Finite Group Databasebeta

← #10596296⋊C3 #13153715

Subgroups — #10596296⋊C3 #13153715

Order 1728.

1109 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3#64153⋊C3C212yesmaximal, complemented
4#64145⋊C3C212yesmaximal, complemented
5#64144⋊C3C214yesmaximal, complemented
6#64135⋊C3C211yesmaximal, characteristic, complemented
7#64133⋊C3C214yesmaximal, complemented
8#10596296C311yesmaximal, characteristic, complemented
9C3×#5825—31nomaximal
10#41×H(3,𝔽3)—24no—
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×D4×H(3,𝔽3)C2212yes—
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2212yes—
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×#634⋊C3—24no—
18C2×#634⋊C3C2212yes—
19C2×#634⋊C3C2212yes—
20#3757⋊C3C2218yes—
21#3757⋊C3C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23#3750⋊C3C2218yes—
24C4×(C3⋊Dic3)⋊C3C2212yes—
25C2×C32×#41—21no—
26C2×C32×#41D311yescharacteristic
27C2×C32×#41C2×C311yescharacteristic
28C22×#634C2×C311yescharacteristic, complemented
29C22×C3×Dic6—31no—
30#64153C2×C312yescomplemented
31#64145C2×C312yescomplemented
32#64144C2×C314yescomplemented
33#64135C2×C311yescharacteristic, complemented
34#64133C2×C314yescomplemented
35C3×#944—32no—
36C3×#936—32no—
37C3×#935—34no—
38C3×#926—31no—
39C3×#924—34no—
40C23×H(3,𝔽3)C2312yes—
41C23×H(3,𝔽3)C2311yescharacteristic
42Q8×H(3,𝔽3)—22no—
43D4×H(3,𝔽3)—24no—
44D4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)—24no—
46C2×C4×H(3,𝔽3)D414yes—
47C2×C4×H(3,𝔽3)C2312yes—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×(C3⋊Dic3)⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2314yes—
53#634⋊C3—24no—
54#634⋊C3—24no—
55C3×#536—91no—
56#5825—31no—
57#5825—91no—
58C32×#41—28no—
59C32×#41—24no—
60C32×#41—24no—
61C2×C32×Q8—21no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×Q8C22×C311yescharacteristic
64C2×C32×D4—22no—
65C2×C32×D4—21no—
66C2×C32×D4C2×D312yes—
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C22×C312yes—
69C2×C32×D4C22×C311yescharacteristic
70C22×C32×C4—22no—
71C22×C32×C4—21no—
72C22×C32×C4C2×D312yes—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C22×C312yes—
75C22×C32×C4C22×C311yescharacteristic
76C22×C3⋊Dic3C22×C312yes—
77C2×#634—24no—
78C2×#634C22×C312yes—
79C2×#634C22×C312yes—
80C22×C3×Dic3—32no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—64no—
84#3757C22×C318yes—
85#3757C22×C312yes—
86C4⋊(C3⋊Dic3)C22×C312yes—
87#3750C22×C318yes—
88C4×C3⋊Dic3C22×C312yes—
89C3×#217—38no—
90C3×#217—32no—
91C3×C4⋊Dic3—32no—
92C3×Dic3⋊C4—38no—
93C3×C4×Dic3—32no—
94C22×H(3,𝔽3)—22no—
95C22×H(3,𝔽3)—22no—
96C22×H(3,𝔽3)—24no—
97C22×H(3,𝔽3)C2×D412yes—
98C22×H(3,𝔽3)C2411yescharacteristic
99C4×H(3,𝔽3)—22no—
100C4×H(3,𝔽3)C2×D412yes—
101C4×H(3,𝔽3)C2×D412yes—
102(C3⋊Dic3)⋊C3—24no—
103(C3⋊Dic3)⋊C3—24no—
104C2×C3×#41—31no—
105C2×C3×#41—31no—
106C2×C3×#41—61no—
107C2×C3×#41C3×D311yescharacteristic
108C22×C3×Q8—91no—
109C22×Dic6—31no—
110C22×Dic6—91no—
111C3×C4⋊Q8—92no—
112C3×#124—92no—
113C3×#123—94no—
114C3×C22⋊Q8—94no—
115C3×#117—91no—
116#944—32no—
117#944—92no—
118#936—32no—
119#936—92no—
120#935—34no—
121#935—94no—
122#926—31no—
123#926—91no—
124#924—34no—
125#924—94no—
126C23×C32—22no—
127C23×C32—21no—
128C23×C32C22×D312yes—
129C23×C32C22×D311yescharacteristic
