Finite Group Databasebeta

← #10592473⋊C3 #13141791

Subgroups — #10592473⋊C3 #13141791

Order 1728.

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

normal subgroups only

This group has 449 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#64147⋊C3C211yesmaximal, characteristic
2C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#63645⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592473C311yesmaximal, characteristic, complemented
9C3×#4774—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64147C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
23C3×#938—31no—
24C3×D3×C4⋊C4—31no—
25C32×#102—21no—
26C32×#102D311yescharacteristic, complemented
27C32×#102C2×C311yescharacteristic, complemented
28#63659C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63645C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×#832—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#818—31no—
35C3×#807—31no—
36C23×H(3,𝔽3)—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#638⋊C3—42no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—42no—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51C3×#480—91no—
52#4774—31no—
53#4774—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C22×C311yescharacteristic
57C2×#638—21no—
58C2×C4×C3⋊D3—21no—
59C2×C4×C3⋊D3C22×C311yescharacteristic
60C2×C3×#67—61no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—61no—
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—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72#3750—21no—
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×#217—61no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×Dic3⋊C4—61no—
78C2×C3×C3⋊C8—31no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)—41no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—21no—
85C2×(C3⋊D3)⋊C3D812yes—
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)#4111yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92C3×#121—91no—
93C2×C3×C4⋊C4—91no—
94#938—31no—
95#938—91no—
96D3×C4⋊C4—31no—
97D3×C4⋊C4—91no—
98C3×#107—91no—
99C3×#107—91no—
100C3×#102—31no—
101C3×#102—31no—
102C3×#102—61no—
103C3×#102—91no—
104C3×#102C3×D311yescharacteristic
105C3×C22⋊C8—91no—
106#832—31no—
107#832—91no—
108(C2×D3)⋊C8—31no—
109(C2×D3)⋊C8—91no—
110#818—31no—
111#818—91no—
112#807—31no—
113#807—91no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C22×C3⋊D3C3×D411yescharacteristic
118C22×C3×D3—31no—
119C32×D4—22no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—41no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C23×C311yescharacteristic
131#638—42no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3C3×D411yescharacteristic
135C4×C3⋊D3—21no—
136C4×C3⋊D3—21no—
137C4×C3⋊D3—42no—
138C3×#67—122no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—122no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—41no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—61no—
150#480—91noSylow
151C2×H(3,𝔽3)—81no—
152C2×H(3,𝔽3)#12311yescharacteristic
153C2×H(3,𝔽3)C2×D811yescharacteristic
154C2×H(3,𝔽3)#13711yescharacteristic
155(C3⋊D3)⋊C3—22no—
156C2×C3×D4—31no—
157C2×C3×D4—31no—
158C2×C3×D4—61no—
159C2×C3×D4—181no—
160C2×C3×D4C2×C3×D311yescharacteristic
161C22×C3×C4—91no—
162C22×C3×C4—181no—
163C2×#67—61no—
164C2×#67—181no—
165C2×C4×D3—31no—
166C2×C4×D3—61no—
167C2×C4×D3—91no—
168C2×C4×D3—181no—
169C2×C3×C8—31no—
170C2×C3×C8—31no—
171C2×C3×C8—61no—
172C2×C3×C8—91no—
173C2×C3×C8C2×C3×D311yescharacteristic
174C3×C4⋊C4—31no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—181no—
180C3×C4⋊C4C2×C3×D311yescharacteristic
181C3×C22⋊C4—181no—
182#217—61no—
183#217—181no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—91no—
187C4⋊Dic3—91no—
188Dic3⋊C4—61no—
189Dic3⋊C4—181no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32—42no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—81no—
201C22×C32—81no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×C3×D411yescharacteristic
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3C3×D812yes—
207C2×C3×D3—32no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4—81no—
215C32×C4#23511yescharacteristic
216C32×C4#23711yescharacteristic
217C32×C4C3×#4111yescharacteristic
