Finite Group Databasebeta

← #13155977

Subgroups — #13155977

Order 1728.

860 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 104 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#11723494C211yesmaximal, characteristic, complemented
2C2×#88790C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4#11722078C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6#10591429—32nomaximal
7#10590551—31nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×D12C2212yes—
10C2×#6289—21no—
11C2×C3×#625—21no—
12C2×C3×#625—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15C32×C4⋊Dic3C2211yescharacteristic
16#88804—21no—
17#88790—24no—
18C2×#3753—32no—
19#64102—31no—
20C3×#940—32no—
21#63914D311yescharacteristic, complemented
22#63894—32no—
23#63894—61no—
24C2×#3714—32no—
25C2×#3714—32no—
26C2×#3714—62no—
27C3×#832—32no—
28C3×#809—32no—
29#63572—31no—
30Dic3⋊(C3⋊C8)—32no—
31#63562D312yescomplemented
32C22×C3⋊(C3⋊D3)—21no—
33C22×C32×D3—22no—
34C2×C33×C4C2311yescharacteristic
35C4⋊(C3⋊(C3⋊D3))—21no—
36C4⋊(C3⋊(C3⋊D3))D412yescomplemented
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C32×Dic3—21no—
40C32×D12—22no—
41C32×D12—22no—
42#6289—42no—
43C3×#625—42no—
44C3×#625—42no—
45C3×C3⋊(C3⋊C8)—22no—
46#5072—91no—
47#4786—92no—
48C2×C32×D4—32no—
49C2×C4⋊(C3⋊D3)—31no—
50C2×C4⋊(C3⋊D3)—32no—
51C2×C4⋊(C3⋊D3)—61no—
52C2×C4⋊(C3⋊D3)—62no—
53C2×C4⋊(C3⋊D3)—121no—
54C2×C4×C3⋊D3—61no—
55C2×C3×#67—62no—
56C2×C3×#67—62no—
57C2×C3×D12—22no—
58C2×C3×D12C2×D312yes—
59C2×C3×D12C2×D312yes—
60C2×#625—21no—
61C2×#625—21no—
62C2×#624—62no—
63C2×#624—62no—
64C32×C4⋊C4—31no—
65#3753—64no—
66#3752—61no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D312yes—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3C2×D312yes—
71C3×C4×Dic3—32no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—32no—
74C2×C3×C3⋊C8—32no—
75C2×C3×C3⋊C8—62no—
76#3722—62no—
77#3722—62no—
78#3714—64no—
79#3714—64no—
80#3714—68no—
81C22×C33C2×D411yescharacteristic
82C2×C3⋊(C3⋊D3)—22no—
83C2×C3⋊(C3⋊D3)—41no—
84C2×C3⋊(C3⋊D3)—41no—
85C2×C32×D3—42no—
86C2×C32×D3—42no—
87C2×C32×D3—42no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3D812yescomplemented
92C32×Dic3—41no—
93C3×#127—91no—
94#940—32no—
95C2×#213—92no—
96C2×#213—92no—
97C2×#213—182no—
98C2×D24—92no—
99#893—92no—
100#893—181no—
101C4×D12—91no—
102C3×#102—92no—
103#832—32no—
104Dic3⋊C8—92no—
105#809—32no—
106C4⋊(C3⋊C8)—91no—
107C23×C32—62no—
108C22×C3⋊D3—61no—
109C22×C3⋊D3—62no—
110C22×C3⋊D3—62no—
111C22×C3⋊D3—122no—
112C22×C3⋊D3—122no—
113C22×C3×D3—22no—
114C22×C3×D3—22no—
115C22×C3×D3—42no—
116C32×D4—62no—
117C32×D4—62no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3#6711yescharacteristic
125C2×C3⋊Dic3#6711yescharacteristic
126C4⋊(C3⋊D3)—32no—
127C4⋊(C3⋊D3)—34no—
128C4⋊(C3⋊D3)—62no—
129C4⋊(C3⋊D3)—64no—
130C4⋊(C3⋊D3)—61no—
131C4⋊(C3⋊D3)—62no—
132C4⋊(C3⋊D3)—62no—
133C4⋊(C3⋊D3)—122no—
134C4⋊(C3⋊D3)—122no—
135C4⋊(C3⋊D3)—122no—
136C4×C3⋊D3—122no—
137C3×#67—124no—
138C3×#67—124no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—32no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C3×D12—24no—
146C3×D12—22no—
147C3×D12—22no—