130C23×C32C23×C312yes—
131C23×C32C23×C311yescharacteristic
132C32×Q8—24no—
133C32×Q8—22no—
134C32×Q8—22no—
135C32×D4—28no—
136C32×D4—24no—
137C32×D4—24no—
138C32×D4—24no—
139C32×D4—22no—
140C32×D4—22no—
141C2×C32×C4—28no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—24no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4#6714yes—
149C2×C32×C4C3×D414yes—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C312yes—
154C2×C32×C4C23×C311yescharacteristic
155C2×C32×C4C23×C311yescharacteristic
156C2×C3⋊Dic3—24no—
157C2×C3⋊Dic3C23×C314yes—
158C2×C3⋊Dic3C23×C314yes—
159#634—24no—
160#634—24no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—64no—
164C3×Dic6—64no—
165C3×Dic6—64no—
166#536—91noSylow
167C2×H(3,𝔽3)—22no—
168C2×H(3,𝔽3)—42no—
169C2×H(3,𝔽3)C22×D411yescharacteristic
170C2×H(3,𝔽3)2−1+412yes—
171C3×#41—24no—
172C3×#41—64no—
173C3×#41—64no—
174C3×#41—68no—
175C2×C3×Q8—31no—
176C2×C3×Q8—31no—
177C2×C3×Q8—61no—
178C2×C3×Q8—92no—
179C2×C3×Q8—92no—
180C2×C3×Q8—184no—
181C2×C3×Q8C2×C3×D311yescharacteristic
182C2×C3×D4—32no—
183C2×C3×D4—31no—
184C2×C3×D4—32no—
185C2×C3×D4—31no—
186C2×C3×D4—62no—
187C2×C3×D4—61no—
188C2×C3×D4C2×C3×D312yes—
189C2×C3×D4C2×C3×D311yescharacteristic
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—32no—
193C22×C3×C4—32no—
194C22×C3×C4—62no—
195C22×C3×C4—61no—
196C22×C3×C4—92no—
197C22×C3×C4C2×C3×D312yes—
198C22×C3×C4C2×C3×D311yescharacteristic
199C22×Dic3—32no—
200C22×Dic3—92no—
201C2×Dic6—32no—
202C2×Dic6—32no—
203C2×Dic6—64no—
204C2×Dic6—92no—
205C2×Dic6—92no—
206C2×Dic6—184no—
207C3×C4⋊C4—98no—
208C3×C4⋊C4—92no—
209C3×C22⋊C4—92no—
210C3×C22⋊C4—98no—
211C3×C42—92no—
212#217—38no—
213#217—32no—
214#217—98no—
215#217—92no—
216C4⋊Dic3—32no—
217C4⋊Dic3—92no—
218Dic3⋊C4—38no—
219Dic3⋊C4—98no—
220C4×Dic3—32no—
221C4×Dic3—92no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—24no—
229C22×C32—21no—
230C22×C32—22no—
231C22×C32—42no—
232C22×C32—44no—
233C22×C32C2×#6712yes—
234C22×C32C2×C3×D412yes—
235C22×C32C23×D311yescharacteristic
236C22×C32C24×C311yescharacteristic
237C32×C4—24no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4C2×#6712yes—
243C32×C4C2×#6712yes—
244C32×C4C2×C3×D412yes—
245C32×C4C2×C3×D412yes—
246C3⋊Dic3—24no—
247C3⋊Dic3—24no—
248C3×Dic3—64no—
249C3×Dic3—64no—
250C2×#41(C3⋊D3)⋊C311yescharacteristic
251C22×Q8—91no—
252C4⋊Q8—92no—
253#124—92no—
254#123—94no—
255C22⋊Q8—94no—
256#117—91no—
257H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
258C23×C3—31no—
259C23×C3—31no—
260C23×C3—32no—
261C23×C3—32no—
262C23×C3—62no—
263C23×C3—61no—
264C23×C3C22×C3×D312yes—
265C23×C3C22×C3×D311yescharacteristic
266C3×Q8—22no—
267C3×Q8—62no—
268C3×Q8—62no—
269C3×Q8—64no—
270C3×Q8—184no—
271C3×Q8—184no—
272C3×D4—24no—
273C3×D4—22no—
274C3×D4—62no—
275C3×D4—62no—
276C3×D4—64no—
277C3×D4—64no—
278C3×D4—68no—
279C3×D4—64no—
280C2×C3×C4—24no—
281C2×C3×C4—34no—
282C2×C3×C4—34no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
289C2×C3×C4—64no—
290C2×C3×C4—64no—
291C2×C3×C4—68no—
292C2×C3×C4—64no—
293C2×C3×C4—61no—
294C2×C3×C4—62no—
295C2×C3×C4—61no—
296C2×C3×C4—94no—
297C2×C3×C4—94no—
298C2×C3×C4—184no—
299C2×C3×C4C3×#6714yes—
300C2×C3×C4C22×C3×D312yes—
301C2×C3×C4C22×C3×D311yescharacteristic
