Finite Group Databasebeta

← #10592486⋊C3 #13141778

Subgroups — #10592486⋊C3 #13141778

Order 1728.

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

normal subgroups only

This group has 484 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, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3C2×#3744⋊C3C211yesmaximal, characteristic, complemented
4#64102⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592486C311yesmaximal, characteristic, complemented
9C3×#4787—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×#634⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—21no—
17#3757⋊C3—21no—
18#3753⋊C3—22no—
19#3752⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3744⋊C3—22no—
23#64145C2×C311yescharacteristic, complemented
24C2×#3753C2×C311yescharacteristic
25C2×#3744C2×C311yescharacteristic, complemented
26#64102C2×C311yescharacteristic, complemented
27C3×#936—31no—
28C2×C3×#213—31no—
29C2×C3×#204—31no—
30C3×#893—31no—
31C32×#102—21no—
32C32×#102D311yescharacteristic, complemented
33C32×#102C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—31no—
37C3×#810—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D412yes—
49C4×(C3⋊D3)⋊C3—42no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#463—91no—
55#4787—31no—
56#4787—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C22×C311yescharacteristic
60C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
61C2×C4×C3⋊D3—21no—
62C2×#634C22×C311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—31no—
66C2×C32×C8—21no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C32×C8C22×C311yescharacteristic
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C4⋊C4C22×C311yescharacteristic
72#3757—21no—
73#3757—21no—
74#3753—22no—
75#3752—21no—
76C4×C3⋊Dic3C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3744—22no—
79C3×#217—61no—
80C3×#217—61no—
81C3×#213—62no—
82C3×(C2×D3)⋊C4—61no—
83C3×C4×Dic3—31no—
84C2×C3×C3⋊C8—31no—
85C3×#204—62no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101C3×#124—91no—
102C3×C4×D4—91no—
103#936—31no—
104#936—91no—
105C2×#213—31no—
106C2×#213—91no—
107C2×#204—31no—
108C2×#204—91no—
109#893—31no—
110#893—91no—
111C3×C4⋊C8—91no—
112C3×#103—91no—
113C3×#102—31no—
114C3×#102—31no—
115C3×#102—61no—
116C3×#102C3×D311yescharacteristic
117Dic3⋊C8—31no—
118Dic3⋊C8—91no—
119#810—31no—
120#810—91no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C22×C3⋊D3—21no—
125C22×C3×D3—61no—
126C32×D4—22no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—41no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C23×C311yescharacteristic
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3C3×D411yescharacteristic
140C2×C3⋊Dic3C3×D411yescharacteristic
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)C3×D412yes—
143C4×C3⋊D3—42no—
144#634—21no—
145#634—21no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—61no—
149C3×D12—32no—
150C3×D12—61no—
151C3×C4×D3—122no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C32×C8—21no—
155C32×C8—21no—
156C32×C8—41no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—61no—
159#463—91noSylow
160C2×H(3,𝔽3)—81no—
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)#13511yescharacteristic
163C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
164(C3⋊D3)⋊C3—42no—
165C2×C3×Q8—91no—
166C2×C3×D4—31no—
167C2×C3×D4—31no—
168C2×C3×D4—61no—
169C2×C3×D4—91no—
170C2×C3×D4C2×C3×D311yescharacteristic
171C22×C3×C4—181no—
172C2×D12—31no—
173C2×D12—91no—
174C2×C4×D3—61no—
175C2×C4×D3—181no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×SD16—182no—
179C3×D8—182no—
180C2×C3×C8—31no—
181C2×C3×C8—31no—
182C2×C3×C8—61no—
183C2×C3×C8—91no—
184C2×C3×C8C2×C3×D311yescharacteristic
185C3×C4⋊C4—31no—
186C3×C4⋊C4—31no—
187C3×C4⋊C4—61no—
188C3×C4⋊C4C2×C3×D311yescharacteristic
189C3×C22⋊C4—181no—
190C3×C22⋊C4—181no—
191C3×C22⋊C4—181no—
192C3×C42—91no—
193#217—61no—
194#217—61no—
195#217—181no—
196#217—181no—
197#213—62no—
198#213—182no—
199(C2×D3)⋊C4—61no—
200(C2×D3)⋊C4—181no—
201C4×Dic3—31no—
202C4×Dic3—91no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205#204—62no—
206#204—182no—
207C22×C32—21no—
208C22×C32—42no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—81no—
216C22×C32—81no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×C3×D411yescharacteristic
219C2×C3⋊D3—22no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3×D3—62no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—81no—
230C32×C4#23511yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4C2×C3×D411yescharacteristic
