Finite Group Databasebeta

← #13160063

Subgroups — #13160063

Order 1728.

1220 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 170 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×D3×#634C211yesmaximal, characteristic, complemented
2C2×#88851C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic
4C2×#88797C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592437—31nomaximal
9#10590951—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6293—21no—
15C2×#6291—21no—
16C2×D3×C3⋊Dic3—21no—
17D3×#634—22no—
18D3×#634—22no—
19C2×C33×C8C2211yescharacteristic
20#88851—22no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88804—21no—
24#88797—22no—
25C22×#634—31no—
26C2×D3×Dic6—34no—
27#64139—31no—
28C2×#3744—31no—
29C2×#3744—31no—
30C2×#3744—34no—
31C2×#3744—64no—
32#63909—34no—
33C2×#3717—34no—
34C32×C22⋊C8—31no—
35#63643—31no—
36#63643D311yescharacteristic, complemented
37C3×(C2×D3)⋊C8D314yescomplemented
38C3×#816—34no—
39#63567—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C32×D3D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46#6308—21no—
47#6308D412yescomplemented
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634D412yescomplemented
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6293—24no—
56#6291—42no—
57D3×C3⋊Dic3—24no—
58C33×C8—21no—
59C32×C3⋊C8—21no—
60#4806—91no—
61#4738—94no—
62C22×C32×C4—31no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—34no—
67C2×C4⋊(C3⋊D3)—64no—
68C2×#634—31no—
69C2×#634—34no—
70C2×#634—62no—
71C2×#634—61no—
72C2×#634—62no—
73C2×#634—64no—
74C2×#634C2×D311yescharacteristic
75C2×C3×C4×D3C2×D314yes—
76C2×C3×Dic6—34no—
77C2×#627—64no—
78C2×#626—64no—
79C2×D3×Dic3—64no—
80D3×Dic6—68no—
81D3×Dic6—68no—
82C2×C32×C8—24no—
83C2×C32×C8—31no—
84C2×C32×C8C2×D314yes—
85C2×C32×C8C2×D311yescharacteristic
86#3752—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3744—62no—
89#3744—68no—
90#3744—616no—
91#3744—62no—
92C3×C4⋊Dic3—34no—
93C2×C3×C3⋊C8C2×D314yes—
94#3722—64no—
95#3717—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C32×D3—21no—
101C2×C32×D3—21no—
102C2×C32×D3SD1612yescomplemented
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—22no—
106C3×C3⋊Dic3—22no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—21no—
109C2×D3×Q8—91no—
110C22×Dic6—94no—
111C2×#215—91no—
112#930—94no—
113C2×#204—94no—
114C2×#204—94no—
115C2×#204—91no—
116C2×#204—184no—
117#895—91no—
118C3×#103—91no—
119C3×C22⋊C8—34no—
120(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
121#816—34no—
122#816—94no—
123#810—91no—
124C23×C32—31no—
125C22×C3⋊D3—61no—
126C22×C3⋊D3—64no—
127C22×C3⋊D3—124no—
128C22×C3×D3D1214yes—
129C2×C32×C4—24no—
130C2×C32×C4—31no—
131C2×C32×C4—61no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135#638—122no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—124no—
142C4⋊(C3⋊D3)—68no—
143C4⋊(C3⋊D3)—61no—
144C4⋊(C3⋊D3)—64no—
145C4⋊(C3⋊D3)—61no—
146C4⋊(C3⋊D3)—64no—
147C4⋊(C3⋊D3)—124no—
148#634—21no—
149#634—32no—
150#634—38no—
151#634—68no—
152#634—64no—