148C3×D12—22no—
149C3×D12—22no—
150C3×D12—42no—
151#625—42no—
152#625—42no—
153#624—124no—
154#624—124no—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—64no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—62no—
159#459—271noSylow
160C2×C33#12111yescharacteristic
161C2×C33C2×D811yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C3⋊(C3⋊D3)—42no—
164C32×D3—82no—
165C2×C3×D4—32no—
166C2×C3×D4—32no—
167C2×C3×D4—62no—
168C2×C3×D4—181no—
169C2×C3×D4—181no—
170C2×#67—62no—
171C2×#67—62no—
172C2×D12—91no—
173C2×D12—92no—
174C2×D12—181no—
175C2×D12—182no—
176C2×D12—361no—
177C2×D12D3212yes—
178C2×C4×D3—181no—
179C2×C4×D3—182no—
180C2×C4×D3—182no—
181C2×C3×C8—92no—
182C3×C4⋊C4—32no—
183C3×C4⋊C4—61no—
184C3×C42—91no—
185#213—184no—
186#213—184no—
187#213—188no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—182no—
190(C2×D3)⋊C4—182no—
191C4⋊Dic3D3211yescharacteristic
192C4×Dic3—32no—
193C4×Dic3—61no—
194C2×C3⋊C8—32no—
195C2×C3⋊C8—32no—
196C2×C3⋊C8—62no—
197D24—184no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—122no—
202C22×C32—122no—
203C22×C32—122no—
204C22×C32D3×D412yes—
205C22×C32C2×#6711yescharacteristic
206C2×C3⋊D3—62no—
207C2×C3⋊D3—64no—
208C2×C3⋊D3—64no—
209C2×C3⋊D3—124no—
210C2×C3⋊D3—124no—
211C2×C3⋊D3—121no—
212C2×C3⋊D3—121no—
213C2×C3⋊D3—122no—
214C2×C3⋊D3—122no—
215C2×C3⋊D3—122no—
216C2×C3⋊D3—122no—
217C2×C3⋊D3—242no—
218C2×C3⋊D3—242no—
219C2×C3⋊D3—242no—
220C2×C3⋊D3—242no—
221C2×C3×D3—44no—
222C2×C3×D3—42no—
223C2×C3×D3—42no—
224C2×C3×D3—42no—
225C2×C3×D3—42no—
226C2×C3×D3—42no—
227C2×C3×D3—42no—
228C2×C3×D3—82no—
229C2×C3×D3—82no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4—121no—
237C32×C4#23512yes—
238C32×C4D3×D412yes—
239C32×C4D3×D411yescharacteristic
240C32×C4#23711yescharacteristic
241C3⋊Dic3—21no—
242C3⋊Dic3#21312yescomplemented
243C3×Dic3—34no—
244C3×Dic3—42no—
245C3×Dic3—42no—
246C3×Dic3—62no—
247C3×Dic3—62no—
248C3×Dic3—121no—
249C2×D8—272no—
250#127—91no—
251C4×D4—271no—
252C4⋊C8—271no—
253#102—92no—
254C33#45911yescharacteristic, complemented, Sylow
255C23×C3—62no—
256C23×C3—62no—
257C23×C3—122no—
258C22×D3—22no—
259C22×D3—182no—
260C22×D3—182no—
261C22×D3—181no—
262C22×D3—362no—
263C22×D3—362no—
264C3×D4—64no—
265C3×D4—62no—
266C3×D4—62no—
267C3×D4—62no—
268C3×D4—62no—
269C3×D4—122no—
270C3×D4—362no—
271C3×D4—362no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—41no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—91no—
278C2×C3×C4—91no—
279C2×C3×C4C2×D3211yescharacteristic
280C2×C3×C4C2×D3212yes—
281#67—124no—
282#67—124no—
283C2×Dic3—21no—
284C2×Dic3—32no—
285C2×Dic3—32no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288D12—22no—
289D12—22no—
290D12—92no—
291D12—94no—
292D12—182no—
293D12—184no—
294D12—182no—
295D12—181no—
296D12—182no—
297D12—362no—
298D12—362no—
299D12—362no—
300C4×D3—364no—
301C4×D3—362no—
302C4×D3—364no—
303C3×C8—182no—
304C3⋊C8—64no—
305C3⋊C8—62no—
306C3⋊C8—62no—
307C2×C32—21no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—22no—
311C2×C32—21no—
312C2×C32—22no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—242no—
317C2×C32#88412yes—
318C2×C32D3×D812yes—