233C32×C4C3×#4111yescharacteristic
234C3⋊Dic3—21no—
235C3⋊Dic3—21no—
236C3⋊Dic3—41no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—121no—
240C2×SD16—91no—
241C2×D8—91no—
242#124—91no—
243C4×D4—91no—
244C4⋊C8—91no—
245#103—91no—
246#102(C3⋊D3)⋊C311yescharacteristic, complemented
247H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—121no—
252C23×C3—181no—
253C22×D3—61no—
254C22×D3—181no—
255C3×Q8—181no—
256C3×Q8—181no—
257C3×D4—21no—
258C3×D4—21no—
259C3×D4—61no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—61no—
263C3×D4—62no—
264C3×D4—92no—
265C3×D4—121no—
266C3×D4—181no—
267C2×C3×C4—21no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—121no—
276C2×C3×C4—181no—
277C2×C3×C4—362no—
278C2×C3×C4C22×C3×D311yescharacteristic
279C2×Dic3—31no—
280C2×Dic3—31no—
281C2×Dic3—61no—
282C2×Dic3—91no—
283C2×Dic3—91no—
284C2×Dic3—181no—
285D12—32no—
286D12—61no—
287D12—92no—
288D12—181no—
289C4×D3—122no—
290C4×D3—362no—
291Dic6—61no—
292Dic6—61no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—21no—
296C3×C8—61no—
297C3×C8—61no—
298C3×C8—121no—
299C3×C8—181no—
300C3⋊C8—61no—
301C3⋊C8—181no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—82no—
306C2×C32—81no—
307C2×C32—81no—
308C2×C32#88411yescharacteristic
309C2×C32#91111yescharacteristic
310C2×C32#91611yescharacteristic
311C2×C32C3×#12111yescharacteristic
312C2×C32C3×#13511yescharacteristic
313C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
314C3⋊D3—42no—
315C3×D3—122no—
316C2×Q8—91no—
317C2×D4—91no—
318C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
319C22×C4—181no—
320SD16—182no—
321D8—182no—
322C2×C8—91no—
323C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
324C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
325C22⋊C4—181no—
326C22⋊C4—181no—
327C22⋊C4—181no—
328C42—91no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—61no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—121no—
341C22×C3—122no—
342C22×C3—182no—
343C22×C3—241no—
344C22×C3—241no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3C3×D3×D411yescharacteristic
348C2×D3—62no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—182no—
352C2×D3—361no—
353C2×D3—361no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—41no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—121no—
362C3×C4—121no—
363C3×C4—181no—
364C3×C4—181no—
365C3×C4—241no—
366C3×C4—361no—
367C3×C4C3×#23511yescharacteristic
368C3×C4C3×D3×D411yescharacteristic
369Dic3—61no—
370Dic3—61no—
371Dic3—121no—
372Dic3—181no—
373Dic3—181no—
374Dic3—361no—
375C32—21no—
376C32#478711yescharacteristic, complemented
377C32C3×#46311yescharacteristic, complemented
378C23—21no—
379C23—181no—
380Q8—181no—
381Q8—181no—
382D4—21no—
383D4—21no—
384D4—92no—
385D4—181no—
386C2×C4—21no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—181no—
390C2×C4—362no—
391C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
392C8—21no—
393C8—181no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—81no—
404C2×C3—241no—
405C2×C3—241no—
406C2×C3—242no—
407C2×C3—362no—
408C2×C3C3×#88411yescharacteristic
409C2×C3C3×#91111yescharacteristic
410C2×C3C3×#91611yescharacteristic
411D3—122no—
412D3—362no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—182no—
417C22—361no—
418C22—361no—
419C22D4×(C3⋊D3)⋊C311yescharacteristic
420C4—41no—
421C4—181no—
422C4—181no—
423C4—361no—
424C4#3828⋊C311yescharacteristic
425C4D4×(C3⋊D3)⋊C311yescharacteristic
426C3—31no—
427C3—31no—
428C3—61no—
429C3C3×#478711yescharacteristic
430C2—81no—
431C2—362no—
432C2#64093⋊C311yescharacteristic
433C2#64120⋊C311yescharacteristic
434C2#64125⋊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×#3753⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#463
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
1. #64145⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
4. #64102⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63641⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592486C3semidirect
23. #64145C2×C3semidirect
25. C2×#3744C2×C3semidirect
26. #64102C2×C3semidirect
32. C32×#102D3semidirect
33. C32×#102C2×C3semidirect
34. #63641C2×C3semidirect
35. #63637C2×C3semidirect
246. #102(C3⋊D3)⋊C3semidirect
247. H(3,𝔽3)#463semidirect
376. C32#4787semidirect
377. C32C3×#463semidirect

Maximal subgroups

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

subgroupindex
#64145⋊C32
C2×#3753⋊C32
C2×#3744⋊C32
#64102⋊C32
H(3,𝔽3)×#1022
#63641⋊C32
#63637⋊C32
#105924863
C3×#47873