Finite Group Databasebeta

← #13169391

Subgroups — #13169391

Order 1728.

1138 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 212 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
1#11723667C211yesmaximal, characteristic
2C4×Dic3×C3⋊D3C211yesmaximal, characteristic
3#11723450C212yesmaximal, complemented
4C3×#63918C212yesmaximal, complemented
5C3×#63849C211yesmaximal, characteristic
6#11722974C214yesmaximal
7#11722940C214yesmaximal, complemented
8#10594445—32nomaximal
9#10593866—31nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×Dic3×C3⋊D3C2212yes—
12C2×C3×#627C2214yes—
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2212yes—
15C32×C4⋊Dic3C2212yes—
16C32×C4×Dic3C2211yescharacteristic
17#88814C2212yes—
18#88813C2212yes—
19#88807C2214yes—
20Dic3⋊(C3⋊Dic3)C2214yes—
21Dic3×C3⋊Dic3C2212yes—
22C3×Dic3⋊Dic3C2214yes—
23C3×#3722C2212yes—
24C3×Dic32C2212yes—
25#64099—32no—
26C42×C3⋊D3—31no—
27C3×#892—32no—
28C3×#888—32no—
29C3×#869—32no—
30#63918D312yescomplemented
31#63893—34no—
32#63887—34no—
33C4×D3×Dic3—32no—
34C4×D3×Dic3—61no—
35#63872—32no—
36#63872—32no—
37#63872—62no—
38#63871—34no—
39#63849—31no—
40#63849D311yescharacteristic
41#63843—34no—
42Dic3×Dic6—34no—
43#63841—32no—
44#63837—34no—
45C22×C3×C3⋊D3C2311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)C2312yes—
48C2×C3×C3⋊Dic3C2314yes—
49C2×C3×C3⋊Dic3C2311yescharacteristic
50C3×C4×C3⋊D3Q814yescomplemented
51C2×C32×Dic3C2314yes—
52C2×C32×Dic3C2312yes—
53Dic3×C3⋊D3—24no—
54C3×#627—28no—
55#5728—92no—
56#5517—91no—
57C2×C4×C3⋊D3—32no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×C4×D3—32no—
60C2×C3×C4×D3—61no—
61C2×C3×Dic6—34no—
62C2×C3×Dic6—34no—
63C2×#627C2×D314yes—
64C2×D3×Dic3—32no—
65C2×D3×Dic3—32no—
66C2×D3×Dic3—62no—
67C32×C4⋊C4—32no—
68C32×C42—31no—
69C4⋊(C3⋊Dic3)—34no—
70C4⋊(C3⋊Dic3)C2×D312yes—
71#3750—34no—
72C4×C3⋊Dic3—31no—
73C4×C3⋊Dic3—32no—
74C4×C3⋊Dic3—32no—
75C4×C3⋊Dic3—62no—
76C4×C3⋊Dic3—61no—
77C4×C3⋊Dic3—121no—
78C3×(C2×D3)⋊C4—34no—
79C3×C4⋊Dic3—22no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—32no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—62no—
84C3×C4⋊Dic3C2×D312yes—
85C3×C4⋊Dic3C2×D312yes—
86C3×Dic3⋊C4—34no—
87C3×C4×Dic3—21no—
88C3×C4×Dic3—34no—
89C3×C4×Dic3C2×D312yes—
90Dic3⋊Dic3—34no—
91Dic3⋊Dic3—34no—
92Dic3⋊Dic3—34no—
93Dic3⋊Dic3—34no—
94Dic3⋊Dic3—64no—
95Dic3⋊Dic3—64no—
96Dic3⋊Dic3C2×D314yes—
97#3722—32no—
98#3722C2×D312yes—
99(C2×D3)⋊Dic3—34no—
100Dic32—34no—
101Dic32—32no—
102Dic32—32no—
103Dic32—32no—
104Dic32—62no—
105Dic32C2×D312yes—
106C22×C33C2411yescharacteristic
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3C2×Q812yes—
109C33×C4C2×Q812yes—
110C3⋊(C3⋊Dic3)—22no—
111C3×C3⋊Dic3—24no—
112C3×C3⋊Dic3C2×Q812yes—
113C32×Dic3—24no—
114C32×Dic3#4114yes—
115C3×C4⋊Q8—92no—
116C3×#117—91no—
117#946—94no—
118Q8×Dic3—94no—
119#926—91no—
120C2×C4×Dic3—91no—
121#897—94no—
122#892—32no—
123#892—92no—
124#890—92no—
125#890—92no—
126#890—92no—
127#890—182no—
128#888—32no—
129#887—94no—
130#878—94no—
