Finite Group Databasebeta

← #10592511⋊C3 #13141923

Subgroups — #10592511⋊C3 #13141923

Order 1728.

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

normal subgroups only

This group has 462 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#64155⋊C3C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3C2×#3743⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592511C311yesmaximal, characteristic, complemented
9C3×#4812—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3755⋊C3—22no—
17#3752⋊C3—21no—
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20#3750⋊C3—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3743⋊C3—22no—
23#64155C2×C311yescharacteristic, complemented
24C2×#3755C2×C311yescharacteristic, complemented
25C2×#3743C2×C311yescharacteristic, complemented
26#64104C2×C311yescharacteristic
27C3×#946—31no—
28C2×C3×#215—31no—
29C2×C3×ΓL(1,𝔽25)—31no—
30C3×#895—31no—
31C32×#103—21no—
32C32×#103D311yescharacteristic, complemented
33C32×#103C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63634C2×C311yescharacteristic, complemented
36C3×#821—31no—
37C3×#807—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—42no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54C3×#468—91no—
55#4812—31no—
56#4812—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C4⋊(C3⋊D3)—21no—
61C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
62C2×C4×C3⋊D3C22×C311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×C4×D3—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#3755—22no—
73#3752—21no—
74#3752—21no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76#3750—21no—
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3743—22no—
79C3×#215—62no—
80C3×(C2×D3)⋊C4—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—31no—
83C3×Dic3⋊C4—61no—
84C2×C3×C3⋊C8—31no—
85C3×ΓL(1,𝔽25)—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—21no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C2×C3×M16—91no—
101C3×C22⋊Q8—91no—
102C3×#121—91no—
103#946—31no—
104#946—91no—
105C2×#215—31no—
106C2×#215—91no—
107C2×ΓL(1,𝔽25)—31no—
108C2×ΓL(1,𝔽25)—91no—
109#895—31no—
110#895—91no—
111C3×#107—91no—
112C3×#103—31no—
113C3×#103—31no—
114C3×#103—61no—
115C3×#103C3×D311yescharacteristic
116C3×#102—91no—
117#821—31no—
118#821—91no—
119#807—31no—
120#807—91no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3C3×D411yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C32×Q8—22no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—41no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3C3×D411yescharacteristic
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—42no—
145C4×C3⋊D3—21no—
146C4×C3⋊D3—21no—
147C2×C3×Dic3—31no—
148C2×C3×Dic3—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—122no—
152C3×C4×D3—61no—
153C3×C4×D3—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)#12111yescharacteristic
161C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
162C2×H(3,𝔽3)#13711yescharacteristic
163(C3⋊D3)⋊C3—41no—
164(C3⋊D3)⋊C3—81no—
165C2×C3×Q8—31no—
166C2×C3×Q8—31no—
167C2×C3×Q8—61no—
168C2×C3×Q8C2×C3×D311yescharacteristic
169C2×C3×D4—91no—
170C2×C3×D4—181no—
171C22×C3×C4—91no—
172C2×D12—31no—
173C2×D12—61no—
174C2×D12—91no—
175C2×D12—181no—
176C2×C4×D3—31no—
177C2×C4×D3—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#215—62no—
194#215—182no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4—61no—
197(C2×D3)⋊C4—181no—
198(C2×D3)⋊C4—181no—
199C4⋊Dic3—31no—
200C4⋊Dic3—91no—
201Dic3⋊C4—61no—
202Dic3⋊C4—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205ΓL(1,𝔽25)—62no—
206ΓL(1,𝔽25)—182no—
207C22×C32—21no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×C3×D411yescharacteristic
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—42no—
225C32×C4—41no—
226C32×C4—41no—
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—61no—
237C3×Dic3—121no—
238C2×SD16—91no—
239C2×M16—91no—
240C22⋊Q8—91no—
241#121—91no—
242#107—91no—
