Finite Group Databasebeta

← #13140242

Subgroups — #13140242

Order 1728.

925 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 925 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6198C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C212yesmaximal, complemented
4C2×#88450C212yesmaximal, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#10591392—32nomaximal
7C23×(C3⋊D3)⋊C3C2212yes—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6198—24no—
13C2×#6198—22no—
14C4⋊#1084—22no—
15#3752⋊C3—24no—
16#88450—24no—
17C22×#626—32no—
18C2×#3752D312yescomplemented
19C2×C3×(C2×D3)⋊C4—32no—
20C2×C3×C4⋊Dic3—32no—
21C2×C3×C4⋊Dic3—61no—
22C2×#3722—32no—
23C2×(C2×D3)⋊Dic3—32no—
24#63579—32no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C22×(C3⋊D3)⋊C3—22no—
27C22×(C3⋊D3)⋊C3—22no—
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3D414yes—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)D412yescomplemented
36C2×#1084—22no—
37C2×#1084—21no—
38C2×#1084—22no—
39C2×#1084D412yes—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43#6198—28no—
44#4991—91no—
45#4961—92no—
46C23×C3⋊D3C2×D312yes—
47C23×C3×D3—32no—
48C22×C32×C4—21no—
49C22×C32×C4C2×D312yes—
50C22×C3⋊Dic3C2×D312yes—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3—61no—
56C2×#626—64no—
57C2×#626—64no—
58C2×#626—64no—
59#3752—24no—
60C3×(C2×D3)⋊C4—64no—
61C3×C4⋊Dic3—64no—
62C3×C4⋊Dic3—64no—
63#3722—64no—
64(C2×D3)⋊Dic3—64no—
65C22×H(3,𝔽3)C2×D412yes—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C22×H(3,𝔽3)#4111yescharacteristic
69C22×H(3,𝔽3)#4112yes—
70C2×(C3⋊D3)⋊C3—24no—
71C2×(C3⋊D3)⋊C3—24no—
72C2×(C3⋊D3)⋊C3—24no—
73C2×(C3⋊D3)⋊C3—42no—
74C2×(C3⋊D3)⋊C3—42no—
75C2×(C3⋊D3)⋊C3—42no—
76C2×(C3⋊D3)⋊C3—42no—
77C4×H(3,𝔽3)—22no—
78#1084—22no—
79#1084—41no—
80(C3⋊Dic3)⋊C3—42no—
81C22×#67—92no—
82C22×D12—91no—
83C2×C3×C4⋊C4—91no—
84C2×C3×C22⋊C4—92no—
85C2×#217—92no—
86C2×(C2×D3)⋊C4—32no—
87C2×(C2×D3)⋊C4—92no—
88C2×(C2×D3)⋊C4—92no—
89C2×(C2×D3)⋊C4—92no—
90C2×C4⋊Dic3—32no—
91C2×C4⋊Dic3—61no—
92C3⋊#95—91no—
93C3⋊#95—92no—
94C23×C32—21no—
95C23×C32C22×D312yes—
96C22×C3⋊D3—22no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—22no—
99C22×C3⋊D3—22no—
100C22×C3⋊D3—22no—
101C22×C3⋊D3—22no—
102C22×C3⋊D3D1214yes—
103C22×C3×D3—34no—
104C22×C3×D3—62no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C22×C3×D3—62no—
109C22×C3×D3—62no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4#6714yescomplemented
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
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—122no—
131C2×C3×Dic3—122no—
132#626—616no—
133#394—271noSylow
134C2×H(3,𝔽3)#12011yescharacteristic
135C2×H(3,𝔽3)#12112yes—
136C2×H(3,𝔽3)#12111yescharacteristic
137C2×H(3,𝔽3)#12312yes—
138C2×H(3,𝔽3)#12411yescharacteristic
139(C3⋊D3)⋊C3—44no—
140C24×C3—92no—
141C23×D3—32no—
142C23×D3—92no—
143C22×C3×C4—32no—
144C22×C3×C4—61no—
145C22×C3×C4—91no—
146C22×C3×C4—92no—
147C22×C3×C4—91no—
148C22×C3×C4D3211yescharacteristic
149C2×#67—184no—
150C2×#67—184no—
151C2×#67—184no—
152C22×Dic3—32no—
153C22×Dic3—32no—
154C22×Dic3—32no—
155C22×Dic3—61no—
156C22×Dic3—61no—
157C22×Dic3—92no—
158C2×D12—182no—
159C2×D12—184no—
160C3×C4⋊C4—182no—
161C3×C22⋊C4—184no—
162#217—184no—
163(C2×D3)⋊C4—64no—
164(C2×D3)⋊C4—184no—
165(C2×D3)⋊C4—184no—
166(C2×D3)⋊C4—184no—
167C4⋊Dic3—64no—
168C4⋊Dic3—64no—
169C22×C32—22no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32C2×D1212yes—
175C22×C32#23512yes—
176C22×C32D3×D412yes—
177C22×C32D3×D412yes—
178C22×C32#23712yes—
179C22×C32#23712yes—
180C22×C32C2×#6712yes—
181C2×C3⋊D3—24no—
182C2×C3⋊D3—24no—
183C2×C3⋊D3—24no—
184C2×C3⋊D3—42no—
185C2×C3⋊D3—42no—
186C2×C3⋊D3—42no—
187C2×C3⋊D3—42no—
188C2×C3×D3—64no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—122no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C2×C3×D3—122no—
195C32×C4—24no—
196C32×C4—24no—
197C3⋊Dic3—42no—
198C3×Dic3—64no—
199C3×Dic3—122no—
200C3×Dic3—122no—
201C3×Dic3—122no—
202C3×Dic3—122no—
203C22×D4—271no—
204C2×C4⋊C4—91no—
205C2×C22⋊C4—92no—
206C2×C22⋊C4—272no—
207#95—271no—
