8—91no—
175C2×C3×C8C2×C3×D311yescharacteristic
176C3×C4⋊C4—31no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—181no—
180C3×C4⋊C4C2×C3×D311yescharacteristic
181C3×C22⋊C4—181no—
182C3×C22⋊C4—181no—
183C3×C42—91no—
184C3×C42—181no—
185#217—61no—
186#217—61no—
187#217—181no—
188#217—181no—
189#214—62no—
190#214—182no—
191Dic3⋊C4—61no—
192Dic3⋊C4—181no—
193C4×Dic3—31no—
194C4×Dic3—61no—
195C4×Dic3—91no—
196C4×Dic3—181no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—91no—
199Dic12—62no—
200Dic12—182no—
201C22×C32—21no—
202C22×C32—42no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—81no—
210C22×C32—81no—
211C22×C32D3×D411yescharacteristic
212C22×C32C2×C3×D411yescharacteristic
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—81no—
218C32×C4#23511yescharacteristic
219C32×C4D3×D411yescharacteristic
220C32×C4C2×C3×D411yescharacteristic
221C32×C4C3×#4111yescharacteristic
222C3⋊Dic3—22no—
223C3⋊Dic3—21no—
224C3⋊Dic3—21no—
225C3⋊Dic3—41no—
226C3×Dic3—62no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—121no—
230C2×Q16—91no—
231C2×SD16—91no—
232#124—91no—
233C4×Q8—91no—
234C4⋊C8—91no—
235#103—91no—
236#102(C3⋊D3)⋊C311yescharacteristic
237H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
238C23×C3—21no—
239C23×C3—61no—
240C23×C3—61no—
241C23×C3—121no—
242C3×Q8—92no—
243C3×Q8—181no—
244C3×Q8—181no—
245C3×Q8—181no—
246C3×D4—21no—
247C3×D4—21no—
248C3×D4—61no—
249C3×D4—61no—
250C3×D4—61no—
251C3×D4—61no—
252C3×D4—62no—
253C3×D4—121no—
254C2×C3×C4—21no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—121no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4C22×C3×D311yescharacteristic
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—91no—
271C2×Dic3—91no—
272C2×Dic3—181no—
273C2×Dic3—181no—
274Dic6—32no—
275Dic6—61no—
276Dic6—61no—
277Dic6—61no—
278Dic6—92no—
279Dic6—181no—
280Dic6—181no—
281Dic6—181no—
282C3×C8—21no—
283C3×C8—61no—
284C3×C8—61no—
285C3×C8—121no—
286C3×C8—181no—
287C3⋊C8—61no—
288C3⋊C8—181no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—82no—
293C2×C32—81no—
294C2×C32—81no—
295C2×C32#88411yescharacteristic
296C2×C32#91211yescharacteristic
297C2×C32#91511yescharacteristic
298C2×C32C3×#12111yescharacteristic
299C2×C32C3×#13511yescharacteristic
300C2×C32C3×#13711yescharacteristic
301C2×Q8—91no—
302C2×Q8—91no—
303C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
304Q16—182no—
305SD16—182no—
306C2×C8—91no—
307C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
308C4⋊C4—181no—
309C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
310C22⋊C4—181no—
311C22⋊C4—181no—
312C42—91no—
313C42—181no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—61no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—122no—
327C22×C3—241no—
328C22×C3—241no—
329C22×C3C3×D3×D411yescharacteristic
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—41no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—121no—
338C3×C4—121no—
339C3×C4—181no—
340C3×C4—181no—
341C3×C4—182no—
342C3×C4—241no—
343C3×C4—361no—
344C3×C4C3×#23511yescharacteristic
345C3×C4C3×D3×D411yescharacteristic
346Dic3—62no—
347Dic3—61no—
348Dic3—61no—
349Dic3—121no—
350Dic3—182no—
351Dic3—181no—
352Dic3—181no—
353Dic3—361no—
354C32—21no—
355C32#476711yescharacteristic, complemented
356C32C3×#46411yescharacteristic, complemented
357C23—21no—
358Q8—92no—
359Q8—181no—
360Q8—181no—
361Q8—181no—
362D4—21no—
363D4—21no—
364C2×C4—21no—
365C2×C4—91no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
370C8—21no—
371C8—181no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—81no—
382C2×C3—241no—
383C2×C3—241no—
384C2×C3—242no—
385C2×C3C3×#88411yescharacteristic
386C2×C3C3×#91211yescharacteristic
387C2×C3C3×#91511yescharacteristic
388C22—41no—
389C22—41no—
390C22—41no—
391C22D4×(C3⋊D3)⋊C311yescharacteristic
392C4—41no—
393C4—181no—
394C4—181no—
395C4—182no—
396C4—361no—
397C4#3828⋊C311yescharacteristic
398C4D4×(C3⋊D3)⋊C311yescharacteristic
399C3—31no—
400C3—31no—
401C3—61no—
402C3C3×#476711yescharacteristic
403C2—81no—
404C2#64093⋊C311yescharacteristic
405C2#64121⋊C311yescharacteristic
406C2#64124⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×H(3,𝔽3)C22
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
3. C2×#3746⋊C3C2semidirect
4. #64096⋊C3C2semidirect
6. #63641⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592466C3semidirect
25. C2×#3746C2×C3semidirect
26. #64096C2×C3semidirect
34. #63641C2×C3semidirect
35. #63637C2×C3semidirect
237. H(3,𝔽3)#464semidirect
355. C32#4767semidirect
356. C32C3×#464semidirect

Maximal subgroups

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

subgroupindex
#64145⋊C32
C2×#3754⋊C32
C2×#3746⋊C32
#64096⋊C32
H(3,𝔽3)×#1022
#63641⋊C32
#63637⋊C32
#105924663
C3×#47673