153#634—61no—
154#634—64no—
155#634—124no—
156#634D1212yescomplemented
157C2×C3×Dic3—64no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3D1214yes—
160C3×C4×D3—24no—
161C3×C4×D3—24no—
162C3×Dic6—38no—
163C3×Dic6—64no—
164#627—616no—
165#626—128no—
166D3×Dic3—616no—
167C32×C8—28no—
168C32×C8—24no—
169C32×C8—21no—
170C32×C8—61no—
171C3×C3⋊C8—24no—
172#448—271noSylow
173C2×C33#12011yescharacteristic
174C2×C33C2×SD1611yescharacteristic
175C2×C33#13711yescharacteristic
176C3⋊(C3⋊D3)—81no—
177C32×D3—22no—
178C2×C3×Q8—91no—
179C22×C3×C4—34no—
180C2×#67—184no—
181C22×Dic3—184no—
182D3×Q8—182no—
183D3×Q8—182no—
184C2×D12—94no—
185C2×D12—91no—
186C2×D12—184no—
187C2×D12—181no—
188C2×C4×D3—181no—
189C2×C4×D3C2×C3⋊D311yescharacteristic
190C2×Dic6—34no—
191C2×Dic6—94no—
192C2×Dic6—91no—
193C2×Dic6—184no—
194C2×Dic6—188no—
195C2×Dic6—188no—
196C2×Dic6—184no—
197C2×Dic6—181no—
198C2×C3×C8—24no—
199C2×C3×C8—34no—
200C2×C3×C8D3214yes—
201C2×C3×C8C2×C3⋊D311yescharacteristic
202C3×C4⋊C4—91no—
203#215—182no—
204(C2×D3)⋊C4—184no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—34no—
207C2×C3⋊C8C2×C3⋊D311yescharacteristic
208#204—188no—
209#204—182no—
210#204—1816no—
211#204—188no—
212C22×C32—24no—
213C22×C32—32no—
214C22×C32—61no—
215C22×C32—61no—
216C22×C32C2×D1214yes—
217C22×C32D3×D411yescharacteristic
218C2×C3⋊D3—128no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—121no—
221C2×C3⋊D3—121no—
222C2×C3⋊D3—121no—
223C2×C3⋊D3—124no—
224C2×C3⋊D3—124no—
225C2×C3⋊D3—244no—
226C2×C3⋊D3—244no—
227C2×C3×D3—24no—
228C2×C3×D3—24no—
229C2×C3×D3#20418yescomplemented
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—61no—
233C32×C4C2×D1211yescharacteristic
234C32×C4D3×D414yes—
235C32×C4D3×D414yes—
236C32×C4D3×D411yescharacteristic
237C3⋊Dic3—22no—
238C3⋊Dic3—41no—
239C3⋊Dic3—62no—
240C3⋊Dic3—68no—
241C3⋊Dic3—128no—
242C3×Dic3—24no—
243C3×Dic3—68no—
244C3×Dic3—124no—
245C22×Q8—271no—
246C2×SD16—271no—
247C2×SD16—271no—
248#121—271no—
249#103—91no—
250#103—271no—
251C22⋊C8—31no—
252C33#44811yescharacteristic, complemented, Sylow
253C23×C3—34no—
254C22×D3—184no—
255C22×D3—181no—
256C22×D3—364no—
257C22×D3C4⋊(C3⋊D3)11yescharacteristic
258C3×Q8—92no—
259C3×Q8—181no—
260C2×C3×C4—24no—
261C2×C3×C4—34no—
262C2×C3×C4—64no—
263C2×C3×C4—64no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4C2×D3214yes—
267C2×C3×C4C22×C3⋊D311yescharacteristic
268#67—368no—
269C2×Dic3—64no—
270C2×Dic3—64no—
271C2×Dic3—1816no—
272C2×Dic3—184no—
273C2×Dic3—181no—
274C2×Dic3—364no—
275C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
276D12—188no—
277D12—184no—
278D12—184no—
279D12—181no—
280D12—181no—
281D12—364no—
282D12—362no—
283C4×D3—21no—
284C4×D3—21no—
285C4×D3—184no—
286Dic6—38no—
287Dic6—64no—
288Dic6—98no—
289Dic6—92no—
290Dic6—188no—
291Dic6—184no—
292Dic6—1816no—
293Dic6—184no—
294Dic6—181no—
295Dic6—364no—
296C3×C8—28no—
297C3×C8—24no—
298C3×C8—21no—
299C3×C8—64no—
300C3⋊C8—21no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—62no—
305C2×C32#88214yes—