218C32×C4C3×#4111yescharacteristic
219C3⋊Dic3—21no—
220C3⋊Dic3—41no—
221C3⋊Dic3—41no—
222C3×Dic3—61no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225#121—91no—
226C2×C4⋊C4—91no—
227#107—91no—
228#107—91no—
229#102—91no—
230#102(C3⋊D3)⋊C311yescharacteristic, complemented
231C22⋊C8—91no—
232H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—91no—
237C23×C3—121no—
238C22×D3—31no—
239C22×D3—91no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—61no—
245C3×D4—61no—
246C3×D4—62no—
247C3×D4—121no—
248C3×D4—362no—
249C2×C3×C4—21no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—91no—
256C2×C3×C4—121no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—362no—
262C2×C3×C4C22×C3×D311yescharacteristic
263#67—122no—
264#67—362no—
265C2×Dic3—31no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—91no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—122no—
274C4×D3—181no—
275C4×D3—181no—
276C4×D3—362no—
277C3×C8—21no—
278C3×C8—61no—
279C3×C8—61no—
280C3×C8—121no—
281C3×C8—181no—
282C3⋊C8—61no—
283C3⋊C8—181no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—82no—
288C2×C32—81no—
289C2×C32—81no—
290C2×C32#88311yescharacteristic
291C2×C32D3×D811yescharacteristic
292C2×C32#91511yescharacteristic
293C2×C32C3×#12311yescharacteristic
294C2×C32C2×C3×D811yescharacteristic
295C2×C32C3×#13711yescharacteristic
296C3⋊D3—22no—
297C3×D3—62no—
298C2×D4—181no—
299C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
300C22×C4—91no—
301C22×C4—181no—
302C2×C8—91no—
303C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
308C22⋊C4—181no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—61no—
315C22×C3—92no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3—122no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—241no—
326C22×C3—241no—
327C22×C3C3×D3×D411yescharacteristic
328C2×D3—32no—
329C2×D3—61no—
330C2×D3—61no—
331C2×D3—92no—
332C2×D3—181no—
333C2×D3—181no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—41no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—121no—
342C3×C4—121no—
343C3×C4—181no—
344C3×C4—241no—
345C3×C4—361no—
346C3×C4—361no—
347C3×C4C3×#23511yescharacteristic
348C3×C4C3×#23711yescharacteristic
349Dic3—61no—
350Dic3—121no—
351Dic3—121no—
352Dic3—181no—
353Dic3—361no—
354Dic3—361no—
355C32—21no—
356C32#477411yescharacteristic, complemented
357C32C3×#48011yescharacteristic, complemented
358C23—21no—
359C23—91no—
360D4—21no—
361D4—21no—
362D4—362no—
363C2×C4—21no—
364C2×C4—91no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—362no—
370C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
371C8—21no—
372C8—181no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—81no—
383C2×C3—182no—
384C2×C3—241no—
385C2×C3—241no—
386C2×C3—242no—
387C2×C3C3×#88311yescharacteristic
388C2×C3C3×D3×D811yescharacteristic
389C2×C3C3×#91511yescharacteristic
390D3—62no—
391D3—182no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—92no—
396C22—181no—
397C22—181no—
398C22D4×(C3⋊D3)⋊C311yescharacteristic
399C4—41no—
400C4—181no—
401C4—361no—
402C4—361no—
403C4#3828⋊C311yescharacteristic
404C4#3830⋊C311yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×#477411yescharacteristic
409C2—81no—
410C2—182no—
411C2#64092⋊C311yescharacteristic
412C2D8×(C3⋊D3)⋊C311yescharacteristic
413C2#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 radical#64147⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#480
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
2. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#102C2semidirect
4. #63659⋊C3C2semidirect
5. #63647⋊C3C2semidirect
6. #63645⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592473C3semidirect
22. C4⋊C4×C3⋊D3C2×C3semidirect
26. C32×#102D3semidirect
27. C32×#102C2×C3semidirect
28. #63659C2×C3semidirect
29. #63647C2×C3semidirect
30. #63645C2×C3semidirect
31. #63634C2×C3semidirect
230. #102(C3⋊D3)⋊C3semidirect
232. H(3,𝔽3)#480semidirect
356. C32#4774semidirect
357. C32C3×#480semidirect

Maximal subgroups

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

subgroupindex
#64147⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63647⋊C32
#63645⋊C32
#63634⋊C32
#105924733
C3×#47743