131C42×D3—92no—
132C42×D3—181no—
133#869—32no—
134C4×Dic6—94no—
135C22×C3⋊D3C22×D311yescharacteristic
136C22×C3×D3—32no—
137C22×C3×D3—61no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—32no—
141C2×C32×C4—34no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D312yes—
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—62no—
149C2×C3⋊Dic3—62no—
150C2×C3⋊Dic3—62no—
151C2×C3⋊Dic3—122no—
152C2×C3⋊Dic3C22×D314yes—
153C2×C3⋊Dic3C22×D311yescharacteristic
154C4×C3⋊D3—64no—
155C4×C3⋊D3Dic614yescomplemented
156C2×C3×Dic3—22no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—34no—
160C2×C3×Dic3—34no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3C22×D314yes—
164C2×C3×Dic3C22×D314yes—
165C2×C3×Dic3C22×D312yes—
166C2×C3×Dic3C22×D312yes—
167C3×C4×D3—38no—
168C3×C4×D3—64no—
169C3×Dic6—68no—
170C3×Dic6—68no—
171#627—28no—
172D3×Dic3—68no—
173D3×Dic3—64no—
174D3×Dic3—64no—
175#533—271noSylow
176C2×C33C22×Q811yescharacteristic
177C2×C33C2×#4112yes—
178C3×C3⋊D3—22no—
179C2×C3×Q8—94no—
180C22×C3×C4—91no—
181C22×Dic3—92no—
182C2×C4×D3—32no—
183C2×C4×D3—61no—
184C2×C4×D3—92no—
185C2×C4×D3—92no—
186C2×C4×D3—182no—
187C2×Dic6—34no—
188C2×Dic6—34no—
189C3×C4⋊C4—32no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—62no—
192C3×C4⋊C4—94no—
193C3×C4⋊C4—92no—
194C3×C22⋊C4—92no—
195C3×C42—32no—
196C3×C42—61no—
197C3×C42—92no—
198#217—92no—
199(C2×D3)⋊C4—34no—
200(C2×D3)⋊C4—94no—
201C4⋊Dic3—32no—
202C4⋊Dic3—32no—
203C4⋊Dic3—34no—
204C4⋊Dic3—62no—
205C4⋊Dic3—94no—
206C4⋊Dic3—94no—
207C4⋊Dic3—94no—
208C4⋊Dic3—184no—
209C4⋊Dic3D3212yes—
210Dic3⋊C4—34no—
211Dic3⋊C4—94no—
212Dic3⋊C4—94no—
213Dic3⋊C4—94no—
214Dic3⋊C4—184no—
215C4×Dic3—34no—
216C4×Dic3—92no—
217C4×Dic3—94no—
218C4×Dic3—92no—
219C4×Dic3—92no—
220C4×Dic3—92no—
221C4×Dic3—91no—
222C4×Dic3—92no—
223C4×Dic3—182no—
224C4×Dic3—181no—
225C4×Dic3—182no—
226C4×Dic3—361no—
227C4×Dic3D3211yescharacteristic
228C22×C32—22no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32C23×D311yescharacteristic
232C22×C32C23×D312yes—
233C2×C3⋊D3—22no—
234C2×C3⋊D3C2×Dic612yes—
235C2×C3×D3—34no—
236C2×C3×D3—62no—
237C2×C3×D3—64no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C32×C4—24no—
241C32×C4—22no—
242C32×C4—34no—
243C32×C4—42no—
244C32×C4—64no—
245C32×C4C2×Dic612yes—
246C32×C4D3×Q814yes—
247C3⋊Dic3—24no—
248C3⋊Dic3—62no—
249C3⋊Dic3—64no—
250C3⋊Dic3—62no—
251C3⋊Dic3—62no—
252C3⋊Dic3—122no—
253C3⋊Dic3—122no—
254C3⋊Dic3—124no—
255C3⋊Dic3C2×Dic612yes—
256C3×Dic3—24no—
257C3×Dic3—24no—
258C3×Dic3—24no—
259C3×Dic3—34no—
260C3×Dic3—44no—
261C3×Dic3—62no—
262C3×Dic3—64no—
263C3×Dic3—64no—
264C3×Dic3—124no—
265C3×Dic3#23714yes—
266C3×Dic3#23914yes—
267C4⋊Q8—92no—
268#125—272no—
269C22⋊Q8—274no—
270C4×Q8—274no—
271#117—91no—
272#117—271no—
273C2×C42—271no—
274C33#53311yescharacteristic, complemented, Sylow
275C23×C3—91no—
276C22×D3—32no—
277C22×D3—61no—
278C3×Q8—188no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—34no—
284C2×C3×C4—34no—
285C2×C3×C4—41no—
286C2×C3×C4—62no—
287C2×C3×C4—64no—
