Finite Group Databasebeta

← #13162026

Subgroups — #13162026

Order 1728.

878 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 143 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2#11723667C211yesmaximal, characteristic
3C2×#88809C211yesmaximal, characteristic
4C3×#63662C212yesmaximal, complemented
5#11722076C212yesmaximal, complemented
6#10591562—31nomaximal
7#10591557—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×Q8×C3⋊D3—22no—
10C3×Q8×C3⋊D3C414yescomplemented
11C2×C3×C4×C3⋊D3—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634—21no—
14C2×C3×#634C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17#88814—21no—
18#88813—21no—
19#88809—22no—
20C2×Q8×C3⋊D3D311yescharacteristic
21C2×C3×D3×Q8—32no—
22C2×C3×D3×Q8—61no—
23#63849—31no—
24#63841—32no—
25C2×#3712—31no—
26C2×#3711—32no—
27#63662D312yescomplemented
28C3×#835—32no—
29C3×#835—32no—
30C3×#835—62no—
31#63567—32no—
32#63566—32no—
33#63565—32no—
34C22×C3×C3⋊D3D411yescharacteristic
35C33×Q8—21no—
36C33×Q8C2×C412yes—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3—24no—
43C3×C4×C3⋊D3C2×C412yes—
44C3×C4×C3⋊D3D412yes—
45C3×#634—21no—
46C3×#634—24no—
47C3×#634C2×C412yes—
48C3×C3⋊(C3⋊C8)—22no—
49#5225—91no—
50#4802—92no—
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—41no—
54C2×C32×Q8C2×D312yes—
55C2×C32×Q8C2×D311yescharacteristic
56Q8×C3⋊D3—22no—
57Q8×C3⋊D3Dic314yescomplemented
58C2×C4×C3⋊D3—21no—
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×#634—21no—
61C2×#634C2×D311yescharacteristic
62C3×D3×Q8—38no—
63C3×D3×Q8—64no—
64C3×D3×Q8—64no—
65C3×D3×Q8—64no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—62no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—62no—
70C2×C3×Dic6—32no—
71C2×C3×Dic6—62no—
72C2×C3×Dic6—61no—
73C2×C3×Dic6—62no—
74C4⋊(C3⋊Dic3)C2×D312yes—
75C2×C3⋊(C3⋊C8)C2×D312yes—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—62no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—32no—
81C2×C3×C3⋊C8—62no—
82#3722—61no—
83(C2×D3)⋊Dic3—62no—
84Dic32—62no—
85Dic32—61no—
86#3712—62no—
87#3711—64no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3C22⋊C412yes—
91C33×C4—22no—
92C33×C4C22×C411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3—42no—
96C3×C3⋊Dic3C22⋊C412yes—
97C22×C3×Q8—91no—
98C2×D3×Q8—32no—
99C2×D3×Q8—61no—
100#926—91no—
101C2×#208—91no—
102C2×ΓL(1,𝔽25)—92no—
103#892—92no—
104C3×#103—92no—
105#835—32no—
106#835—32no—
107#835—62no—
108#835—92no—
109#816—92no—
110#810—92no—
111C22×C3⋊D3#6711yescharacteristic
112C22×C3×D3—32no—
113C22×C3×D3—61no—
114C32×Q8—21no—
115C32×Q8—22no—
116C32×Q8—22no—
117C32×Q8—24no—
118C32×Q8—22no—
119C32×Q8—42no—
120C32×Q8—42no—
121C32×Q8—42no—
122C32×Q8C4×D314yes—
123C32×Q8C2×Dic312yes—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4—42no—
131C2×C32×C4—42no—
132C2×C32×C4C22×D312yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3#6711yescharacteristic
137C4×C3⋊D3—24no—
138C4×C3⋊D3C2×Dic312yes—
139C4×C3⋊D3#6712yes—
140#634—21no—
141#634—24no—
142#634C2×Dic312yes—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—122no—
150C3×C4×D3—34no—
151C3×C4×D3—34no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×C4×D3—68no—
155C3×C4×D3—68no—
156C3×Dic6—34no—
157C3×Dic6—62no—
158C3×Dic6—68no—
159C3×Dic6—68no—
160C3×Dic6—62no—
161C3×Dic6—62no—
162C3⋊(C3⋊C8)—22no—
163C3×C3⋊C8—62no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—122no—
166#419—271noSylow
167C2×C33C2×C22⋊C411yescharacteristic
168C2×C33#13712yes—
169C3×C3⋊D3—22no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—41no—
173C2×C3×Q8—94no—
174C2×C3×Q8—91no—
175C2×C3×Q8—181no—
176C2×C3×Q8—182no—
177C2×C3×Q8D3212yes—
178C2×C3×Q8D3211yescharacteristic
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181D3×Q8—38no—
182D3×Q8—64no—
183D3×Q8—64no—
184D3×Q8—64no—
185C2×C4×D3—32no—
186C2×C4×D3—61no—
187C2×C4×D3—62no—
188C2×C4×D3—62no—
189C2×Dic6—32no—
190C2×Dic6—61no—
191C2×Dic6—62no—
192C2×Dic6—62no—
193C2×C3×C8—92no—
194C3×C4⋊C4—92no—
195#217—181no—
196(C2×D3)⋊C4—182no—
197C4⋊Dic3—32no—
198C4⋊Dic3—32no—
199C4⋊Dic3—62no—
200C4×Dic3—182no—
201C4×Dic3—181no—
202#208—182no—
203C2×C3⋊C8—32no—
204C2×C3⋊C8—32no—
205C2×C3⋊C8—62no—
206ΓL(1,𝔽25)—184no—
