Finite Group Databasebeta

← #13140150

Subgroups — #13140150

Order 1728.

839 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#88450C212yesmaximal
3#11720574C211yesmaximal, characteristic, complemented
4#63658⋊C3C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591294—31nomaximal
7#10591219—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—24no—
14C2×C4×(C3⋊D3)⋊C3C414yescomplemented
15#88450—24no—
16C22×C4×C3⋊D3D311yescharacteristic
17C22×C3×C4×D3—31no—
18C2×#3722—32no—
19C2×(C2×D3)⋊Dic3—32no—
20#63658D311yescharacteristic, complemented
21C3×C3⋊#95—31no—
22C3×C3⋊#95—31no—
23C3×C3⋊#95—31no—
24C3×C3⋊#95—61no—
25#63579—32no—
26#63579—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3—21no—
30C22×(C3⋊D3)⋊C3—21no—
31C22×(C3⋊D3)⋊C3C2×C412yes—
32C22×(C3⋊D3)⋊C3D414yes—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)C2×C412yes—
35C2×#1084—22no—
36C2×#1084—22no—
37C2×#1084—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3C2×C412yes—
43C4×(C3⋊D3)⋊C3—28no—
44#5211—91no—
45#4664—91no—
46#4664—91no—
47C23×C3⋊D3C2×D311yescharacteristic
48C23×C3×D3—31no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D312yes—
53C22×C3⋊Dic3C2×D311yescharacteristic
54C2×C4×C3⋊D3—24no—
55C2×C4×C3⋊D3Dic314yescomplemented
56C22×C3×Dic3—31no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—62no—
61C2×C3×C4×D3—34no—
62C2×C3×C4×D3—64no—
63#3722—64no—
64(C2×D3)⋊Dic3—64no—
65C22×H(3,𝔽3)C22×C411yescharacteristic
66C22×H(3,𝔽3)C2×D412yes—
67C22×H(3,𝔽3)#4112yes—
68C22×H(3,𝔽3)#4111yescharacteristic
69C22×H(3,𝔽3)#4111yescharacteristic
70C2×(C3⋊D3)⋊C3—22no—
71C2×(C3⋊D3)⋊C3—22no—
72C2×(C3⋊D3)⋊C3—24no—
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3C22⋊C414yes—
75C4×H(3,𝔽3)—22no—
76#1084—42no—
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3—42no—
79C23×C3×C4—91no—
80C22×C4×D3—31no—
81C22×C4×D3—91no—
82C2×#217—92no—
83C2×(C2×D3)⋊C4—92no—
84C2×(C2×D3)⋊C4—92no—
85C3×#95—91no—
86C3×#95—91no—
87C3⋊#95—31no—
88C3⋊#95—31no—
89C3⋊#95—31no—
90C3⋊#95—61no—
91C3⋊#95—91no—
92C3⋊#95—92no—
93C3⋊#95—92no—
94C3⋊#95—92no—
95C23×C32—21no—
96C23×C32C22×D311yescharacteristic
97C23×C32C22×D311yescharacteristic
98C22×C3⋊D3—21no—
99C22×C3⋊D3—22no—
100C22×C3⋊D3—21no—
101C22×C3⋊D3C2×Dic312yes—
102C22×C3⋊D3#6714yes—
103C22×C3×D3—32no—
104C22×C3×D3—34no—
105C22×C3×D3—62no—
106C22×C3×D3—61no—
107C22×C3×D3—61no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—42no—
112C2×C32×C4C4×D312yes—
113C2×C32×C4C2×Dic312yes—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3C2×Dic312yes—
119C4×C3⋊D3—28no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—122no—
133C2×C3×Dic3—122no—
134C3×C4×D3—68no—
135#381—271noSylow
136C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
137C2×H(3,𝔽3)#11711yescharacteristic
138C2×H(3,𝔽3)#11711yescharacteristic
139C2×H(3,𝔽3)#12312yes—
140C2×H(3,𝔽3)#12312yes—
141(C3⋊D3)⋊C3—24no—
142C24×C3—91no—
143C23×D3—31no—
144C23×D3—91no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—61no—
148C22×C3×C4—94no—
149C22×C3×C4—91no—
150C22×C3×C4—92no—
151C22×C3×C4—92no—
152C22×C3×C4—184no—
153C22×C3×C4D3211yescharacteristic
154C22×Dic3—31no—
155C22×Dic3—32no—
156C22×Dic3—32no—
157C22×Dic3—32no—
158C22×Dic3—62no—
159C22×Dic3—91no—
160C22×Dic3—92no—
161C2×C4×D3—34no—
162C2×C4×D3—64no—
163C2×C4×D3—94no—
164C2×C4×D3—184no—
165#217—184no—
166(C2×D3)⋊C4—184no—
167(C2×D3)⋊C4—184no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32C2×C4×D311yescharacteristic
174C22×C32#23511yescharacteristic
175C22×C32#23512yes—
176C22×C32#23512yes—
177C22×C32#23511yescharacteristic
178C22×C32D3×D412yes—
179C22×C32#23711yescharacteristic
180C22×C32#23911yescharacteristic
181C22×C32C22×Dic311yescharacteristic
182C22×C32C2×#6712yes—
183C2×C3⋊D3—22no—
184C2×C3⋊D3—22no—
185C2×C3⋊D3—24no—
186C2×C3⋊D3—24no—
187C2×C3⋊D3#21714yes—
188C2×C3×D3—34no—
189C2×C3×D3—62no—
190C2×C3×D3—64no—
191C2×C3×D3—62no—
192C2×C3×D3—64no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—42no—
196C3⋊Dic3—22no—
197C3⋊Dic3—42no—
198C3×Dic3—62no—
199C3×Dic3—124no—
200C3×Dic3—122no—
201C3×Dic3—122no—
202C3×Dic3—122no—
203C23×C4—91no—
204C2×C22⋊C4—272no—
205#95—91no—
