Finite Group Databasebeta

← #13147753

Subgroups — #13147753

Order 1728.

836 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C4×C4⋊#272C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721114C212yesmaximal
5#11721094C212yesmaximal, complemented
6#11721089C212yesmaximal, complemented
7#11721088C214yesmaximal, complemented
8#11721082C212yesmaximal, complemented
9#10593879—31nomaximal
10#10593863—31nomaximal
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×#634⋊C3C2212yes—
14C2×#634⋊C3C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#88542C2212yes—
17C4⋊#1084C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2212yes—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21#3750⋊C3C2212yes—
22C4×(C3⋊Dic3)⋊C3C2212yes—
23#88451C2214yes—
24#88451C2214yes—
25#88448C2214yes—
26#88447C2214yes—
27#64078D311yescharacteristic, complemented
28C4×#634D311yescharacteristic, complemented
29C3×C4×D12—31no—
30C3×C4×D12—31no—
31C3×C4×D12—61no—
32C3×#869—31no—
33C3×C4×Dic6—31no—
34#63863—32no—
35#63861—34no—
36#63860—32no—
37#63858—34no—
38#63857—32no—
39#63856—32no—
40#63844—32no—
41#63843—32no—
42#63841—32no—
43#63841—32no—
44C22×#272C2312yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084C2312yes—
49C4⋊#272—22no—
50C4×#272—24no—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54#634⋊C3—22no—
55#634⋊C3—24no—
56#5732—91no—
57#5584—91no—
58#5542—91no—
59C2×#634C2×D311yescharacteristic
60C2×#634C2×D312yes—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—62no—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—31no—
69C32×C42—21no—
70C32×C42C2×D311yescharacteristic
71C32×C42C2×D311yescharacteristic
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)C2×D312yes—
74C4⋊(C3⋊Dic3)C2×D312yes—
75#3750C2×D312yes—
76C4×C3⋊Dic3C2×D312yes—
77C3×(C2×D3)⋊C4—32no—
78C3×(C2×D3)⋊C4—32no—
79C3×(C2×D3)⋊C4—62no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—32no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—61no—
86C3×Dic3⋊C4—32no—
87C3×C4×Dic3—32no—
88#3724—34no—
89Dic3⋊Dic3—34no—
90(C2×D3)⋊Dic3—34no—
91(C2×D3)⋊Dic3—34no—
92(C2×D3)⋊Dic3—34no—
93(C2×D3)⋊Dic3—34no—
94Dic32—34no—
95Dic32—34no—
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×#272—22no—
98C2×#272—22no—
99C2×#272—22no—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)#4112yes—
102C4×H(3,𝔽3)#4112yes—
103#1084—22no—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—24no—
106(C3⋊Dic3)⋊C3—22no—
107C3×C4⋊Q8—91no—
108C3×C4×Q8—91no—
109C3×C4×D4—91no—
110#947—92no—
111#946—92no—
112Q8×Dic3—92no—
113#936—92no—
114#926—92no—
115#924—92no—
116#898—94no—
117#898—94no—
118#897—92no—
119#892—92no—
120#892—92no—
121#887—92no—
122#879—94no—
123#875—92no—
124C4×D12—31no—
125C4×D12—31no—
126C4×D12—61no—
127#869—31no—
128#869—91no—
129C4×Dic6—31no—
130C4×Dic6—91no—
131C4×Dic6—92no—
132C22×C3×D3—32no—
133C22×C3×D3—32no—
134C22×C3×D3—62no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3C22×D312yes—
145C2×C3⋊Dic3C22×D314yes—
146C2×C3⋊Dic3C22×D312yes—
147#634—22no—
148#634—24no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C3×D12—64no—
156C3×D12—62no—
157C3×D12—62no—
158C3×C4×D3—64no—
159C3×C4×D3—64no—
160C3×C4×D3—124no—
161C3×Dic6—64no—
162C3×Dic6—62no—
163#556—271noSylow
164C2×H(3,𝔽3)C2×#4111yescharacteristic
165C2×H(3,𝔽3)C2×#4111yescharacteristic
166C2×H(3,𝔽3)2−1+411yescharacteristic
167#272—42no—
168C2×C3×Q8—92no—
169C2×C3×Q8—91no—
170C2×C3×D4—91no—
171C22×C3×C4—92no—
172C2×D12—31no—
173C2×D12—31no—
174C2×D12—61no—
175C2×C4×D3—32no—
176C2×C4×D3—32no—
177C2×C4×D3—62no—
178C2×Dic6—31no—
179C2×Dic6—32no—
180C2×Dic6—92no—
181C2×Dic6—91no—
182C3×C4⋊C4—91no—
183C3×C4⋊C4—92no—
184C3×C4⋊C4—92no—
185C3×C4⋊C4—92no—
186C3×C4⋊C4—91no—
187C3×C22⋊C4—92no—
188C3×C42—31no—
189C3×C42—31no—
190C3×C42—61no—
191C3×C42—92no—
192C3×C42D3211yescharacteristic
193#217—94no—
194#217—94no—
195(C2×D3)⋊C4—32no—
196(C2×D3)⋊C4—32no—
197(C2×D3)⋊C4—62no—
198(C2×D3)⋊C4—94no—
199(C2×D3)⋊C4—94no—
200(C2×D3)⋊C4—94no—
201(C2×D3)⋊C4—94no—
202C4⋊Dic3—31no—
203C4⋊Dic3—31no—
204C4⋊Dic3—31no—
205C4⋊Dic3—32no—
206C4⋊Dic3—32no—
207C4⋊Dic3—61no—