302C2×C3×C4C22×C3×D311yescharacteristic
303C2×Dic3—34no—
304C2×Dic3—34no—
305C2×Dic3—64no—
306C2×Dic3—94no—
307C2×Dic3—94no—
308C2×Dic3—184no—
309Dic6—64no—
310Dic6—64no—
311Dic6—184no—
312Dic6—184no—
313C2×C32—22no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—42no—
318C2×C32—44no—
319C2×C32—42no—
320C2×C32—42no—
321C2×C32#101012yes—
322C2×C32C22×#6711yescharacteristic
323C2×C32C22×C3×D411yescharacteristic
324C2×C32C3×2−1+412yes—
325#41—24no—
326C2×Q8—92no—
327C2×Q8—92no—
328C2×Q8—184no—
329C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
330C2×D4C2×(C3⋊D3)⋊C312yes—
331C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
332C22×C4—92no—
333C22×C4C2×(C3⋊D3)⋊C312yes—
334C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
335C4⋊C4—92no—
336C4⋊C4—98no—
337C22⋊C4—92no—
338C22⋊C4—98no—
339C42—92no—
340C22×C3—22no—
341C22×C3—24no—
342C22×C3—22no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3—62no—
350C22×C3—62no—
351C22×C3—64no—
352C22×C3—64no—
353C22×C3—64no—
354C22×C3—61no—
355C22×C3—62no—
356C22×C3—122no—
357C22×C3—124no—
358C22×C3C2×C3×#6712yes—
359C22×C3C23×C3×D311yescharacteristic
360C3×C4—22no—
361C3×C4—32no—
362C3×C4—32no—
363C3×C4—32no—
364C3×C4—32no—
365C3×C4—62no—
366C3×C4—62no—
367C3×C4—64no—
368C3×C4—62no—
369C3×C4—62no—
370C3×C4—184no—
371C3×C4—184no—
372C3×C4C2×C3×#6712yes—
373C3×C4C2×C3×#6712yes—
374Dic3—64no—
375Dic3—64no—
376Dic3—184no—
377Dic3—184no—
378C32—21no—
379C32#582511yescharacteristic, complemented
380C32C3×#53611yescharacteristic, complemented
381C23C22×(C3⋊D3)⋊C312yes—
382C23C22×(C3⋊D3)⋊C311yescharacteristic
383Q8—22no—
384Q8—184no—
385Q8—184no—
386D4—24no—
387D4—22no—
388C2×C4—24no—
389C2×C4—94no—
390C2×C4—94no—
391C2×C4—184no—
392C2×C4#638⋊C314yes—
393C2×C4C22×(C3⋊D3)⋊C312yes—
394C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
395C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
396C2×C3—22no—
397C2×C3—32no—
398C2×C3—32no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—42no—
402C2×C3—62no—
403C2×C3—62no—
404C2×C3—62no—
405C2×C3—61no—
406C2×C3—122no—
407C2×C3—122no—
408C2×C3—122no—
409C2×C3—124no—
410C2×C3C3×#101012yes—
411C2×C3C22×C3×#6711yescharacteristic
412C22—22no—
413C22—24no—
414C22—22no—
415C22C2×#638⋊C312yes—
416C22C23×(C3⋊D3)⋊C311yescharacteristic
417C4—22no—
418C4—184no—
419C4—184no—
420C4C2×#638⋊C312yes—
421C4C2×#638⋊C312yes—
422C3—31no—
423C3—31no—
424C3—61no—
425C3C3×#582511yescharacteristic
426C2—22no—
427C2—42no—
428C2#1172263912yes—
429C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting, prime radicalC2×#41×H(3,𝔽3)C2
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. #64153⋊C3C2semidirect
4. #64145⋊C3C2semidirect
5. #64144⋊C3C2semidirect
6. #64135⋊C3C2semidirect
7. #64133⋊C3C2semidirect
8. #10596296C3semidirect
28. C22×#634C2×C3semidirect
30. #64153C2×C3semidirect
31. #64145C2×C3semidirect
32. #64144C2×C3semidirect
33. #64135C2×C3semidirect
34. #64133C2×C3semidirect
257. H(3,𝔽3)#536semidirect
379. C32#5825semidirect
380. C32C3×#536semidirect

Maximal subgroups

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

subgroupindex
C2×#41×H(3,𝔽3)2
C22×#634⋊C32
#64153⋊C32
#64145⋊C32
#64144⋊C32
#64135⋊C32
#64133⋊C32
#105962963
C3×#58253