Finite Group Databasebeta

← #13159458

Subgroups — #13159458

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×#89015C211yesmaximal, characteristic, complemented
2#11723936C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63637C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592480—31nomaximal
9#10590617—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6291—21no—
16C2×D3×C3⋊Dic3—21no—
17#89015—22no—
18#89015—22no—
19C33×C4⋊C4C2211yescharacteristic
20#88859—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88797—22no—
24#88790—22no—
25C2×#3830—31no—
26C2×#3796—34no—
27C2×#3753—31no—
28C2×#3744—31no—
29#64104—31no—
30#64104—34no—
31#64104—64no—
32C2×#3717—34no—
33C2×#3714—34no—
34C32×#102—31no—
35#63647—31no—
36#63637D311yescharacteristic, complemented
37C3×#810—34no—
38C3×#809D314yescomplemented
39#63557—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)D411yescharacteristic
42C22×C32×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—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634D412yescomplemented
54C32×D12—21no—
55C32×D12D412yescomplemented
56#6291—24no—
57D3×C3⋊Dic3—24no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4810—91no—
61#4781—94no—
62C2×C32×D4—31no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65#3830—62no—
66#3830—62no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C4⋊(C3⋊D3)—68no—
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—64no—
76C2×#634C2×D311yescharacteristic
77C2×C3×D12C2×D314yes—
78C2×C3×Dic6—34no—
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#3753—62no—
88#3752—61no—
89#3752—64no—
90#3752—68no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92#3744—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×C32×D3—22no—
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107C33×C4—41no—
108C33×C4C2×D411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C3⋊(C3⋊Dic3)—21no—
111C3×C3⋊Dic3—22no—
112C2×#239—91no—
113C2×#237—94no—
114C2×#215—91no—
115C2×#213—94no—
116C2×D24—91no—
117C2×#204—94no—
118#895—94no—
119#895—91no—
120#895—184no—
121C3×#103—91no—
122C3×#102—34no—
123(C2×D3)⋊C8—94no—
124(C2×D3)⋊C8—91no—
125#810—34no—
126#809C3⋊D311yescharacteristic, complemented
127C23×C32—61no—
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—24no—
135C32×D4—32no—
136C32×D4—61no—
137C2×C32×C4—28no—
138C2×C32×C4—24no—
139C2×C32×C4—21no—
140C2×C32×C4—24no—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D311yescharacteristic
143#638—64no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—31no—
146C2×C3⋊Dic3—34no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—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)—128no—
157C4⋊(C3⋊D3)—122no—
158C4×C3⋊D3—61no—
159C4×C3⋊D3—61no—
160C4×C3⋊D3—64no—
161C4×C3⋊D3—64no—
162C4×C3⋊D3—124no—
163C4×C3⋊D3—124no—
164#634—21no—
165#634D1212yescomplemented
166C2×C3×Dic3—64no—
167C3×D12—24no—
168C3×D12D1218yescomplemented
169C3×Dic6—38no—
170C3×Dic6—64no—
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—
183C32×D3—42no—
184C2×C3×Q8—91no—
185C2×C3×D4—34no—
186C2×#67—184no—
187C22×Dic3—184no—
188#239—182no—
189#239—182no—
190#237—188no—
191#237—188no—
192C2×D12—94no—
193C2×D12—91no—
194C2×D12—184no—
195C2×D12—181no—
196C2×D12—184no—
197C2×D12—181no—
198C2×D12—188no—
199C2×D12C2×C3⋊D311yescharacteristic
200C2×C4×D3—94no—
201C2×C4×D3—91no—
202C2×C4×D3—184no—
203C2×C4×D3—181no—
204C2×Dic6—34no—
205C2×C3×C8—34no—
206C2×C3×C8—91no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4D3214yes—
209C3×C4⋊C4C2×C3⋊D311yescharacteristic
210#215—182no—
211#213—188no—
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—182no—
218#204—188no—
219C22×C32—24no—
220C22×C32—62no—
221C22×C32—121no—
222C22×C32—121no—
223C22×C32D3×D414yes—
224C22×C32D3×D411yescharacteristic
225C2×C3⋊D3—68no—
226C2×C3⋊D3—61no—
227C2×C3⋊D3—64no—
228C2×C3⋊D3—61no—
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—124no—
240C2×C3⋊D3—121no—
241C2×C3⋊D3—124no—
242C2×C3⋊D3—121no—
243C2×C3×D3—28no—
