Finite Group Databasebeta

← #13159433

Subgroups — #13159433

Order 1728.

1324 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 160 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×#89019C211yesmaximal, characteristic, complemented
2#11723936C211yesmaximal, characteristic
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63637C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592505—31nomaximal
9#10590606—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6293—21no—
16C2×#6289—21no—
17#89019—22no—
18#89019—22no—
19C33×C4⋊C4C2211yescharacteristic
20#88859—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88797—22no—
24#88795—22no—
25C2×Q8×C3⋊D3—31no—
26C2×#3797—34no—
27C2×#3755—31no—
28C2×#3744—31no—
29#64104—31no—
30#64104—34no—
31#64104—64no—
32C2×#3717—34no—
33C2×#3717—34no—
34C32×#103—31no—
35#63647—31no—
36#63637D311yescharacteristic, complemented
37C3×#810—34no—
38C3×#810D314yescomplemented
39#63557—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)D411yescharacteristic
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))—42no—
48C4×C3⋊(C3⋊D3)—21no—
49C4×C3⋊(C3⋊D3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634D412yescomplemented
53C2×C32×Dic3—21no—
54C32×Dic6—21no—
55C32×Dic6D412yescomplemented
56#6293—24no—
57#6289—24no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4806—94no—
61#4806—91no—
62C2×C32×Q8—31no—
63Q8×C3⋊D3—62no—
64Q8×C3⋊D3—62no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—34no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—64no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C4⋊(C3⋊D3)—68no—
71C2×C4×C3⋊D3—31no—
72C2×C4×C3⋊D3—34no—
73C2×C4×C3⋊D3—61no—
74C2×C4×C3⋊D3—64no—
75C2×#634—61no—
76C2×#634C2×D311yescharacteristic
77C2×C3×Dic6—34no—
78C2×C3×Dic6C2×D314yes—
79C2×#627—64no—
80C2×#624—64no—
81#3797—68no—
82#3797—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#3744—62no—
93C2×C3×C3⋊C8—34no—
94C2×C3×C3⋊C8C2×D314yes—
95#3717—68no—
96#3717—68no—
97C22×C33C2×D411yescharacteristic
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3⋊(C3⋊D3)SD1612yescomplemented
104C33×C4—41no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3⋊(C3⋊Dic3)—21no—
108C3×C3⋊Dic3—22no—
109C32×Dic3—22no—
110C2×D3×Q8—94no—
111C2×D3×Q8—91no—
112C2×#215—94no—
113C2×#215—91no—
114C2×#204—94no—
115C2×#204—91no—
116#895—94no—
117#895—91no—
118#895—184no—
119C3×#103—34no—
120C3×#103—91no—
121(C2×D3)⋊C8—94no—
122(C2×D3)⋊C8—91no—
123#810—34no—
124#810C3⋊D311yescharacteristic, complemented
125C22×C3⋊D3—31no—
126C22×C3⋊D3—34no—
127C22×C3⋊D3—64no—
128C22×C3⋊D3—68no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—61no—
131C32×Q8—32no—
132C32×Q8—61no—
133C2×C32×C4—28no—
134C2×C32×C4—24no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—61no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—31no—
142C2×C3⋊Dic3—34no—
143C2×C3⋊Dic3—64no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)—61no—
146C4⋊(C3⋊D3)—64no—
147C4⋊(C3⋊D3)—64no—
148C4⋊(C3⋊D3)—124no—
149C4⋊(C3⋊D3)—124no—
150C4⋊(C3⋊D3)—122no—
151C4⋊(C3⋊D3)—128no—
152C4⋊(C3⋊D3)—1216no—
153C4×C3⋊D3—64no—
154C4×C3⋊D3—61no—