306C2×C32#88211yescharacteristic
307C2×C32C2×#20414yes—
308C2×C32#90914yes—
309C2×C32D3×SD1611yescharacteristic
310C2×C32#91811yescharacteristic
311C3⋊D3—248no—
312C3⋊D3—244no—
313C3⋊D3—241no—
314C3×D3—28no—
315C2×Q8—91no—
316C2×Q8—271no—
317C2×Q8—541no—
318C2×Q8—542no—
319C2×Q8—542no—
320C2×D4—271no—
321C2×D4—541no—
322C22×C4—31no—
323C22×C4—541no—
324SD16—542no—
325SD16—542no—
326C2×C8—31no—
327C2×C8D3×C3⋊D311yescharacteristic
328C4⋊C4—91no—
329C22⋊C4—541no—
330C22×C3—24no—
331C22×C3—38no—
332C22×C3—64no—
333C22×C3—64no—
334C22×C3D3×D1214yes—
335C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
336C2×D3—21no—
337C2×D3—21no—
338C2×D3—368no—
339C2×D3—364no—
340C2×D3—364no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—364no—
345C2×D3—724no—
346C2×D3—724no—
347C2×D3#374412yescomplemented
348C3×C4—24no—
349C3×C4—24no—
350C3×C4—64no—
351C3×C4—182no—
352C3×C4—361no—
353C3×C4D3×D1214yes—
354C3×C4#381114yes—
355C3×C4D4×C3⋊D311yescharacteristic
356C3×C4D4×C3⋊D311yescharacteristic
357Dic3—21no—
358Dic3—68no—
359Dic3—124no—
360Dic3—182no—
361Dic3—188no—
362Dic3—368no—
363C32—24no—
364C32#473814yescomplemented
365C32#480611yescharacteristic, complemented
366C23—31no—
367C23—541no—
368Q8—92no—
369Q8—181no—
370Q8—272no—
371Q8—544no—
372Q8—541no—
373D4—541no—
374D4—541no—
375D4—1082no—
376C2×C4—31no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—544no—
382C2×C4—541no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—21no—
385C8—61no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—68no—
390C2×C3D3×#20414yes—
391C2×C3#6380114yes—
392C2×C3#6392314yes—
393C2×C3#6409111yescharacteristic
394C2×C3C2×#374411yescharacteristic
395C2×C3#6411811yescharacteristic
396D3—22no—
397D3—728no—
398D3—721no—
399D3—724no—
400C22—32no—
401C22—61no—
402C22—61no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22D3×C4⋊(C3⋊D3)11yescharacteristic
407C4—61no—
408C4—182no—
409C4—361no—
410C4—542no—
411C4C3⋊D3×D1211yescharacteristic
412C4#8903911yescharacteristic
413C3—24no—
414C3#1059095114yescomplemented
415C3#1059243711yescharacteristic, complemented
416C2—62no—
417C2—2161no—
418C2D3×#374411yescharacteristic
419C2#1172332711yescharacteristic
420C2#1172354211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723516C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#448
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×D3×#634C2semidirect
2. C2×#88851C2semidirect
4. C2×#88797C2semidirect
5. C3×#63643C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722042C2semidirect
36. #63643D3semidirect
37. C3×(C2×D3)⋊C8D3semidirect
47. #6308D4semidirect
51. C3×#634D4semidirect
102. C2×C32×D3SD16semidirect
120. (C2×D3)⋊C8C3⋊D3semidirect
156. #634D12semidirect
229. C2×C3×D3#204semidirect
252. C33#448semidirect
347. C2×D3#3744semidirect
364. C32#4738semidirect
365. C32#4806semidirect
414. C3#10590951semidirect
415. C3#10592437semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C2×#888512
#117235162
C2×#887972
C3×#636432
C32×(C2×D3)⋊C82
#117220422
#105924373
#105909513