Finite Group Databasebeta

← #13159456

Subgroups — #13159456

Order 1728.

1348 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89017C211yesmaximal, characteristic, complemented
2#11723936C211yesmaximal, characteristic, complemented
3C2×#88795C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592509—31nomaximal
9#10590617—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6292—21no—
16C2×Dic3×C3⋊D3—21no—
17#89017—22no—
18#89017—22no—
19C33×C4⋊C4C2211yescharacteristic
20#88859—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88795—22no—
24#88791—22no—
25C2×#3832—31no—
26C2×#3796—34no—
27C2×#3755—31no—
28C2×#3745—31no—
29#64104—31no—
30#64104—34no—
31#64104—64no—
32C2×#3717—34no—
33C2×#3714—34no—
34C32×#103—31no—
35#63647—31no—
36#63636D311yescharacteristic, complemented
37C3×#810D314yescomplemented
38C3×#809—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)D411yescharacteristic
42C22×C3×C3⋊D3—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))—42no—
49C4×C3⋊(C3⋊D3)—21no—
50C4×C3⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)D412yescomplemented
53C2×C32×Dic3—21no—
54C32×Dic6—21no—
55C32×Dic6D412yescomplemented
56#6292—24no—
57Dic3×C3⋊D3—24no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4810—94no—
61#4781—91no—
62C2×C32×Q8—31no—
63#3832—62no—
64#3832—62no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—34no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4⋊(C3⋊D3)—68no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—61no—
76C2×C4×C3⋊D3—64no—
77C2×C3×D12—34no—
78C2×C3×Dic6C2×D314yes—
79C2×#626—64no—
80C2×D3×Dic3—64no—
81#3796—68no—
82#3796—68no—
83C2×C32×C8—31no—
84C32×C4⋊C4—24no—
85C32×C4⋊C4C2×D314yes—
86C32×C4⋊C4C2×D311yescharacteristic
87#3755—62no—
88#3752—61no—
89#3752—64no—
90#3752—68no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92#3745—62no—
93C2×C3×C3⋊C8—34no—
94C2×C3×C3⋊C8C2×D314yes—
95#3717—68no—
96#3714—68no—
97C22×C33C2×D411yescharacteristic
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3⋊(C3⋊D3)—41no—
104C2×C3×C3⋊D3—22no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C33×C4—41no—
108C33×C4C2×D411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C3⋊(C3⋊Dic3)—21no—
111C32×Dic3—22no—
112C2×#239—94no—
113C2×#237—91no—
114C2×#215—94no—
115C2×#213—91no—
116C2×D24—94no—
117C2×#204—91no—
118#895—94no—
119#895—91no—
120#895—184no—
121C3×#103—34no—
122C3×#102—91no—
123(C2×D3)⋊C8—94no—
124(C2×D3)⋊C8—91no—
125#810C3⋊D311yescharacteristic, complemented
126#809—34no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3—31no—
129C22×C3⋊D3—34no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—68no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—61no—
134C22×C3×D3—64no—
135C32×Q8—32no—
136C32×Q8—61no—
137C2×C32×C4—28no—
138C2×C32×C4—24no—
139C2×C32×C4—21no—
140C2×C32×C4—24no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3—31no—
145C2×C3⋊Dic3—34no—
146C2×C3⋊Dic3—64no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—61no—
151C4⋊(C3⋊D3)—64no—
152C4⋊(C3⋊D3)—64no—
153C4⋊(C3⋊D3)—124no—
154C4⋊(C3⋊D3)—124no—
155C4⋊(C3⋊D3)—1216no—
156C4⋊(C3⋊D3)—122no—
157C4⋊(C3⋊D3)—128no—
158C4⋊(C3⋊D3)D1212yescomplemented
