Finite Group Databasebeta

← #13178696

Subgroups — #13178696

Order 1728.

1262 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1262 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C3×#64149C211yesmaximal, characteristic
2C3×#64144C212yesmaximal, complemented
3#11723726C214yesmaximal, complemented
4#11723718C214yesmaximal, complemented
5#11723686C212yesmaximal, complemented
6#11723665C212yesmaximal, complemented
7#10595212—31nomaximal
8#10595200—32nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638C2214yes—
11C22×C3×C3⋊Dic3C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6296C2212yes—
14C2×#6294C2212yes—
15C3×#3757C2212yes—
16C3×#3757C2214yes—
17C3×#3750C2214yes—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19#88815C2212yes—
20#88814C2212yes—
21#88814C2214yes—
22#88813C2214yes—
23#64149D311yescharacteristic
24#64144D312yescomplemented
25C3×#940—32no—
26C3×#940—61no—
27C3×#935—32no—
28C3×#935—32no—
29C3×#935—62no—
30#67⋊Dic3—34no—
31Dic3×#67—34no—
32#63959—34no—
33#63959—34no—
34#63953—34no—
35#63952—34no—
36#63897—32no—
37#63884—34no—
38#63838—32no—
39#63837—34no—
40C23×C33C2312yes—
41C22×C3×C3⋊D3C2312yes—
42C33×D4—41no—
43C2×C33×C4C2311yescharacteristic
44C3×#638—28no—
45C2×C3×C3⋊Dic3—42no—
46C2×C3×C3⋊Dic3C2314yes—
47C2×C3×C3⋊Dic3C2314yes—
48C2×C3×C3⋊Dic3C2312yes—
49C3×C4⋊(C3⋊D3)—41no—
50#6296—24no—
51#6294—42no—
52#5713—91no—
53#5664—92no—
54C2×C32×D4—22no—
55C2×C32×D4—21no—
56C2×C32×D4—41no—
57C2×C32×D4C2×D311yescharacteristic
58C2×C32×D4C2×D312yes—
59C2×#638C2×D314yes—
60C22×C3⋊Dic3C2×D312yes—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×#67—34no—
63C2×C3×#67—34no—
64C2×C3×#67—64no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—62no—
68C2×C3×D12—32no—
69C2×C3×D12—61no—
70C2×#627—34no—
71C2×#627—32no—
72C2×#624—32no—
73C2×D3×Dic3—34no—
74#3757C2×D312yes—
75#3757C2×D314yes—
76#3750C2×D314yes—
77C4×C3⋊Dic3C2×D311yescharacteristic
78C3×#217—34no—
79C3×#217—32no—
80C3×#217—32no—
81C3×#217—34no—
82C3×#217—62no—
83C3×#217—64no—
84C3×Dic3⋊C4—34no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—64no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—61no—
89#3724—32no—
90Dic3⋊Dic3—34no—
91#3722—34no—
92#3722—32no—
93(C2×D3)⋊Dic3—34no—
94(C2×D3)⋊Dic3—34no—
95(C2×D3)⋊Dic3—34no—
96Dic32—34no—
97Dic32—34no—
98Dic32—34no—
99C22×C33—24no—
100C22×C33—22no—
101C22×C33C2411yescharacteristic
102C2×C3×C3⋊D3—24no—
103C2×C3×C3⋊D3—22no—
104C33×C4—21no—
105C3×C3⋊Dic3—24no—
106C3×C3⋊Dic3—24no—
107C3×C3⋊Dic3#4114yes—
108C3×#127—91no—
109C3×#123—92no—
110#940—32no—
111#940—61no—
112#936—94no—
113#935—32no—
114#935—32no—
115#935—62no—
116#935—92no—
117D4×Dic3—94no—
118C4×#67—94no—
119#894—94no—
120#887—94no—
121#885—94no—
122#885—94no—
123#67⋊C4—94no—
124C4×Dic6—92no—
125C23×C32—22no—
126C23×C32—22no—
127C23×C32—22no—
128C23×C32—42no—
129C23×C32C22×D312yes—
130C23×C32C22×D312yes—
131C23×C32C22×D312yes—
132C22×C3⋊D3C22×D312yes—
