Finite Group Databasebeta

← #10592867⋊C3 #13143371

Subgroups — #10592867⋊C3 #13143371

Order 1728.

523 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3C2×#3756⋊C3C211yesmaximal, characteristic, complemented
4#64145⋊C3C211yesmaximal, characteristic, complemented
5C2×#3753⋊C3C211yesmaximal, characteristic
6C2×#3744⋊C3C211yesmaximal, characteristic, complemented
7C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#10592867C311yesmaximal, characteristic, complemented
9C3×#5168—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—22no—
17#3756⋊C3—22no—
18#3753⋊C3—22no—
19#3752⋊C3—22no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3744⋊C3—22no—
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic, complemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64156C2×C311yescharacteristic, complemented
27C2×#3756C2×C311yescharacteristic, complemented
28#64145C2×C311yescharacteristic, complemented
29C2×#3753C2×C311yescharacteristic
30C2×#3744C2×C311yescharacteristic, complemented
31C3×#947—31no—
32C2×C3×#216—31no—
33C3×#936—31no—
34C2×C3×#213—31no—
35C2×C3×#204—31no—
36C8×C3⋊Dic3C2×C311yescharacteristic, complemented
37C3×C8×Dic3—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54C3×#495—91no—
55#5168—31no—
56#5168—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C32×D4—21no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×#634C22×C311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×Dic6—31no—
67C32×SD16—24no—
68C32×SD16—22no—
69C32×SD16—22no—
70C2×C32×C8—21no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C22×C311yescharacteristic
73#3757—22no—
74#3756—22no—
75#3753—22no—
76#3752—22no—
77C4×C3⋊Dic3C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79#3744—22no—
80C3×#217—62no—
81C3×#216—62no—
82C3×#213—62no—
83C3×(C2×D3)⋊C4—62no—
84C3×C4×Dic3—31no—
85C2×C3×C3⋊C8—31no—
86C3×#204—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—42no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—42no—
92C4×H(3,𝔽3)—41no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)C2×D411yescharacteristic
95(C3⋊Dic3)⋊C3—22no—
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—31no—
99C2×C3×SD16—31no—
100C2×C3×SD16—61no—
101C2×C3×SD16—91no—
102C2×C3×SD16C3×D311yescharacteristic
103C2×C3×D8—91no—
104C3×#124—91no—
105C3×#124—91no—
106#947—31no—
107#947—91no—
108C2×#216—31no—
109C2×#216—91no—
110#936—31no—
111#936—91no—
112C2×#213—31no—
113C2×#213—91no—
114C2×#204—31no—
115C2×#204—91no—
116C3×C4×C8—91no—
117C8×Dic3—31no—
118C8×Dic3—91no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—41no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C32×Q8—22no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—41no—
130C32×D4—22no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—41no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C23×C311yescharacteristic
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3C3×D412yes—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146#634—21no—
147#634—21no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C32×C8—22no—
155C32×C8#6712yes—
156C32×C8C3×D412yes—
157C3⋊(C3⋊C8)C3×D412yes—
158C3×C3⋊C8—32no—
159#495—91noSylow
160C2×H(3,𝔽3)—81no—
161C2×H(3,𝔽3)#12711yescharacteristic
162C2×H(3,𝔽3)#13512yes—
163(C3⋊D3)⋊C3—81no—
164C2×C3×Q8—31no—
165C2×C3×Q8—31no—
166C2×C3×Q8—61no—
167C2×C3×Q8—91no—
168C2×C3×Q8C2×C3×D311yescharacteristic
169C2×C3×D4—31no—
170C2×C3×D4—31no—
171C2×C3×D4—61no—
172C2×C3×D4—91no—
173C2×C3×D4C2×C3×D311yescharacteristic
174C2×D12—31no—
175C2×D12—91no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×Q16—182no—
179C3×SD16—22no—
180C3×SD16—62no—
181C3×SD16—62no—
182C3×SD16—64no—
183C3×SD16—182no—
184C3×D8—182no—
185C2×C3×C8—31no—
186C2×C3×C8—31no—
187C2×C3×C8—61no—
188C2×C3×C8—91no—
189C2×C3×C8C2×C3×D311yescharacteristic
190C3×C22⋊C4—182no—
191C3×C22⋊C4—182no—
192C3×C42—91no—
193#217—62no—
194#217—182no—
195#216—62no—
196#216—182no—
197#213—62no—
198#213—182no—
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—182no—
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—42no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—42no—
213C22×C32—82no—
214C22×C32D3×D411yescharacteristic
215C22×C32C2×C3×D411yescharacteristic
216C2×C3⋊D3—41no—
217C2×C3⋊D3—42no—