159C4×C3⋊D3—64no—
160C4×C3⋊D3—61no—
161C4×C3⋊D3—61no—
162C4×C3⋊D3—64no—
163C4×C3⋊D3—64no—
164C4×C3⋊D3—124no—
165C4×C3⋊D3—124no—
166C2×C3×Dic3—24no—
167C3×D12—38no—
168C3×D12—64no—
169C3×Dic6—24no—
170C3×Dic6D1218yescomplemented
171#626—616no—
172D3×Dic3—616no—
173C32×C8—61no—
174C3⋊(C3⋊C8)—21no—
175C3×C3⋊C8—24no—
176C3×C3⋊C8—64no—
177#449—271noSylow
178C2×C33#12011yescharacteristic
179C2×C33#13511yescharacteristic
180C2×C33AGL(1,ℤ/8)11yescharacteristic
181C3⋊(C3⋊D3)—41no—
182C3⋊(C3⋊D3)—81no—
183C3×C3⋊D3—42no—
184C2×C3×Q8—34no—
185C2×C3×D4—91no—
186C2×#67—181no—
187C22×Dic3—181no—
188#239—188no—
189#239—188no—
190#237—182no—
191#237—182no—
192C2×D12—34no—
193C2×D12—94no—
194C2×D12—91no—
195C2×D12—184no—
196C2×D12—184no—
197C2×D12—184no—
198C2×D12—181no—
199C2×D12—188no—
200C2×C4×D3—94no—
201C2×C4×D3—91no—
202C2×C4×D3—184no—
203C2×C4×D3—184no—
204C2×Dic6C2×C3⋊D311yescharacteristic
205C2×C3×C8—34no—
206C2×C3×C8—91no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4D3214yes—
209C3×C4⋊C4C2×C3⋊D311yescharacteristic
210#215—188no—
211#213—182no—
212(C2×D3)⋊C4—184no—
213(C2×D3)⋊C4—181no—
214(C2×D3)⋊C4—188no—
215C2×C3⋊C8—34no—
216C2×C3⋊C8C2×C3⋊D311yescharacteristic
217D24—188no—
218#204—182no—
219C22×C32—24no—
220C22×C32D3×D414yes—
221C22×C32D3×D411yescharacteristic
222C2×C3⋊D3—22no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3—68no—
226C2×C3⋊D3—61no—
227C2×C3⋊D3—61no—
228C2×C3⋊D3—64no—
229C2×C3⋊D3—61no—
230C2×C3⋊D3—64no—
231C2×C3⋊D3—64no—
232C2×C3⋊D3—124no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—128no—
235C2×C3⋊D3—128no—
236C2×C3⋊D3—128no—
237C2×C3⋊D3—124no—
238C2×C3⋊D3—121no—
239C2×C3⋊D3—121no—
240C2×C3⋊D3—124no—
241C2×C3⋊D3—121no—
242C2×C3⋊D3—124no—
243C2×C3×D3—68no—
244C2×C3×D3—124no—
245C2×C3×D3—124no—
246C32×C4—24no—
247C32×C4—24no—
248C32×C4—48no—
249C32×C4—41no—
250C32×C4—44no—
251C32×C4—62no—
252C32×C4C2×D1214yes—
253C32×C4C2×D1211yescharacteristic
254C32×C4D3×D414yes—
255C32×C4D3×D411yescharacteristic
256C3⋊Dic3—61no—
257C3⋊Dic3—64no—
258C3⋊Dic3—124no—
259C3×Dic3—28no—
260C2×#41—271no—
261C2×SD16—271no—
262C2×D8—271no—
263#121—271no—
264#103—31no—
265#102—91no—
266C22⋊C8—271no—
267C33#44911yescharacteristic, complemented, Sylow
268C23×C3—181no—
269C22×D3—64no—
270C22×D3—94no—
271C22×D3—91no—
272C22×D3—184no—
273C22×D3—188no—
274C22×D3—184no—
275C22×D3—181no—
276C3×Q8—38no—
277C3×Q8—64no—
278C3×D4—92no—
279C3×D4—181no—
280C2×C3×C4—28no—
281C2×C3×C4—24no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—64no—
285C2×C3×C4C2×D3214yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—184no—
288C2×Dic3—21no—
289C2×Dic3—94no—
290C2×Dic3—91no—
291C2×Dic3—184no—
292C2×Dic3—184no—
293D12—38no—
294D12—64no—
295D12—1816no—
296D12—184no—
297D12—184no—
298D12—181no—
299D12—181no—
300D12—364no—
301D12—364no—
302D12—3616no—
303D12—362no—
304D12—368no—
305C4×D3—1816no—
306C4×D3—184no—
307C4×D3—184no—
308C4×D3—181no—
309C4×D3—181no—
310C4×D3—364no—
311C4×D3—364no—
312Dic6—21no—
313Dic6C4⋊(C3⋊D3)12yescomplemented
314C3×C8—64no—
315C3×C8—181no—
316C3⋊C8—21no—
317C3⋊C8—64no—
318C2×C32—24no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32#88214yes—