155C4×C3⋊D3—61no—
156C4×C3⋊D3—64no—
157C4×C3⋊D3—64no—
158C4×C3⋊D3—124no—
159C4×C3⋊D3—124no—
160#634—21no—
161#634—64no—
162#634D1212yescomplemented
163C2×C3×Dic3—24no—
164C2×C3×Dic3—64no—
165C3×Dic6—24no—
166C3×Dic6—38no—
167C3×Dic6—64no—
168C3×Dic6D1218yescomplemented
169#627—616no—
170#624—616no—
171C32×C8—61no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—24no—
174C3×C3⋊C8—64no—
175#448—271noSylow
176C2×C33#12011yescharacteristic
177C2×C33C2×SD1611yescharacteristic
178C2×C33#13711yescharacteristic
179C3⋊(C3⋊D3)—22no—
180C3⋊(C3⋊D3)—81no—
181C2×C3×Q8—34no—
182C2×C3×Q8—91no—
183D3×Q8—182no—
184D3×Q8—188no—
185D3×Q8—188no—
186D3×Q8—182no—
187C2×D12—94no—
188C2×D12—91no—
189C2×D12—184no—
190C2×D12—184no—
191C2×D12—181no—
192C2×D12—188no—
193C2×C4×D3—94no—
194C2×C4×D3—91no—
195C2×C4×D3—184no—
196C2×C4×D3—181no—
197C2×C4×D3—184no—
198C2×Dic6—34no—
199C2×Dic6—181no—
200C2×Dic6—184no—
201C2×Dic6C2×C3⋊D311yescharacteristic
202C2×C3×C8—34no—
203C2×C3×C8—91no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4D3214yes—
206C3×C4⋊C4C2×C3⋊D311yescharacteristic
207#215—182no—
208#215—188no—
209(C2×D3)⋊C4—184no—
210(C2×D3)⋊C4—181no—
211(C2×D3)⋊C4—188no—
212C2×C3⋊C8—34no—
213C2×C3⋊C8C2×C3⋊D311yescharacteristic
214#204—188no—
215#204—182no—
216C22×C32—24no—
217C22×C32D3×D414yes—
218C22×C32D3×D411yescharacteristic
219C2×C3⋊D3—32no—
220C2×C3⋊D3—38no—
221C2×C3⋊D3—68no—
222C2×C3⋊D3—61no—
223C2×C3⋊D3—61no—
224C2×C3⋊D3—64no—
225C2×C3⋊D3—64no—
226C2×C3⋊D3—124no—
227C2×C3⋊D3—124no—
228C2×C3⋊D3—128no—
229C2×C3⋊D3—128no—
230C2×C3⋊D3—128no—
231C2×C3⋊D3—124no—
232C2×C3⋊D3—121no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—121no—
235C2×C3⋊D3—124no—
236C2×C3⋊D3—121no—
237C32×C4—24no—
238C32×C4—24no—
239C32×C4—48no—
240C32×C4—44no—
241C32×C4—41no—
242C32×C4—62no—
243C32×C4C2×D1214yes—
244C32×C4C2×D1211yescharacteristic
245C32×C4D3×D414yes—
246C32×C4D3×D411yescharacteristic
247C3⋊Dic3—22no—
248C3⋊Dic3—61no—
249C3⋊Dic3—64no—
250C3⋊Dic3—124no—
251C3×Dic3—28no—
252C3×Dic3—68no—
253C22×Q8—271no—
254C2×SD16—271no—
255C2×SD16—271no—
256#121—271no—
257#103—31no—
258#103—91no—
259C22⋊C8—271no—
260C33#44811yescharacteristic, complemented, Sylow
261C22×D3—94no—
262C22×D3—91no—
263C22×D3—184no—
264C22×D3—188no—
265C22×D3—184no—
266C22×D3—181no—
267C3×Q8—38no—
268C3×Q8—64no—
269C3×Q8—92no—
270C3×Q8—181no—
271C2×C3×C4—28no—
272C2×C3×C4—24no—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—64no—
276C2×C3×C4—181no—
277C2×C3×C4C2×D3214yes—
278C2×C3×C4C22×C3⋊D311yescharacteristic
279C2×Dic3—21no—
280C2×Dic3—64no—
281C2×Dic3—94no—
282C2×Dic3—91no—
283C2×Dic3—184no—
284D12—184no—
285D12—184no—
286D12—181no—
287D12—181no—
288D12—364no—
289D12—364no—
290D12—362no—
291D12—368no—
292D12—3616no—
293C4×D3—184no—
294C4×D3—1816no—
295C4×D3—184no—
296C4×D3—184no—
297C4×D3—181no—
298C4×D3—181no—
299C4×D3—364no—
300C4×D3—364no—
301Dic6—21no—
302Dic6—38no—
303Dic6—64no—
304Dic6—184no—
305Dic6—1816no—
306Dic6C4⋊(C3⋊D3)12yescomplemented
307C3×C8—64no—
308C3×C8—181no—
309C3⋊C8—21no—
310C3⋊C8—64no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—24no—
314C2×C32#88214yes—