288C2×C3×C4—91no—
289C2×C3×C4—94no—
290C2×C3×C4—94no—
291C2×C3×C4C2×D3212yes—
292C2×C3×C4C2×D3211yescharacteristic
293C2×Dic3—32no—
294C2×Dic3—34no—
295C2×Dic3—34no—
296C2×Dic3—61no—
297C2×Dic3—64no—
298C2×Dic3—92no—
299C2×Dic3—92no—
300C2×Dic3—92no—
301C2×Dic3—182no—
302C2×Dic3—182no—
303C2×Dic3—182no—
304C2×Dic3—184no—
305C2×Dic3—362no—
306C2×Dic3C2×D3214yes—
307C2×Dic3C2×D3212yes—
308C4×D3—38no—
309C4×D3—64no—
310C4×D3—184no—
311C4×D3—184no—
312C4×D3—188no—
313Dic6—68no—
314Dic6—68no—
315C2×C32—22no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—41no—
321C2×C32—42no—
322C2×C32C22×Dic611yescharacteristic
323C2×C32C2×#23712yes—
324C2×C32C2×D3×Q812yes—
325C2×C32C2×#23912yes—
326C2×C32D3×#4112yes—
327C3⋊D3—22no—
328C3×D3—64no—
329C3×D3—64no—
330C2×Q8—94no—
331C22×C4—91no—
332C22×C4—272no—
333C4⋊C4—32no—
334C4⋊C4—94no—
335C4⋊C4—92no—
336C4⋊C4—274no—
337C4⋊C4—274no—
338C22⋊C4—92no—
339C22⋊C4—272no—
340C42—31no—
341C42—92no—
342C42—272no—
343C42—272no—
344C42—271no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—41no—
348C22×C3—92no—
349C22×C3—182no—
350C22×C3C22×D3212yes—
351C22×C3C22×D3211yescharacteristic
352C2×D3—34no—
353C2×D3—62no—
354C2×D3—64no—
355C2×D3—62no—
356C2×D3—62no—
357C3×C4—22no—
358C3×C4—24no—
359C3×C4—34no—
360C3×C4—34no—
361C3×C4—42no—
362C3×C4—64no—
363C3×C4—64no—
364C3×C4—64no—
365C3×C4—92no—
366C3×C4—124no—
367C3×C4—184no—
368C3×C4D3×Dic614yes—
369C3×C4#379712yes—
370Dic3—24no—
371Dic3—34no—
372Dic3—62no—
373Dic3—64no—
374Dic3—64no—
375Dic3—124no—
376Dic3—184no—
377Dic3—182no—
378Dic3—182no—
379Dic3—182no—
380Dic3—362no—
381Dic3—362no—
382Dic3—364no—
383Dic3#379614yes—
384C32—21no—
385C32—22no—
386C32—41no—
387C32#551711yescharacteristic, complemented
388C32#572812yescomplemented
389C23—91no—
390Q8—188no—
391C2×C4—32no—
392C2×C4—34no—
393C2×C4—91no—
394C2×C4—94no—
395C2×C4—94no—
396C2×C4—272no—
397C2×C4—544no—
398C2×C4D3311yescharacteristic
399C2×C3—22no—
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—42no—
405C2×C3—41no—
406C2×C3—182no—
407C2×C3C2×D3×Dic612yes—
408C2×C3C2×#379612yes—
409C2×C3C2×#379711yescharacteristic
410C2×C3D3×#23712yes—
411C2×C3D3×#23912yes—
412D3—64no—
413D3—64no—
414C22—92no—
415C22—182no—
416C22C2×D3311yescharacteristic
417C4—34no—
418C4—64no—
419C4—92no—
420C4—184no—
421C4—542no—
422C4#8895712yes—
423C3—21no—
424C3—22no—
425C3—41no—
426C3#1059386611yescharacteristic, complemented
427C3#1059444512yescomplemented
428C2—182no—
429C2D3×#379612yes—
430C2C2×#8895711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723450C2semidirect
4. C3×#63918C2semidirect
7. #11722940C2semidirect
30. #63918D3semidirect
50. C3×C4×C3⋊D3Q8semidirect
155. C4×C3⋊D3Dic6semidirect
274. C33#533semidirect
387. C32#5517semidirect
388. C32#5728semidirect
426. C3#10593866semidirect
427. C3#10594445semidirect

Maximal subgroups

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

subgroupindex
#117236672
C4×Dic3×C3⋊D32
#117234502
C3×#639182
C3×#638492
#117229742
#117229402
#105944453
#105938663