319C2×C32#91412yes—
320C2×C32C2×#21311yescharacteristic
321C2×C32#93211yescharacteristic
322C2×C32#93811yescharacteristic
323C3⋊D3—122no—
324C3⋊D3—124no—
325C3⋊D3—124no—
326C3⋊D3—244no—
327C3⋊D3—244no—
328C3×D3—84no—
329C3×D3—82no—
330C3×D3—82no—
331C2×D4—32no—
332C2×D4—181no—
333C2×D4—181no—
334C2×D4—271no—
335C22×C4—541no—
336D8—544no—
337C2×C8—92no—
338C4⋊C4—31no—
339C22⋊C4—541no—
340C42—91no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—41no—
344C22×C3—124no—
345C22×C3—122no—
346C22×C3—122no—
347C22×C3—122no—
348C22×C3—122no—
349C22×C3—122no—
350C22×C3—122no—
351C22×C3—242no—
352C22×C3—242no—
353C22×C3D3×#6712yes—
354C22×C3#381111yescharacteristic
355C2×D3—42no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—184no—
359C2×D3—182no—
360C2×D3—184no—
361C2×D3—364no—
362C2×D3—364no—
363C2×D3—362no—
364C2×D3—362no—
365C2×D3—362no—
366C2×D3—361no—
367C2×D3—362no—
368C2×D3—361no—
369C2×D3—722no—
370C2×D3—722no—
371C2×D3—722no—
372C2×D3—722no—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—22no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—92no—
380C3×C4—122no—
381C3×C4—122no—
382C3×C4—181no—
383C3×C4#379811yescharacteristic
384C3×C4#380211yescharacteristic
385C3×C4#380412yes—
386C3×C4#381112yes—
387Dic3—34no—
388Dic3—41no—
389Dic3—62no—
390Dic3—62no—
391Dic3—121no—
392C32—21no—
393C32—22no—
394C32—41no—
395C32#478612yescomplemented
396C32#507211yescharacteristic, complemented
397C23—62no—
398C23—541no—
399D4—62no—
400D4—62no—
401D4—272no—
402D4—362no—
403D4—362no—
404D4—541no—
405C2×C4—61no—
406C2×C4—91no—
407C2×C4—91no—
408C2×C4—1082no—
409C2×C4D3311yescharacteristic
410C8—182no—
411C2×C3—21no—
412C2×C3—22no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—21no—
416C2×C3—22no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—244no—
421C2×C3—242no—
422C2×C3—242no—
423C2×C3#6391211yescharacteristic
424C2×C3D3×#21312yes—
425C2×C3#6392611yescharacteristic
426C2×C3#6393412yes—
427C2×C3#6394111yescharacteristic
428C2×C3#6395812yes—
429D3—82no—
430D3—364no—
431D3—362no—
432D3—364no—
433D3—724no—
434D3—724no—
435C22—122no—
436C22—122no—
437C22—122no—
438C22—542no—
439C22—1081no—
440C22—1081no—
441C22#8895211yescharacteristic
442C4—92no—
443C4—121no—
444C4—181no—
445C4#8895411yescharacteristic
446C4#8895911yescharacteristic
447C3—21no—
448C3—22no—
449C3—41no—
450C3#1059055111yescharacteristic, complemented
451C3#1059142912yescomplemented
452C2—242no—
453C2—1082no—
454C2#1172283011yescharacteristic
455C2#1172287311yescharacteristic
456C2#1172296411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723494C2semidirect
2. C2×#88790C2semidirect
3. C3×#63914C2semidirect
4. #11722078C2semidirect
5. C3×#63562C2semidirect
21. #63914D3semidirect
31. #63562D3semidirect
36. C4⋊(C3⋊(C3⋊D3))D4semidirect
91. C3×C3⋊Dic3D8semidirect
242. C3⋊Dic3#213semidirect
254. C33#459semidirect
395. C32#4786semidirect
396. C32#5072semidirect
450. C3#10590551semidirect
451. C3#10591429semidirect

Maximal subgroups

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

subgroupindex
#117234942
C2×#887902
C3×#639142
#117220782
C3×#635622
#105914293
#105905513