244C2×C3×D3—44no—
245C2×C3×D3—44no—
246C32×C4—24no—
247C32×C4—24no—
248C32×C4—48no—
249C32×C4—44no—
250C32×C4—41no—
251C32×C4C2×D1214yes—
252C32×C4C2×D1211yescharacteristic
253C32×C4D3×D414yes—
254C32×C4D3×D411yescharacteristic
255C3⋊Dic3—22no—
256C3⋊Dic3—61no—
257C3⋊Dic3—64no—
258C3⋊Dic3—124no—
259C3×Dic3—68no—
260C2×#41—271no—
261C2×SD16—271no—
262C2×D8—271no—
263#121—271no—
264#103—91no—
265#102—31no—
266C22⋊C8—271no—
267C33#44911yescharacteristic, complemented, Sylow
268C23×C3—64no—
269C22×D3—21no—
270C22×D3—94no—
271C22×D3—91no—
272C22×D3—184no—
273C22×D3—188no—
274C22×D3—184no—
275C22×D3—181no—
276C3×Q8—92no—
277C3×Q8—181no—
278C3×D4—38no—
279C3×D4—64no—
280C2×C3×C4—28no—
281C2×C3×C4—24no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3214yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—1816no—
288C2×Dic3—64no—
289C2×Dic3—94no—
290C2×Dic3—91no—
291C2×Dic3—184no—
292C2×Dic3—1816no—
293D12—21no—
294D12—184no—
295D12—184no—
296D12—184no—
297D12—181no—
298D12—181no—
299D12—364no—
300D12—364no—
301D12—3616no—
302D12—362no—
303D12—368no—
304D12C4⋊(C3⋊D3)12yescomplemented
305C4×D3—184no—
306C4×D3—184no—
307C4×D3—184no—
308C4×D3—181no—
309C4×D3—181no—
310C4×D3—364no—
311C4×D3—364no—
312Dic6—38no—
313Dic6—64no—
314C3×C8—64no—
315C3×C8—181no—
316C3⋊C8—21no—
317C3⋊C8—64no—
318C2×C32—24no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32—122no—
322C2×C32#88214yes—
323C2×C32#88211yescharacteristic
324C2×C32#91114yes—
325C2×C32#91411yescharacteristic
326C2×C32#91614yes—
327C2×C32#91911yescharacteristic
328C3⋊D3—128no—
329C3⋊D3—124no—
330C3⋊D3—121no—
331C3⋊D3—248no—
332C3⋊D3—244no—
333C3⋊D3—241no—
334C3×D3—48no—
335#41—542no—
336#41—542no—
337C2×Q8—91no—
338C2×D4—31no—
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—68no—
352C22×C3—124no—
353C22×C3—124no—
354C22×C3#381114yes—
355C22×C3D4×C3⋊D311yescharacteristic
356C2×D3—22no—
357C2×D3—41no—
358C2×D3—41no—
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—182no—
383C3×C4D3×D1214yes—
384C3×C4D3×D1214yes—
385C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
386C3×C4D4×C3⋊D311yescharacteristic
387Dic3—68no—
388Dic3—181no—
389Dic3—184no—
390Dic3—364no—
391C32—24no—
392C32#478114yescomplemented
393C32#481011yescharacteristic, complemented
394C23—61no—
395C23—271no—
396C23—541no—
397Q8—92no—
398Q8—181no—
399D4—32no—
400D4—61no—
401D4—544no—
402D4—541no—
403D4—541no—
404D4—1082no—
405C2×C4—21no—
406C2×C4—181no—
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—128no—
418C2×C3#6392314yes—
419C2×C3#6393714yes—
420C2×C3#6395014yes—
421C2×C3#6409111yescharacteristic
422C2×C3#6412011yescharacteristic
423C2×C3#6412511yescharacteristic
424D3—42no—
425D3—368no—
426D3—364no—
427D3—361no—
428D3—728no—
429D3—724no—
430D3—721no—
431C22—62no—
432C22—121no—
433C22—121no—
434C22—541no—
435C22—541no—
436C22—541no—
437C22—1081no—
438C22—1081no—
439C22—1081no—
440C22#8903911yescharacteristic
441C4—41no—
442C4—182no—
443C4—541no—
444C4D3×C4⋊(C3⋊D3)11yescharacteristic
445C4C3⋊D3×D1211yescharacteristic
446C3—24no—
447C3#1059061714yescomplemented
448C3#1059248011yescharacteristic, complemented
449C2—122no—
450C2—1081no—
451C2—2161no—
452C2#1172354211yescharacteristic
453C2#1172356411yescharacteristic
454C2#1172358711yescharacteristic
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×#89015C2semidirect
2. #11723936C2semidirect
3. C2×#88797C2semidirect
4. C2×#88790C2semidirect
5. C3×#63637C2semidirect
6. C32×#809C2semidirect
7. #11722024C2semidirect
36. #63637D3semidirect
38. C3×#809D3semidirect
53. C3×#634D4semidirect
55. C32×D12D4semidirect
126. #809C3⋊D3semidirect
165. #634D12semidirect
168. C3×D12D12semidirect
267. C33#449semidirect
304. D12C4⋊(C3⋊D3)semidirect
392. C32#4781semidirect
393. C32#4810semidirect
447. C3#10590617semidirect
448. C3#10592480semidirect

Maximal subgroups

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

subgroupindex
C2×#890152
#117239362
C2×#887972
C2×#887902
C3×#636372
C32×#8092
#117220242
#105924803
#105906173