133C22×C3×D3—34no—
134C22×C3×D3—62no—
135C32×D4—41no—
136C32×D4—42no—
137C32×D4—42no—
138C32×D4—44no—
139C32×D4—44no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D311yescharacteristic
145#638—28no—
146C2×C3⋊Dic3—42no—
147C2×C3⋊Dic3C22×D314yes—
148C2×C3⋊Dic3C22×D314yes—
149C2×C3⋊Dic3C22×D312yes—
150C4⋊(C3⋊D3)—41no—
151C3×#67—68no—
152C3×#67—68no—
153C3×#67—128no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3—64no—
162C2×C3×Dic3—122no—
163C2×C3×Dic3—122no—
164C2×C3×Dic3—124no—
165C3×D12—122no—
166C3×D12—122no—
167#627—68no—
168#627—64no—
169#624—122no—
170D3×Dic3—124no—
171#559—271noSylow
172C2×C33—42no—
173C2×C33C2×#4112yes—
174C2×C33H(5,𝔽2)11yescharacteristic
175C3×C3⋊D3—42no—
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—41no—
179C2×C3×D4—94no—
180C2×C3×D4—91no—
181C2×C3×D4D3211yescharacteristic
182C2×C3×D4D3212yes—
183C22×C3×C4—92no—
184C2×#67—34no—
185C2×#67—34no—
186C2×#67—64no—
187C22×Dic3—32no—
188C22×Dic3—32no—
189C22×Dic3—62no—
190C22×Dic3—92no—
191C2×D12—32no—
192C2×D12—61no—
193C2×C4×D3—94no—
194C2×Dic6—92no—
195C2×Dic6—94no—
196C3×C4⋊C4—94no—
197C3×C22⋊C4—94no—
198C3×C22⋊C4—92no—
199C3×C42—91no—
200#217—32no—
201#217—34no—
202#217—34no—
203#217—32no—
204#217—64no—
205#217—62no—
206#217—94no—
207#217—92no—
208(C2×D3)⋊C4—94no—
209(C2×D3)⋊C4—94no—
210(C2×D3)⋊C4—94no—
211C4⋊Dic3—92no—
212Dic3⋊C4—34no—
213Dic3⋊C4—34no—
214Dic3⋊C4—64no—
215Dic3⋊C4—94no—
216C4×Dic3—32no—
217C4×Dic3—61no—
218C4×Dic3—94no—
219C4×Dic3—94no—
220C4×Dic3—94no—
221C22×C32—22no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—22no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32—42no—
232C22×C32—42no—
233C22×C32—44no—
234C22×C32—44no—
235C22×C32—44no—
236C22×C32—44no—
237C22×C32—84no—
238C22×C32C23×D311yescharacteristic
239C22×C32C23×D312yes—
240C2×C3⋊D3—24no—
241C2×C3⋊D3—22no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244C2×C3×D3—64no—
245C2×C3×D3—122no—
246C2×C3×D3—124no—
247C32×C4—21no—
248C32×C4—22no—
249C32×C4—22no—
250C32×C4—42no—
251C32×C4—42no—
252C3⋊Dic3—24no—
253C3⋊Dic3—24no—
254C3⋊Dic3#23714yes—
255C3×Dic3—38no—
256C3×Dic3—64no—
257C3×Dic3—64no—
258C3×Dic3—64no—
259C3×Dic3—64no—
260C3×Dic3—64no—
261C3×Dic3—124no—
262C3×Dic3—124no—
263#127—91no—
264#124—274no—
265#123—92no—
266#123—272no—
267C4×Q8—272no—
268C4×D4—274no—
269C33#55911yescharacteristic, complemented, Sylow
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—42no—
274C23×C3—92no—
275C23×C3C2×D3212yes—
276C23×C3C2×D3212yes—
277C23×C3C2×D3212yes—
278C22×D3—34no—
279C22×D3—62no—
280C3×D4—42no—
281C3×D4—41no—
282C3×D4—42no—
283C3×D4—44no—
284C3×D4—44no—
285C3×D4—188no—
286C3×D4—361no—
287C2×C3×C4—21no—
288C2×C3×C4—22no—
289C2×C3×C4—41no—
290C2×C3×C4—92no—
291C2×C3×C4—94no—
292C2×C3×C4—94no—
293C2×C3×C4—362no—
294C2×C3×C4C2×D3212yes—
295C2×C3×C4C2×D3211yescharacteristic
296#67—68no—
297#67—68no—
298#67—128no—
299C2×Dic3—34no—
300C2×Dic3—34no—
301C2×Dic3—34no—