206#95—91no—
207#95—272no—
208H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—61no—
212C23×C3—94no—
213C23×C3—92no—
214C23×C3—181no—
215C23×C3—182no—
216C23×C3—181no—
217C23×C3C2×D3211yescharacteristic
218C22×D3—32no—
219C22×D3—34no—
220C22×D3—61no—
221C22×D3—62no—
222C22×D3—61no—
223C22×D3—92no—
224C22×D3—94no—
225C22×D3—181no—
226C22×D3—182no—
227C22×D3—181no—
228C2×C3×C4—22no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—92no—
235C2×C3×C4—122no—
236C2×C3×C4—188no—
237C2×C3×C4—182no—
238C2×C3×C4—182no—
239C2×C3×C4—182no—
240C2×C3×C4—182no—
241C2×C3×C4—182no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4D3×Dic312yes—
245C2×Dic3—32no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—64no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—92no—
258C2×Dic3—122no—
259C2×Dic3—122no—
260C2×Dic3—182no—
261C2×Dic3—182no—
262C2×Dic3—182no—
263C2×Dic3—182no—
264C4×D3—68no—
265C4×D3—188no—
266C2×C32—21no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32#87211yescharacteristic
272C2×C32D3×C22⋊C411yescharacteristic
273C2×C32#88312yes—
274C2×C32#89211yescharacteristic
275C2×C32#89412yes—
276C2×C32#92611yescharacteristic
277C2×C32#92611yescharacteristic
278C2×C32#92912yes—
279C2×C32#92912yes—
280C2×C32C2×#21711yescharacteristic
281C3⋊D3—24no—
282C3×D3—64no—
283C24—91no—
284C22×C4—91no—
285C22×C4—94no—
286C22×C4—92no—
287C22×C4—92no—
288C22×C4—184no—
289C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C22⋊C4—544no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—32no—
294C22×C3—32no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—32no—
300C22×C3—32no—
301C22×C3—61no—
302C22×C3—62no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—62no—
306C22×C3—94no—
307C22×C3—182no—
308C22×C3—184no—
309C22×C3—182no—
310C22×C3—184no—
311C22×C3#379511yescharacteristic
312C22×C3#379812yes—
313C22×C3#379911yescharacteristic
314C22×C3C2×D3×Dic311yescharacteristic
315C22×C3D3×#6712yes—
316C2×D3—34no—
317C2×D3—62no—
318C2×D3—64no—
319C2×D3—62no—
320C2×D3—64no—
321C2×D3—94no—
322C2×D3—182no—
323C2×D3—184no—
324C2×D3—182no—
325C2×D3—184no—
326C3×C4—22no—
327C3×C4—62no—
328C3×C4—62no—
329C3×C4—122no—
330C3×C4—182no—
331C3×C4—362no—
332C3×C4—362no—
333Dic3—62no—
334Dic3—122no—
335Dic3—122no—
336Dic3—124no—
337Dic3—122no—
338Dic3—182no—
339Dic3—362no—
340C32—21no—
341C32#466411yescharacteristic, complemented
342C32#521111yescharacteristic, complemented
343C23—92no—
344C23—94no—
345C23—181no—
346C23—182no—
347C23—181no—
348C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C2×C4—22no—
350C2×C4—92no—
351C2×C4—188no—
352C2×C4—182no—
353C2×C4—182no—
354C2×C4—182no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4((C3⋊D3)⋊C3)⋊C412yes—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—32no—
365C2×C3—32no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—32no—
369C2×C3—32no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—62no—
373C2×C3—61no—
374C2×C3—62no—
375C2×C3—184no—
376C2×C3#6384111yescharacteristic
377C2×C3#6385012yes—
378C2×C3#6385511yescharacteristic
379C2×C3#6390512yes—
380C2×C3D3×#21711yescharacteristic
381D3—64no—
382D3—184no—
383C22—94no—
384C22—182no—
385C22—184no—
386C22—182no—
387C22—184no—
388C22#8864811yescharacteristic
389C22#8865012yes—
390C22#8865211yescharacteristic
391C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
392C22#8867612yes—
393C4—22no—
394C4—182no—
395C4—362no—
396C4—362no—
397C3—31no—
398C3—31no—
399C3—61no—
400C3#1059129411yescharacteristic
401C2—184no—
402C2#1172108111yescharacteristic
403C2#1172110012yes—
404C2#1172110511yescharacteristic
405C2#1172113912yes—
406C2#1172131011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#381
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11720574C2semidirect
4. #63658⋊C3C2semidirect
5. #11720334C2semidirect
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
20. #63658D3semidirect
55. C2×C4×C3⋊D3Dic3semidirect
208. H(3,𝔽3)#381semidirect
341. C32#4664semidirect
342. C32#5211semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×#884502
#117205742
#63658⋊C32
#117203342
#105912943
#105912193