208H(3,𝔽3)#39411yescharacteristic, complemented, Sylow
209C23×C3—32no—
210C23×C3—61no—
211C23×C3—94no—
212C23×C3—182no—
213C23×C3—182no—
214C23×C3—182no—
215C23×C3—182no—
216C23×C3—182no—
217C23×C3—182no—
218C23×C3C2×D3211yescharacteristic
219C22×D3—34no—
220C22×D3—62no—
221C22×D3—62no—
222C22×D3—62no—
223C22×D3—62no—
224C22×D3—62no—
225C22×D3—62no—
226C22×D3—94no—
227C22×D3—182no—
228C22×D3—182no—
229C22×D3—182no—
230C22×D3—182no—
231C22×D3—182no—
232C22×D3—182no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—34no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—92no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4—182no—
245C2×C3×C4—182no—
246C2×C3×C4—182no—
247C2×C3×C4—181no—
248C2×C3×C4#62512yes—
249#67—1816no—
250C2×Dic3—34no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—62no—
261C2×Dic3—122no—
262C2×Dic3—122no—
263C2×Dic3—182no—
264C2×Dic3—182no—
265C2×Dic3—182no—
266D12—188no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32#88212yes—
273C2×C32#88312yes—
274C2×C32#88512yes—
275C2×C32#88612yes—
276C2×C32#93012yes—
277C2×C32#93812yes—
278C2×C32#93912yes—
279C3⋊D3—44no—
280C3×D3—124no—
281C24—92no—
282C2×D4—542no—
283C2×D4—544no—
284C22×C4—91no—
285C22×C4—92no—
286C22×C4—91no—
287C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C4⋊C4—182no—
289C22⋊C4—184no—
290C22⋊C4—544no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—32no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—32no—
297C22×C3—32no—
298C22×C3—62no—
299C22×C3—62no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—184no—
304C22×C3—184no—
305C22×C3—184no—
306C22×C3—362no—
307C22×C3—362no—
308C22×C3—362no—
309C22×C3—362no—
310C22×C3#379512yes—
311C22×C3D3×D1212yes—
312C22×C3#380511yescharacteristic
313C22×C3C2×#62511yescharacteristic
314C22×C3#381111yescharacteristic
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—64no—
318C2×D3—122no—
319C2×D3—122no—
320C2×D3—122no—
321C2×D3—122no—
322C2×D3—184no—
323C2×D3—184no—
324C2×D3—184no—
325C2×D3—362no—
326C2×D3—362no—
327C2×D3—362no—
328C2×D3—362no—
329C3×C4—22no—
330C3×C4—64no—
331C3×C4—64no—
332C3×C4—182no—
333C3×C4—362no—
334C3×C4—361no—
335Dic3—64no—
336Dic3—122no—
337Dic3—122no—
338Dic3—122no—
339Dic3—122no—
340Dic3—362no—
341C32—21no—
342C32#496112yescomplemented
343C23—94no—
344C23—182no—
345C23—182no—
346C23—182no—
347C23—182no—
348C23—182no—
349C23—182no—
350C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351D4—548no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—92no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—181no—
361C2×C4#619812yescomplemented
362C2×C3—32no—
363C2×C3—32no—
364C2×C3—32no—
365C2×C3—32no—
366C2×C3—32no—
367C2×C3—32no—
368C2×C3—32no—
369C2×C3—61no—
370C2×C3—62no—
371C2×C3—62no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—364no—
375C2×C3#6389112yes—
376C2×C3#6390812yes—
377C2×C3#6391311yescharacteristic
378C2×C3#6392311yescharacteristic
379C2×C3#6398011yescharacteristic
380D3—124no—
381D3—364no—
382C22—184no—
383C22—184no—
384C22—184no—
385C22—362no—
386C22—362no—
387C22—362no—
388C22—362no—
389C22#8864812yes—
390C22#8865412yes—
391C22#8867211yescharacteristic
392C22C2×#619811yescharacteristic
393C22#8867711yescharacteristic
394C4—22no—
395C4—182no—
396C4—362no—
397C4—361no—
398C3—32no—
399C3—61no—
400C3#1059139411yescharacteristic
401C2—364no—
402C2#1172113112yes—
403C2#1172114112yes—
404C2#1172114311yescharacteristic
405C2#1172116311yescharacteristic
406C2#1172131811yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC22×#6198C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#394
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1084C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×#88450C2semidirect
5. #11720334C2semidirect
18. C2×#3752D3semidirect
35. C2×C4×H(3,𝔽3)D4semidirect
115. C2×C32×C4#67semidirect
208. H(3,𝔽3)#394semidirect
342. C32#4961semidirect
361. C2×C4#6198semidirect

Maximal subgroups

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

subgroupindex
C22×#61982
C2×C4⋊#10842
C2×#3752⋊C32
C2×#884502
#117203342
#105913923