302C2×Dic3—34no—
303C2×Dic3—34no—
304C2×Dic3—62no—
305C2×Dic3—64no—
306C2×Dic3—64no—
307C2×Dic3—122no—
308C2×Dic3—122no—
309C2×Dic3—124no—
310C2×Dic3—362no—
311D12—122no—
312D12—122no—
313C4×D3—364no—
314Dic6—188no—
315Dic6—184no—
316C2×C32—24no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—21no—
320C2×C32—42no—
321C2×C32—41no—
322C2×C32—42no—
323C2×C32—44no—
324C2×C32—42no—
325C2×C32—42no—
326C2×C32—82no—
327C2×C32—82no—
328C2×C32—84no—
329C2×C32C2×#23712yes—
330C2×C32#100412yes—
331C2×C32#100411yescharacteristic
332C2×C32D3×#4114yes—
333C3⋊D3—42no—
334C3×D3—124no—
335C3×D3—242no—
336C2×Q8—272no—
337C2×D4—94no—
338C2×D4—91no—
339C2×D4#111311yescharacteristic
340C22×C4—92no—
341C22×C4—272no—
342C4⋊C4—94no—
343C4⋊C4—272no—
344C22⋊C4—94no—
345C22⋊C4—92no—
346C22⋊C4—274no—
347C22⋊C4—272no—
348C42—91no—
349C42—274no—
350C22×C3—24no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—21no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—41no—
366C22×C3—84no—
367C22×C3—184no—
368C22×C3—182no—
369C22×C3C22×D3212yes—
370C22×C3C22×D3211yescharacteristic
371C2×D3—64no—
372C2×D3—64no—
373C2×D3—64no—
374C2×D3—122no—
375C2×D3—124no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—21no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—94no—
382C3×C4—184no—
383C3×C4—184no—
384Dic3—38no—
385Dic3—64no—
386Dic3—64no—
387Dic3—64no—
388Dic3—64no—
389Dic3—64no—
390Dic3—124no—
391Dic3—124no—
392C32—22no—
393C32—21no—
394C32—41no—
395C32#566412yescomplemented
396C32#571311yescharacteristic, complemented
397C23—92no—
398C23C2×#111312yes—
399Q8—544no—
400D4—41no—
401D4—188no—
402D4—361no—
403C2×C4—92no—
404C2×C4—94no—
405C2×C4—94no—
406C2×C4—362no—
407C2×C4—1082no—
408C2×C4C2×#111311yescharacteristic
409C2×C3—22no—
410C2×C3—24no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—44no—
414C2×C3—42no—
415C2×C3—42no—
416C2×C3—42no—
417C2×C3—42no—
418C2×C3—41no—
419C2×C3—82no—
420C2×C3—82no—
421C2×C3—84no—
422C2×C3—362no—
423C2×C3D3×#23714yes—
424C2×C3#6431212yes—
425C2×C3#6433012yes—
426C2×C3#6433011yescharacteristic
427D3—124no—
428D3—242no—
429C22—24no—
430C22—22no—
431C22—184no—
432C22—182no—
433C22C22×#111311yescharacteristic
434C4—21no—
435C4—94no—
436C4—184no—
437C4—184no—
438C3—21no—
439C3—22no—
440C3—41no—
441C3#1059520012yescomplemented
442C3#1059521211yescharacteristic, complemented
443C2—42no—
444C2—362no—
445C2#1172462212yes—
446C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC3×#64149C2
hypercenterC2×D4#1113
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64144C2semidirect
3. #11723726C2semidirect
4. #11723718C2semidirect
5. #11723686C2semidirect
6. #11723665C2semidirect
24. #64144D3semidirect
269. C33#559semidirect
395. C32#5664semidirect
396. C32#5713semidirect
441. C3#10595200semidirect
442. C3#10595212semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C3×#641442
#117237262
#117237182
#117236862
#117236652
#105952123
#105952003