218C2×C3×D3—122no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—42no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4D3×D411yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C32×C4C2×C3×D411yescharacteristic
228C32×C4C2×C3×D411yescharacteristic
229C3⋊Dic3—22no—
230C3⋊Dic3—41no—
231C3×Dic3—62no—
232C3×Dic3—121no—
233C2×Q16—91no—
234C2×SD16—91no—
235C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
236C2×D8—91no—
237#124—91no—
238#124—91no—
239C4×C8—91no—
240H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
241C23×C3—21no—
242C23×C3—61no—
243C23×C3—61no—
244C23×C3—121no—
245C23×C3—181no—
246C22×D3—61no—
247C22×D3—181no—
248C3×Q8—21no—
249C3×Q8—21no—
250C3×Q8—61no—
251C3×Q8—61no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—62no—
255C3×Q8—121no—
256C3×Q8—181no—
257C3×Q8—181no—
258C3×D4—21no—
259C3×D4—21no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—61no—
263C3×D4—61no—
264C3×D4—62no—
265C3×D4—121no—
266C3×D4—181no—
267C3×D4—181no—
268C2×C3×C4—21no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—92no—
275C2×C3×C4—121no—
276C2×C3×C4—181no—
277C2×C3×C4C22×C3×D311yescharacteristic
278C2×Dic3—32no—
279C2×Dic3—61no—
280C2×Dic3—92no—
281C2×Dic3—181no—
282D12—61no—
283D12—61no—
284D12—181no—
285D12—181no—
286Dic6—61no—
287Dic6—61no—
288Dic6—181no—
289Dic6—181no—
290C3×C8—32no—
291C3×C8—32no—
292C3×C8—62no—
293C3×C8—92no—
294C3×C8C3×#6712yes—
295C3⋊C8—32no—
296C3⋊C8—92no—
297C2×C32—22no—
298C2×C32—21no—
299C2×C32—82no—
300C2×C32—81no—
301C2×C32—81no—
302C2×C32#91612yes—
303C2×C32#94011yescharacteristic
304C2×C32C3×#12711yescharacteristic
305C2×C32C3×#13512yes—
306C3⋊D3—81no—
307C3×D3—241no—
308C2×Q8—91no—
309C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
310C2×D4—91no—
311C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
312Q16—182no—
313SD16—22no—
314SD16—182no—
315D8—182no—
316C2×C8—91no—
317C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
318C22⋊C4—182no—
319C22⋊C4—182no—
320C42—91no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—42no—
324C22×C3—41no—
325C22×C3—61no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—122no—
329C22×C3—122no—
330C22×C3—122no—
331C22×C3—242no—
332C22×C3—361no—
333C22×C3—362no—
334C22×C3C3×D3×D411yescharacteristic
335C2×D3—122no—
336C2×D3—121no—
337C2×D3—361no—
338C2×D3—362no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—41no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—121no—
347C3×C4—121no—
348C3×C4—122no—
349C3×C4—182no—
350C3×C4—361no—
351C3×C4C3×D3×D411yescharacteristic
352C3×C4C2×C3×#6711yescharacteristic
353Dic3—62no—
354Dic3—121no—
355Dic3—182no—
356Dic3—361no—
357C32—21no—
358C32#516811yescharacteristic, complemented
359C32C3×#49511yescharacteristic, complemented
360C23—21no—
361C23—181no—
362Q8—21no—
363Q8—21no—
364Q8—181no—
365Q8—181no—
366D4—21no—
367D4—21no—
368D4—181no—
369D4—181no—
370C2×C4—21no—
371C2×C4—92no—
372C2×C4—181no—
373C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
374C8—92no—
375C8#638⋊C312yes—
376C2×C3—32no—
377C2×C3—32no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—62no—
381C2×C3—61no—
382C2×C3—81no—
383C2×C3—241no—
384C2×C3—241no—
385C2×C3—242no—
386C2×C3—721no—
387C2×C3C3×#91612yes—
388C2×C3C3×#94011yescharacteristic
389D3—241no—
390D3—721no—
391C22—42no—
392C22—41no—
393C22—362no—
394C22—361no—
395C22D4×(C3⋊D3)⋊C311yescharacteristic
396C4—41no—
397C4—182no—
398C4—361no—
399C4D4×(C3⋊D3)⋊C311yescharacteristic
400C4C2×#638⋊C311yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3C3×#516811yescharacteristic
405C2—81no—
406C2—721no—
407C2#64125⋊C312yes—
408C2#64149⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radicalC2×#3753⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×SD16×H(3,𝔽3)C2semidirect
2. #64156⋊C3C2semidirect
3. C2×#3756⋊C3C2semidirect
4. #64145⋊C3C2semidirect
6. C2×#3744⋊C3C2semidirect
7. C8×(C3⋊Dic3)⋊C3C2semidirect
8. #10592867C3semidirect
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64156C2×C3semidirect
27. C2×#3756C2×C3semidirect
28. #64145C2×C3semidirect
30. C2×#3744C2×C3semidirect
36. C8×C3⋊Dic3C2×C3semidirect
235. C2×SD16(C3⋊D3)⋊C3semidirect
240. H(3,𝔽3)#495semidirect
358. C32#5168semidirect
359. C32C3×#495semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
#64156⋊C32
C2×#3756⋊C32
#64145⋊C32
C2×#3753⋊C32
C2×#3744⋊C32
C8×(C3⋊Dic3)⋊C32
#105928673
C3×#51683