Finite Group Databasebeta

← #13155416

Subgroups — #13155416

Order 1728.

774 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 126 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×#89040C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic
3C3×#63849C211yesmaximal, characteristic
4#11722039C212yesmaximal, complemented
5C3×#63567C212yesmaximal, complemented
6#10590691—31nomaximal
7#10590600—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×#6296—21no—
11C2×#6294—21no—
12#89040—22no—
13#89040C414yescomplemented
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C32×C4⋊Dic3C2212yes—
16C2×C32×C3⋊C8C2211yescharacteristic
17#88787—22no—
18C3×#3722—21no—
19C3×Dic32—21no—
20C2×#3797—31no—
21C2×D3×Dic6—32no—
22C2×#3743—31no—
23C3×#892—32no—
24#63849D311yescharacteristic
25C2×#3711—32no—
26C2×#3711—61no—
27#63637—32no—
28C3×#816—32no—
29C3×#810—32no—
30#63567D312yescomplemented
31#63565—32no—
32#63565—32no—
33#63565—62no—
34C22×C3×C3⋊D3D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3C2×C412yes—
39C3×C4×C3⋊D3D412yes—
40C3×#634—22no—
41C3×#634C2×C414yes—
42C2×C32×Dic3—22no—
43#6296—24no—
44#6294—24no—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C32×C3⋊C8—21no—
47#5004—91no—
48#4802—92no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×#634C2×D312yes—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—61no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—62no—
56C2×#627—62no—
57C2×#627—61no—
58C2×#624—61no—
59C2×D3×Dic3—62no—
60#3797—34no—
61#3797—62no—
62D3×Dic6—38no—
63D3×Dic6—64no—
64C2×C32×C8—31no—
65C32×C4⋊C4—32no—
66C2×C3⋊(C3⋊C8)—32no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—61no—
69C2×C3⋊(C3⋊C8)—62no—
70C2×C3⋊(C3⋊C8)—121no—
71#3743—62no—
72C3×(C2×D3)⋊C4—62no—
73C3×C4⋊Dic3—22no—
74C3×C4⋊Dic3C2×D312yes—
75C3×C4⋊Dic3C2×D312yes—
76C3×C4×Dic3—62no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8C2×D312yes—
79#3722—21no—
80Dic32—21no—
81#3711—64no—
82#3711—64no—
83C22×C33C2×D411yescharacteristic
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3C22⋊C412yes—
86C33×C4C22×C411yescharacteristic
87C33×C4C2×D411yescharacteristic
88C3×C3⋊Dic3—24no—
89C3×C3⋊Dic3C22⋊C412yes—
90C32×Dic3—42no—
91C2×D3×Q8—92no—
92C22×Dic6—91no—
93C3×#117—91no—
94C2×#208—91no—
95C2×ΓL(1,𝔽25)—92no—
96C2×ΓL(1,𝔽25)—181no—
97#892—32no—
98C3×#103—92no—
99#835—92no—
100#816—32no—
101#810—32no—
102#810—92no—
103#810—92no—
104#810—182no—
105C22×C3⋊D3#6711yescharacteristic
106C22×C3×D3—32no—
107C22×C3×D3—61no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D312yes—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3#6711yescharacteristic
116C4×C3⋊D3C4×D312yes—
117C4×C3⋊D3D1212yes—
118#634—22no—
119#634C4×D314yes—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—122no—
128C3×C4×D3—34no—
129C3×C4×D3—34no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×Dic6—34no—
133C3×Dic6—34no—
134C3×Dic6—64no—
135C3×Dic6—62no—
136C3×Dic6—62no—
137C3×Dic6—122no—
138#627—68no—
139#627—64no—
140#624—64no—
141D3×Dic3—68no—
142C32×C8—61no—
143C3⋊(C3⋊C8)—61no—
144C3⋊(C3⋊C8)—62no—
145C3⋊(C3⋊C8)—62no—
146C3⋊(C3⋊C8)—122no—
147C3⋊(C3⋊C8)—122no—
148C3×C3⋊C8—22no—
149C3×C3⋊C8—22no—
150#419—271noSylow
151C2×C33C2×C22⋊C411yescharacteristic
152C2×C33#13712yes—
153C3×C3⋊D3—22no—
154C2×C3×Q8—92no—
155C22×C3×C4—91no—
156C22×Dic3—181no—
157D3×Q8—98no—
158D3×Q8—184no—
159C2×C4×D3—32no—
160C2×C4×D3—61no—
161C2×C4×D3—182no—
162C2×Dic6—32no—
163C2×Dic6—32no—
164C2×Dic6—62no—
165C2×Dic6—94no—
166C2×Dic6—182no—
167C2×Dic6—182no—
168C2×Dic6—181no—
169C2×C3×C8—32no—
170C2×C3×C8—61no—
171C3×C4⋊C4—32no—
172C3×C4⋊C4—32no—
173C3×C4⋊C4—62no—
174C3×C22⋊C4—181no—
175C3×C42—181no—
176(C2×D3)⋊C4—62no—
177C4⋊Dic3D3212yes—
178C4×Dic3—62no—
179#208—182no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—92no—
182C2×C3⋊C8—181no—
183C2×C3⋊C8—182no—
184C2×C3⋊C8—361no—
185C2×C3⋊C8D3211yescharacteristic
186ΓL(1,𝔽25)—184no—
