Finite Group Databasebeta

← #10592479⋊C3 #13141797

Subgroups — #10592479⋊C3 #13141797

Order 1728.

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

normal subgroups only

This group has 470 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
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3C2×#3743⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63643⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592479C311yesmaximal, characteristic, complemented
9C3×#4780—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—21no—
17#3754⋊C3—22no—
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3743⋊C3—22no—
23#64147C2×C311yescharacteristic
24C2×#3754C2×C311yescharacteristic, complemented
25C2×#3743C2×C311yescharacteristic, complemented
26#64106C2×C311yescharacteristic, complemented
27C3×#938—31no—
28C2×C3×#214—31no—
29C2×C3×ΓL(1,𝔽25)—31no—
30C3×#897—31no—
31C32×#102—21no—
32C32×#102D311yescharacteristic, complemented
33C32×#102C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35#63634C2×C311yescharacteristic, complemented
36C3×#816—31no—
37C3×#807—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54C3×#468—91no—
55#4780—31no—
56#4780—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C22×C311yescharacteristic
60C2×#638—21no—
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C2×#634C22×C311yescharacteristic
63C2×C3×#67—61no—
64C2×C3×C4×D3—31no—
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#3754—22no—
74#3752—21no—
75C4⋊(C3⋊Dic3)—21no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3743—22no—
79C3×#217—61no—
80C3×#214—62no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—61no—
84C2×C3×C3⋊C8—31no—
85C3×ΓL(1,𝔽25)—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—21no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
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—41no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C2×C3×M16—91no—
101C3×C22⋊Q8—91no—
102C3×#121—91no—
103#938—31no—
104#938—91no—
105C2×#214—31no—
106C2×#214—91no—
107C2×ΓL(1,𝔽25)—31no—
108C2×ΓL(1,𝔽25)—91no—
109#897—31no—
110#897—91no—
111C3×#107—91no—
112C3×#103—91no—
113C3×#102—31no—
114C3×#102—31no—
115C3×#102—61no—
116C3×#102C3×D311yescharacteristic
117#816—31no—
118#816—91no—
119#807—31no—
120#807—91no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C22×C3⋊D3C3×D411yescharacteristic
125C22×C3×D3—31no—
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
138#638—42no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3C3×D411yescharacteristic
142C4×C3⋊D3—21no—
143C4×C3⋊D3—21no—
144#634—21no—
145#634—21no—
146C3×#67—122no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C3×C4×D3—61no—
151C3×C4×D3—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C32×C8—21no—
155C32×C8—21no—
156C32×C8—41no—
157C3⋊(C3⋊C8)C3×D412yescomplemented
158C3×C3⋊C8—32no—
159#468—91noSylow
160C2×H(3,𝔽3)—81no—
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
163C2×H(3,𝔽3)#13711yescharacteristic
164(C3⋊D3)⋊C3—41no—
165C2×C3×Q8—91no—
166C2×C3×D4—31no—
167C2×C3×D4—31no—
168C2×C3×D4—61no—
169C2×C3×D4—181no—
170C2×C3×D4C2×C3×D311yescharacteristic
171C22×C3×C4—91no—
172C2×#67—61no—
173C2×#67—181no—
174C2×C4×D3—31no—
175C2×C4×D3—91no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×SD16—182no—
179C3×M16—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⋊C4—91no—
189C3×C4⋊C4—181no—
190C3×C4⋊C4C2×C3×D311yescharacteristic
191C3×C22⋊C4—181no—
192C3×C22⋊C4—181no—
193#217—61no—
194#217—181no—
195#214—62no—
196#214—182no—
197(C2×D3)⋊C4—61no—
198(C2×D3)⋊C4—181no—
199C4⋊Dic3—31no—
200C4⋊Dic3—61no—
201C4⋊Dic3—91no—
202C4⋊Dic3—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205ΓL(1,𝔽25)—62no—
206ΓL(1,𝔽25)—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—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—21no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
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—41no—