208C4⋊Dic3—91no—
209C4⋊Dic3—92no—
210C4⋊Dic3—92no—
211C4⋊Dic3—94no—
212Dic3⋊C4—32no—
213Dic3⋊C4—92no—
214Dic3⋊C4—94no—
215Dic3⋊C4—94no—
216C4×Dic3—32no—
217C4×Dic3—92no—
218C4×Dic3—94no—
219C4×Dic3—94no—
220C4×Dic3—94no—
221C22×C32—21no—
222C22×C32C23×D311yescharacteristic
223C22×C32C23×D311yescharacteristic
224C2×C3×D3—64no—
225C2×C3×D3—62no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—122no—
232C2×C3×D3—122no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—41no—
238C32×C4#23512yes—
239C32×C4#23712yes—
240C32×C4#23912yes—
241C32×C4#23912yes—
242C3⋊Dic3—22no—
243C3⋊Dic3—24no—
244C3⋊Dic3—22no—
245C3×Dic3—62no—
246C3×Dic3—64no—
247C3×Dic3—62no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—122no—
251C4⋊Q8—91no—
252#126—274no—
253#124—272no—
254C22⋊Q8—272no—
255C4×Q8—91no—
256C4×Q8—272no—
257C4×D4—91no—
258#117—272no—
259H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
260C23×C3—92no—
261C22×D3—32no—
262C22×D3—32no—
263C22×D3—62no—
264C3×Q8—184no—
265C3×Q8—182no—
266C3×D4—182no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—92no—
277C2×C3×C4—94no—
278C2×C3×C4—92no—
279C2×C3×C4—92no—
280C2×C3×C4—184no—
281C2×C3×C4C2×D3211yescharacteristic
282C2×C3×C4C2×D3211yescharacteristic
283C2×C3×C4C2×D3211yescharacteristic
284C2×Dic3—32no—
285C2×Dic3—32no—
286C2×Dic3—32no—
287C2×Dic3—32no—
288C2×Dic3—34no—
289C2×Dic3—62no—
290C2×Dic3—92no—
291C2×Dic3—92no—
292C2×Dic3—94no—
293D12—62no—
294D12—62no—
295D12—64no—
296C4×D3—64no—
297C4×D3—64no—
298C4×D3—124no—
299Dic6—64no—
300Dic6—62no—
301Dic6—184no—
302Dic6—182no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32C2×#23511yescharacteristic
307C2×C32C2×#23711yescharacteristic
308C2×C32C2×#23911yescharacteristic
309C2×C32C2×#23911yescharacteristic
310C2×C32#100711yescharacteristic
311C2×C32#101011yescharacteristic
312C3×D3—124no—
313C3×D3—122no—
314C3×D3—122no—
315C2×Q8—91no—
316C2×Q8—92no—
317C2×D4—91no—
318C22×C4—92no—
319C4⋊C4—91no—
320C4⋊C4—92no—
321C4⋊C4—91no—
322C4⋊C4—92no—
323C4⋊C4—92no—
324C4⋊C4—274no—
325C22⋊C4—92no—
326C22⋊C4—274no—
327C22⋊C4—274no—
328C42—92no—
329C42—274no—
330C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—182no—
335C22×C3—182no—
336C22×C3—182no—
337C22×C3C22×D3211yescharacteristic
338C2×D3—62no—
339C2×D3—62no—
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—64no—
345C2×D3—122no—
346C2×D3—122no—
347C3×C4—21no—
348C3×C4—32no—
349C3×C4—32no—
350C3×C4—32no—
351C3×C4—32no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—62no—
355C3×C4—62no—
356C3×C4—121no—
357C3×C4—184no—
358C3×C4—182no—
359C3×C4—182no—
360C3×C4—182no—
361C3×C4#379612yes—
362C3×C4#379912yes—
363Dic3—62no—
364Dic3—62no—
365Dic3—64no—
366Dic3—62no—
367Dic3—62no—
368Dic3—122no—
369Dic3—184no—
370Dic3—182no—
371Dic3—182no—
372C32—21no—
373C32#558411yescharacteristic, complemented
374C32#573211yescharacteristic, complemented
375C23—92no—
376Q8—184no—
377Q8—182no—
378D4—182no—
379C2×C4—92no—
380C2×C4—92no—
381C2×C4—92no—
382C2×C4—94no—
383C2×C4—184no—
384C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
386C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—362no—
397C2×C3C2×#379611yescharacteristic
398C2×C3#6429711yescharacteristic
399C2×C3C2×#379911yescharacteristic
400D3—122no—
401D3—122no—
402D3—124no—
403C22—182no—
404C22—182no—
405C22—182no—
406C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
407C4—21no—
408C4—184no—
409C4—182no—
410C4—182no—
411C4—182no—
412C4#8864912yes—
413C4#8865212yes—
414C3—31no—
415C3—31no—
416C3—61no—
417C3#1059386511yescharacteristic
418C2—362no—
419C2C2×#8864911yescharacteristic
420C2#1172248211yescharacteristic
421C2C2×#8865211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. C4×#634⋊C3C2semidirect
5. #11721094C2semidirect
6. #11721089C2semidirect
7. #11721088C2semidirect
8. #11721082C2semidirect
27. #64078D3semidirect
28. C4×#634D3semidirect
259. H(3,𝔽3)#556semidirect
373. C32#5584semidirect
374. C32#5732semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#64078⋊C32
C4×#634⋊C32
#117211142
#117210942
#117210892
#117210882
#117210822
#105938793
#105938633