315C2×C32#88211yescharacteristic
316C2×C32D3×SD1614yes—
317C2×C32D3×SD1611yescharacteristic
318C2×C32#91814yes—
319C2×C32#91811yescharacteristic
320C3⋊D3—62no—
321C3⋊D3—68no—
322C3⋊D3—128no—
323C3⋊D3—248no—
324C3⋊D3—244no—
325C3⋊D3—241no—
326C2×Q8—31no—
327C2×Q8—91no—
328C2×Q8—541no—
329C2×Q8—542no—
330C2×Q8—542no—
331C2×D4—271no—
332C2×D4—541no—
333C22×C4—271no—
334C22×C4—541no—
335SD16—542no—
336SD16—542no—
337C2×C8—31no—
338C2×C8—91no—
339C4⋊C4D3×C3⋊D311yescharacteristic
340C22⋊C4—541no—
341C22×C3—24no—
342C22×C3#381114yes—
343C22×C3D4×C3⋊D311yescharacteristic
344C2×D3—92no—
345C2×D3—98no—
346C2×D3—188no—
347C2×D3—181no—
348C2×D3—181no—
349C2×D3—184no—
350C2×D3—184no—
351C2×D3—364no—
352C2×D3—364no—
353C2×D3—368no—
354C2×D3—368no—
355C2×D3—368no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—364no—
360C2×D3—364no—
361C2×D3—364no—
362C3×C4—24no—
363C3×C4—24no—
364C3×C4—48no—
365C3×C4—41no—
366C3×C4—44no—
367C3×C4—68no—
368C3×C4—182no—
369C3×C4D3×D1214yes—
370C3×C4D3×D1214yes—
371C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
372C3×C4D4×C3⋊D311yescharacteristic
373Dic3—22no—
374Dic3—68no—
375Dic3—181no—
376Dic3—184no—
377Dic3—364no—
378C32—24no—
379C32#480614yescomplemented
380C32#480611yescharacteristic, complemented
381C23—271no—
382C23—541no—
383Q8—32no—
384Q8—61no—
385Q8—92no—
386Q8—181no—
387Q8—544no—
388D4—541no—
389D4—541no—
390D4—1082no—
391C2×C4—21no—
392C2×C4—61no—
393C2×C4—181no—
394C2×C4—271no—
395C2×C4—544no—
396C2×C4—541no—
397C2×C4—541no—
398C2×C4C2×D3×C3⋊D311yescharacteristic
399C8—61no—
400C8—181no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—24no—
404C2×C3#6392314yes—
405C2×C3#6393014yes—
406C2×C3#6394514yes—
407C2×C3#6409111yescharacteristic
408C2×C3SD16×C3⋊D311yescharacteristic
409C2×C3#6412711yescharacteristic
410D3—188no—
411D3—182no—
412D3—368no—
413D3—728no—
414D3—724no—
415D3—721no—
416C22—272no—
417C22—541no—
418C22—541no—
419C22—1081no—
420C22—1081no—
421C22—1081no—
422C22#8903911yescharacteristic
423C4—41no—
424C4—62no—
425C4—182no—
426C4—541no—
427C4D3×C4⋊(C3⋊D3)11yescharacteristic
428C4C3⋊D3×D1211yescharacteristic
429C3—24no—
430C3#1059060614yescomplemented
431C3#1059250511yescharacteristic, complemented
432C2—542no—
433C2—2161no—
434C2#1172354211yescharacteristic
435C2#1172355311yescharacteristic
436C2#1172357811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723936C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#448
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×#89019C2semidirect
3. C2×#88797C2semidirect
4. C2×#88795C2semidirect
5. C3×#63637C2semidirect
6. C32×#810C2semidirect
7. #11722024C2semidirect
36. #63637D3semidirect
38. C3×#810D3semidirect
52. C3×#634D4semidirect
55. C32×Dic6D4semidirect
103. C2×C3⋊(C3⋊D3)SD16semidirect
124. #810C3⋊D3semidirect
162. #634D12semidirect
168. C3×Dic6D12semidirect
260. C33#448semidirect
306. Dic6C4⋊(C3⋊D3)semidirect
379. C32#4806semidirect
380. C32#4806semidirect
430. C3#10590606semidirect
431. C3#10592505semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
#117239362
C2×#887972
C2×#887952
C3×#636372
C32×#8102
#117220242
#105925053
#105906063