322C2×C32#88211yescharacteristic
323C2×C32#91111yescharacteristic
324C2×C32#91414yes—
325C2×C32#91611yescharacteristic
326C2×C32#91914yes—
327C3⋊D3—42no—
328C3⋊D3—128no—
329C3⋊D3—121no—
330C3⋊D3—124no—
331C3⋊D3—248no—
332C3⋊D3—241no—
333C3⋊D3—244no—
334C3×D3—128no—
335#41—542no—
336#41—542no—
337C2×Q8—31no—
338C2×D4—91no—
339C2×D4—271no—
340C2×D4—541no—
341C2×D4—541no—
342C22×C4—271no—
343C22×C4—541no—
344SD16—542no—
345D8—542no—
346C2×C8—31no—
347C2×C8—91no—
348C4⋊C4D3×C3⋊D311yescharacteristic
349C22⋊C4—541no—
350C22×C3—24no—
351C22×C3—182no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3#381114yes—
355C22×C3D4×C3⋊D311yescharacteristic
356C2×D3—68no—
357C2×D3—124no—
358C2×D3—124no—
359C2×D3—188no—
360C2×D3—181no—
361C2×D3—181no—
362C2×D3—181no—
363C2×D3—184no—
364C2×D3—184no—
365C2×D3—184no—
366C2×D3—364no—
367C2×D3—364no—
368C2×D3—368no—
369C2×D3—368no—
370C2×D3—368no—
371C2×D3—361no—
372C2×D3—361no—
373C2×D3—361no—
374C2×D3—364no—
375C2×D3—364no—
376C2×D3—364no—
377C3×C4—24no—
378C3×C4—24no—
379C3×C4—48no—
380C3×C4—41no—
381C3×C4—44no—
382C3×C4—68no—
383C3×C4D3×D1214yes—
384C3×C4D3×D1214yes—
385C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
386C3×C4D4×C3⋊D311yescharacteristic
387Dic3—22no—
388Dic3—181no—
389Dic3—184no—
390Dic3—364no—
391C32—24no—
392C32#478111yescharacteristic, complemented
393C32#481014yescomplemented
394C23—181no—
395C23—271no—
396C23—541no—
397Q8—32no—
398Q8—61no—
399D4—92no—
400D4—181no—
401D4—544no—
402D4—541no—
403D4—541no—
404D4—1082no—
405C2×C4—21no—
406C2×C4—61no—
407C2×C4—271no—
408C2×C4—544no—
409C2×C4—541no—
410C2×C4—541no—
411C2×C4C2×D3×C3⋊D311yescharacteristic
412C8—61no—
413C8—181no—
414C2×C3—24no—
415C2×C3—24no—
416C2×C3—24no—
417C2×C3—362no—
418C2×C3#6392314yes—
419C2×C3#6393714yes—
420C2×C3#6395014yes—
421C2×C3#6409111yescharacteristic
422C2×C3#6412311yescharacteristic
423C2×C3#6412811yescharacteristic
424D3—128no—
425D3—368no—
426D3—364no—
427D3—361no—
428D3—728no—
429D3—724no—
430D3—721no—
431C22—182no—
432C22—361no—
433C22—361no—
434C22—541no—
435C22—541no—
436C22—541no—
437C22—1081no—
438C22—1081no—
439C22—1081no—
440C22#8903911yescharacteristic
441C4—41no—
442C4—62no—
443C4—541no—
444C4D3×C4⋊(C3⋊D3)11yescharacteristic
445C4C3⋊D3×D1211yescharacteristic
446C3—24no—
447C3#1059061714yescomplemented
448C3#1059250911yescharacteristic, complemented
449C2—362no—
450C2—1081no—
451C2—2161no—
452C2#1172354211yescharacteristic
453C2#1172356611yescharacteristic
454C2#1172358911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89017C2semidirect
2. #11723936C2semidirect
3. C2×#88795C2semidirect
4. C2×#88791C2semidirect
5. C3×#63636C2semidirect
6. C32×#810C2semidirect
7. #11722024C2semidirect
36. #63636D3semidirect
37. C3×#810D3semidirect
52. C3×C4⋊(C3⋊D3)D4semidirect
55. C32×Dic6D4semidirect
125. #810C3⋊D3semidirect
158. C4⋊(C3⋊D3)D12semidirect
170. C3×Dic6D12semidirect
267. C33#449semidirect
313. Dic6C4⋊(C3⋊D3)semidirect
392. C32#4781semidirect
393. C32#4810semidirect
447. C3#10590617semidirect
448. C3#10592509semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
#117239362
C2×#887952
C2×#887912
C3×#636362
C32×#8102
#117220242
#105925093
#105906173