243#103(C3⋊D3)⋊C311yescharacteristic, complemented
244#102—91no—
245H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
246C23×C3—91no—
247C23×C3—181no—
248C22×D3—31no—
249C22×D3—61no—
250C22×D3—91no—
251C22×D3—181no—
252C3×Q8—21no—
253C3×Q8—21no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—61no—
258C3×Q8—62no—
259C3×Q8—121no—
260C3×D4—181no—
261C3×D4—181no—
262C3×D4—362no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—91no—
273C2×C3×C4—121no—
274C2×C3×C4—121no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4C22×C3×D311yescharacteristic
279C2×Dic3—31no—
280C2×Dic3—61no—
281C2×Dic3—91no—
282C2×Dic3—181no—
283D12—61no—
284D12—61no—
285D12—122no—
286D12—181no—
287D12—181no—
288D12—362no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—181no—
292C4×D3—181no—
293C3×C8—21no—
294C3×C8—61no—
295C3×C8—61no—
296C3×C8—92no—
297C3×C8—121no—
298C3⋊C8—32no—
299C3⋊C8—92no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32#88411yescharacteristic
304C2×C32#91411yescharacteristic
305C2×C32#91811yescharacteristic
306C2×C32C3×#12111yescharacteristic
307C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
308C2×C32C3×#13711yescharacteristic
309C3⋊D3—41no—
310C3⋊D3—81no—
311C3×D3—121no—
312C3×D3—241no—
313C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
314C2×D4—91no—
315C2×D4—181no—
316C22×C4—91no—
317SD16—182no—
318M16—182no—
319C2×C8—91no—
320C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
321C4⋊C4—91no—
322C4⋊C4—181no—
323C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
324C22⋊C4—181no—
325C22⋊C4—181no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3C3×D3×D411yescharacteristic
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—361no—
346C2×D3—361no—
347C2×D3—361no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—121no—
357C3×C4—121no—
358C3×C4—121no—
359C3×C4—121no—
360C3×C4—122no—
361C3×C4—181no—
362C3×C4—241no—
363C3×C4—361no—
364C3×C4C3×#23511yescharacteristic
365C3×C4C3×D3×D411yescharacteristic
366Dic3—61no—
367Dic3—121no—
368Dic3—181no—
369Dic3—361no—
370C32—21no—
371C32#481211yescharacteristic, complemented
372C32C3×#46811yescharacteristic, complemented
373C23—91no—
374C23—181no—
375Q8—21no—
376Q8—21no—
377D4—181no—
378D4—181no—
379D4—362no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—91no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
387C8—21no—
388C8—92no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—61no—
398C2×C3—361no—
399C2×C3—721no—
400C2×C3C3×#88411yescharacteristic
401C2×C3C3×#91411yescharacteristic
402C2×C3C3×#91811yescharacteristic
403D3—121no—
404D3—241no—
405D3—361no—
406D3—721no—
407C22—181no—
408C22—181no—
409C22—181no—
410C22—361no—
411C22—361no—
412C22—361no—
413C22D4×(C3⋊D3)⋊C311yescharacteristic
414C4—41no—
415C4—41no—
416C4—181no—
417C4—361no—
418C4#3828⋊C311yescharacteristic
419C4D4×(C3⋊D3)⋊C311yescharacteristic
420C3—31no—
421C3—31no—
422C3—61no—
423C3C3×#481211yescharacteristic
424C2—361no—
425C2—721no—
426C2#64093⋊C311yescharacteristic
427C2#64123⋊C311yescharacteristic
428C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radical#64104⋊C3C2
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #64155⋊C3C2semidirect
2. C2×#3755⋊C3C2semidirect
3. C2×#3743⋊C3C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #63648⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592511C3semidirect
23. #64155C2×C3semidirect
24. C2×#3755C2×C3semidirect
25. C2×#3743C2×C3semidirect
32. C32×#103D3semidirect
33. C32×#103C2×C3semidirect
34. #63648C2×C3semidirect
35. #63634C2×C3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
157. C3⋊(C3⋊C8)C3×D4semidirect
243. #103(C3⋊D3)⋊C3semidirect
245. H(3,𝔽3)#468semidirect
371. C32#4812semidirect
372. C32C3×#468semidirect

Maximal subgroups

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

subgroupindex
#64155⋊C32
C2×#3755⋊C32
C2×#3743⋊C32
#64104⋊C32
H(3,𝔽3)×#1032
#63648⋊C32
#63634⋊C32
#105925113
C3×#48123