207C22×C32—22no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32D3×D412yes—
211C22×C32C2×#6711yescharacteristic
212C2×C3⋊D3—22no—
213C2×C3⋊D3#21712yes—
214C2×C3×D3—34no—
215C2×C3×D3—62no—
216C2×C3×D3—64no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C32×C4—22no—
220C32×C4—24no—
221C32×C4—24no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4—22no—
225C32×C4—21no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4—44no—
229C32×C4—44no—
230C32×C4C2×C4×D312yes—
231C32×C4D3×D412yes—
232C32×C4C22×Dic311yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C3⋊Dic3—22no—
235C3⋊Dic3—42no—
236C3⋊Dic3#21712yes—
237C3×Dic3—34no—
238C3×Dic3—62no—
239C3×Dic3—64no—
240C3×Dic3—64no—
241C3×Dic3—122no—
242C3×Dic3—122no—
243C3×Dic3—242no—
244C22×Q8—91no—
245C2×M16—271no—
246#117—271no—
247#103—92no—
248#103—272no—
249C33#41911yescharacteristic, complemented, Sylow
250C23×C3—91no—
251C22×D3—32no—
252C22×D3—61no—
253C3×Q8—22no—
254C3×Q8—22no—
255C3×Q8—21no—
256C3×Q8—22no—
257C3×Q8—24no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×Q8—42no—
261C3×Q8—92no—
262C3×Q8—184no—
263C3×Q8—181no—
264C3×Q8D3×Dic314yes—
265C3×Q8#62412yes—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—21no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—91no—
275C2×C3×C4—92no—
276C2×C3×C4—92no—
277C2×C3×C4—184no—
278C2×C3×C4—181no—
279C2×C3×C4—182no—
280C2×C3×C4C2×D3212yes—
281C2×C3×C4C2×D3211yescharacteristic
282C2×Dic3—32no—
283C2×Dic3—61no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—122no—
289C4×D3—34no—
290C4×D3—34no—
291C4×D3—62no—
292C4×D3—62no—
293C4×D3—68no—
294C4×D3—68no—
295Dic6—34no—
296Dic6—62no—
297Dic6—68no—
298Dic6—68no—
299Dic6—62no—
300Dic6—62no—
301C3×C8—182no—
302C3⋊C8—62no—
303C3⋊C8—62no—
304C3⋊C8—122no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32—42no—
312C2×C32D3×C22⋊C412yes—
313C2×C32#91512yes—
314C2×C32#91812yes—
315C2×C32#94212yes—
316C2×C32C2×#21711yescharacteristic
317C3⋊D3—22no—
318C3×D3—64no—
319C3×D3—64no—
320C2×Q8—94no—
321C2×Q8—91no—
322C2×Q8—181no—
323C2×Q8—182no—
324C2×Q8#111311yescharacteristic
325C22×C4—91no—
326C22×C4—181no—
327M16—542no—
328C2×C8—92no—
329C4⋊C4—92no—
330C22⋊C4—541no—
331C42—541no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—41no—
335C22×C3—92no—
336C22×C3—182no—
337C22×C3D3×#6712yes—
338C22×C3#381111yescharacteristic
339C2×D3—34no—
340C2×D3—62no—
341C2×D3—64no—
342C2×D3—62no—
343C2×D3—62no—
344C3×C4—24no—
345C3×C4—24no—
346C3×C4—22no—
347C3×C4—21no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—44no—
352C3×C4—44no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—92no—
356C3×C4—182no—
357C3×C4—362no—
358C3×C4C2×D3×Dic312yes—
359C3×C4C2×#62411yescharacteristic
360C3×C4D3×#6712yes—
361C3×C4#381111yescharacteristic
362Dic3—34no—
363Dic3—62no—
364Dic3—64no—
365Dic3—64no—
366Dic3—122no—
367Dic3—122no—
368Dic3—242no—
369C32—22no—
370C32—21no—
371C32—41no—
372C32#480212yescomplemented
373C32#522511yescharacteristic, complemented
374C23—91no—
375Q8—21no—
376Q8—92no—
377Q8—184no—
378Q8—181no—
379Q8#629412yes—
380C2×C4—21no—
381C2×C4—91no—
382C2×C4—92no—
383C2×C4—92no—
384C2×C4—184no—
385C2×C4—181no—
386C2×C4—182no—
387C2×C4C2×#111311yescharacteristic
388C8—182no—
389C2×C3—22no—
390C2×C3—22no—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—22no—
394C2×C3—42no—
395C2×C3—41no—
396C2×C3—182no—
397C2×C3#6394612yes—
398C2×C3#6394812yes—
399C2×C3#6395112yes—
400C2×C3D3×#21712yes—
401C2×C3#6397411yescharacteristic
402D3—64no—
403D3—64no—
404C22—92no—
405C22—182no—
406C22#8904911yescharacteristic
407C4—22no—
408C4—92no—
409C4—182no—
410C4—362no—
411C4C2×#629411yescharacteristic
412C4#8904911yescharacteristic
413C3—21no—
414C3—22no—
415C3—41no—
416C3#1059155712yescomplemented
417C3#1059156211yescharacteristic, complemented
418C2—182no—
419C2#1172371212yes—
420C2#1172372311yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. C3×#63662C2semidirect
5. #11722076C2semidirect
10. C3×Q8×C3⋊D3C4semidirect
27. #63662D3semidirect
57. Q8×C3⋊D3Dic3semidirect
249. C33#419semidirect
372. C32#4802semidirect
373. C32#5225semidirect
416. C3#10591557semidirect
417. C3#10591562semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
#117236672
C2×#888092
C3×#636622
#117220762
#105915623
#105915573