187ΓL(1,𝔽25)—184no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—41no—
191C22×C32D3×D412yes—
192C22×C32C2×#6711yescharacteristic
193C2×C3⋊D3—22no—
194C2×C3⋊D3(C2×D3)⋊C412yes—
195C2×C3×D3—34no—
196C2×C3×D3—62no—
197C2×C3×D3—62no—
198C2×C3×D3—62no—
199C2×C3×D3—122no—
200C32×C4—22no—
201C32×C4—21no—
202C32×C4—22no—
203C32×C4—21no—
204C32×C4—41no—
205C32×C4—41no—
206C32×C4—122no—
207C32×C4C2×C4×D312yes—
208C32×C4C2×C4×D311yescharacteristic
209C32×C4C2×D1211yescharacteristic
210C32×C4D3×D412yes—
211C3⋊Dic3—24no—
212C3⋊Dic3(C2×D3)⋊C412yes—
213C3×Dic3—34no—
214C3×Dic3—42no—
215C3×Dic3—42no—
216C3×Dic3—62no—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—82no—
220C3×Dic3—124no—
221C22×Q8—271no—
222C2×M16—271no—
223#117—91no—
224#103—92no—
225#103—272no—
226C33#41911yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C22×D3—32no—
229C22×D3—61no—
230C3×Q8—94no—
231C3×Q8—182no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4—91no—
238C2×C3×C4—92no—
239C2×C3×C4—92no—
240C2×C3×C4—122no—
241C2×C3×C4—182no—
242C2×C3×C4C2×D3212yes—
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—22no—
245C2×Dic3—32no—
246C2×Dic3—61no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C2×Dic3—122no—
250C2×Dic3—184no—
251C4×D3—34no—
252C4×D3—34no—
253C4×D3—62no—
254C4×D3—62no—
255C4×D3—188no—
256Dic6—34no—
257Dic6—34no—
258Dic6—64no—
259Dic6—62no—
260Dic6—62no—
261Dic6—122no—
262Dic6—188no—
263Dic6—184no—
264C3×C8—62no—
265C3×C8—62no—
266C3⋊C8—21no—
267C3⋊C8—182no—
268C3⋊C8—182no—
269C3⋊C8—181no—
270C3⋊C8—362no—
271C3⋊C8—362no—
272C2×C32—22no—
273C2×C32—22no—
274C2×C32—21no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—41no—
278C2×C32—42no—
279C2×C32D3×C22⋊C412yes—
280C2×C32#91512yes—
281C2×C32#91812yes—
282C2×C32C2×(C2×D3)⋊C411yescharacteristic
283C2×C32#95112yes—
284C3⋊D3—22no—
285C3×D3—64no—
286C3×D3—122no—
287C2×Q8—92no—
288C2×Q8—274no—
289C2×Q8—541no—
290C2×Q8—542no—
291C22×C4—91no—
292C22×C4—541no—
293M16—542no—
294C2×C8—31no—
295C2×C8—271no—
296C4⋊C4—32no—
297C22⋊C4—181no—
298C42—181no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—41no—
302C22×C3—92no—
303C22×C3—182no—
304C22×C3#380211yescharacteristic
305C22×C3D3×#6712yes—
306C2×D3—34no—
307C2×D3—62no—
308C2×D3—62no—
309C2×D3—62no—
310C2×D3—122no—
311C3×C4—22no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—22no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—92no—
318C3×C4—122no—
319C3×C4—122no—
320C3×C4—184no—
321C3×C4—242no—
322C3×C4C4×D3212yes—
323C3×C4C4×D3211yescharacteristic
324C3×C4D3×D1212yes—
325C3×C4#381111yescharacteristic
326Dic3—34no—
327Dic3—42no—
328Dic3—62no—
329Dic3—64no—
330Dic3—64no—
331Dic3—124no—
332C32—22no—
333C32—21no—
334C32—41no—
335C32#480212yescomplemented
336C32#500411yescharacteristic, complemented
337C23—91no—
338Q8—94no—
339Q8—182no—
340Q8—544no—
341C2×C4—62no—
342C2×C4—91no—
343C2×C4—92no—
344C2×C4—92no—
345C2×C4—182no—
346C2×C4—544no—
347C2×C4D3311yescharacteristic
348C8—61no—
349C8—541no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—22no—
355C2×C3—42no—
356C2×C3—41no—
357C2×C3—182no—
358C2×C3#6381112yes—
359C2×C3D3×(C2×D3)⋊C412yes—
360C2×C3#6392111yescharacteristic
361C2×C3#6392912yes—
362C2×C3#6394312yes—
363D3—64no—
364D3—122no—
365C22—92no—
366C22—182no—
367C22#8895311yescharacteristic
368C4—92no—
369C4—122no—
370C4—184no—
371C4#8894911yescharacteristic
372C4#8895511yescharacteristic
373C3—21no—
374C3—22no—
375C3—41no—
376C3#1059060012yescomplemented
377C3#1059069111yescharacteristic, complemented
378C2—182no—
379C2#1172289112yes—
380C2#1172290511yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722039C2semidirect
5. C3×#63567C2semidirect
13. #89040C4semidirect
30. #63567D3semidirect
226. C33#419semidirect
335. C32#4802semidirect
336. C32#5004semidirect
376. C3#10590600semidirect
377. C3#10590691semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C2×#887872
C3×#638492
#117220392
C3×#635672
#105906913
#105906003