236C3⋊Dic3—41no—
237C3×Dic3—61no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C2×SD16—91no—
241C2×M16—91no—
242C22⋊Q8—91no—
243#121—91no—
244#107—91no—
245#103—91no—
246#102(C3⋊D3)⋊C311yescharacteristic, complemented
247H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—91no—
252C23×C3—121no—
253C22×D3—31no—
254C22×D3—91no—
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—121no—
265C3×D4—362no—
266C2×C3×C4—21no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—91no—
273C2×C3×C4—121no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4C22×C3×D311yescharacteristic
279#67—122no—
280#67—362no—
281C2×Dic3—31no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—91no—
285C2×Dic3—181no—
286C2×Dic3—181no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—181no—
290C4×D3—181no—
291Dic6—61no—
292Dic6—61no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—21no—
296C3×C8—61no—
297C3×C8—61no—
298C3×C8—92no—
299C3×C8—121no—
300C3⋊C8—32no—
301C3⋊C8—92no—
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#91511yescharacteristic
311C2×C32C3×#12111yescharacteristic
312C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
313C2×C32C3×#13711yescharacteristic
314C3⋊D3—41no—
315C3×D3—121no—
316C2×Q8—91no—
317C2×D4—181no—
318C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
319C22×C4—91no—
320SD16—182no—
321M16—182no—
322C2×C8—91no—
323C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
324C4⋊C4—91no—
325C4⋊C4—181no—
326C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
327C22⋊C4—181no—
328C22⋊C4—181no—
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—181no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3—241no—
346C22×C3—241no—
347C22×C3C3×D3×D411yescharacteristic
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—181no—
352C2×D3—181no—
353C2×D3—181no—
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—241no—
365C3×C4—361no—
366C3×C4—361no—
367C3×C4C3×#23511yescharacteristic
368C3×C4C3×D3×D411yescharacteristic
369Dic3—61no—
370Dic3—121no—
371Dic3—121no—
372Dic3—181no—
373Dic3—361no—
374Dic3—361no—
375C32—21no—
376C32#478011yescharacteristic, complemented
377C32C3×#46811yescharacteristic, complemented
378C23—21no—
379C23—91no—
380Q8—181no—
381Q8—181no—
382D4—21no—
383D4—21no—
384D4—362no—
385C2×C4—21no—
386C2×C4—91no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
392C8—21no—
393C8—92no—
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—361no—
408C2×C3C3×#88411yescharacteristic
409C2×C3C3×#91111yescharacteristic
410C2×C3C3×#91511yescharacteristic
411D3—121no—
412D3—361no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—181no—
417C22—181no—
418C22—181no—
419C22D4×(C3⋊D3)⋊C311yescharacteristic
420C4—41no—
421C4—181no—
422C4—361no—
423C4—361no—
424C4#3828⋊C311yescharacteristic
425C4D4×(C3⋊D3)⋊C311yescharacteristic
426C3—31no—
427C3—31no—
428C3—61no—
429C3C3×#478011yescharacteristic
430C2—81no—
431C2—361no—
432C2#64093⋊C311yescharacteristic
433C2#64120⋊C311yescharacteristic
434C2#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×#468
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. C2×#3754⋊C3C2semidirect
3. C2×#3743⋊C3C2semidirect
4. #64106⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63643⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592479C3semidirect
24. C2×#3754C2×C3semidirect
25. C2×#3743C2×C3semidirect
26. #64106C2×C3semidirect
32. C32×#102D3semidirect
33. C32×#102C2×C3semidirect
34. #63643C2×C3semidirect
35. #63634C2×C3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
157. C3⋊(C3⋊C8)C3×D4semidirect
246. #102(C3⋊D3)⋊C3semidirect
247. H(3,𝔽3)#468semidirect
376. C32#4780semidirect
377. C32C3×#468semidirect

Maximal subgroups

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

subgroupindex
#64147⋊C32
C2×#3754⋊C32
C2×#3743⋊C32
#64106⋊C32
H(3,𝔽3)×#1022
#63643⋊C32
#63634⋊C32
#105924793
C3×#47803