Finite Group Databasebeta

← #13160070

Subgroups — #13160070

Order 1728.

1116 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 210 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×#89021C211yesmaximal, characteristic
2C2×C32×D3×C8C211yesmaximal, characteristic, complemented
3#11723424C211yesmaximal, characteristic, complemented
4#11722186C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592812—31nomaximal
9#10590954—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6290—21no—
14C2×Dic3×C3⋊D3—21no—
15#89021—22no—
16#89021C414yescomplemented
17C2×C33×C8C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C32×D3×C8—22no—
22(C2×D3)⋊(C3⋊Dic3)—21no—
23Dic3×C3⋊Dic3—21no—
24C2×#3828—31no—
25C2×#3795—34no—
26C22×C32×C8—31no—
27#64135—31no—
28C2×C3×D3×C8D314yescomplemented
29#63841—34no—
30#63648—31no—
31#63648D311yescharacteristic, complemented
32#63643—31no—
33#63643—31no—
34#63643—34no—
35#63643—64no—
36C3×#821—34no—
37#63567—34no—
38#63563—34no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308C2×C412yes—
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)C2×C412yes—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3C2×C412yes—
51C32×C4×D3D412yes—
52#6290—24no—
53Dic3×C3⋊D3—24no—
54C33×C8—21no—
55C32×C3⋊C8—21no—
56#5113—94no—
57#4772—91no—
58C22×C32×C4—31no—
59C2×#638—61no—
60#3828—34no—
61#3828—62no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C4×C3⋊D3—61no—
64C2×#634—31no—
65C2×#634—34no—
66C2×#634—64no—
67C2×C3×D12—34no—
68C2×C3×C4×D3C2×D314yes—
69C2×#625—64no—
70C2×D3×Dic3—64no—
71#3795—316no—
72#3795—68no—
73C2×C32×C8—24no—
74C2×C32×C8—31no—
75C2×C32×C8—62no—
76C2×C32×C8C2×D314yes—
77C2×C32×C8C2×D311yescharacteristic
78#3757—61no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—34no—
81C4⋊(C3⋊Dic3)—64no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C4×C3⋊Dic3—61no—
84C3×C4⋊Dic3—34no—
85C2×C3×C3⋊C8C2×D314yes—
86C3×D3×C8—28no—
87(C2×D3)⋊Dic3—64no—
88Dic32—64no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C32×D3—21no—
94C2×C32×D3—21no—
95C2×C32×D3C22⋊C412yes—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—22no—
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3—41no—
101C32×Dic3C22⋊C412yes—
102C2×#237—91no—
103C2×#235—94no—
104C22×C3×C8—34no—
105#926—94no—
106C2×D3×C8C3⋊D311yescharacteristic, complemented
107#892—91no—
108C3×#102—91no—
109#832—91no—
110#821—34no—
111#821—94no—
112#816—94no—
113#816—94no—
114#816—91no—
115#816—184no—
116#810—91no—
117C23×C32—31no—
118C22×C3⋊D3—21no—
119C22×C3×D3—64no—
120C22×C3×D3D1214yes—
121C2×C32×C4—24no—
122C2×C32×C4—31no—
123C2×C32×C4—32no—
124C2×C32×C4—32no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127#638—64no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—61no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—124no—
134C2×C3⋊Dic3—124no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)C4×D312yes—
137C4×C3⋊D3—64no—
138#634—32no—
139#634—38no—
140#634—68no—
141#634—61no—
142#634—64no—
143#634—124no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3D1214yes—
146C3×D12—38no—
147C3×D12—64no—
148C3×C4×D3C4×D318yes—
149C3×C4×D3#6718yes—
150#625—616no—
151D3×Dic3—616no—
152C32×C8—21no—
153C32×C8—24no—
154C32×C8—44no—
155C32×C8—61no—
156C3×C3⋊C8—24no—
157#416—271noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33#13511yescharacteristic
161C3×C3⋊D3—42no—
162C32×D3—22no—
163C2×C3×D4—91no—
164C22×C3×C4—34no—
165C2×#67—184no—
166C2×#67—181no—
167C22×Dic3—181no—
168#237—94no—
169#237—182no—
170#235—916no—
171#235—188no—
172C2×D12—34no—
173C2×C4×D3—184no—
174C2×C4×D3C2×C3⋊D311yescharacteristic
175C2×Dic6—94no—
176C2×Dic6—91no—
177C2×Dic6—184no—
178C2×C3×C8—24no—
179C2×C3×C8—34no—
180C2×C3×C8—68no—
181C2×C3×C8D3214yes—
182C2×C3×C8C2×C3⋊D311yescharacteristic
183C3×C4⋊C4—91no—
184#217—184no—
185(C2×D3)⋊C4—181no—
186C4⋊Dic3—34no—
187C4⋊Dic3—94no—
188C4⋊Dic3—91no—
189C4⋊Dic3—184no—
190C4×Dic3—184no—
191C4×Dic3—181no—
192C2×C3⋊C8C2×C3⋊D311yescharacteristic
193D3×C8—22no—
194C22×C32—24no—
195C22×C32—32no—
196C22×C32—61no—
197C22×C32—61no—
198C22×C32C2×D1214yes—
199C22×C32D3×D411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—24no—
204C2×C3×D3—24no—
205C2×C3×D3—68no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C2×C3×D3(C2×D3)⋊C418yes—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—32no—
212C32×C4C2×C4×D314yes—
213C32×C4C2×C4×D311yescharacteristic
214C32×C4D3×D411yescharacteristic
215C32×C4C2×#6714yes—
216C3⋊Dic3—41no—
217C3⋊Dic3—62no—
218C3⋊Dic3—68no—
219C3⋊Dic3—128no—
220C3⋊Dic3—121no—
221C3⋊Dic3—124no—
222C3⋊Dic3—244no—
223C3×Dic3—124no—
224C3×Dic3(C2×D3)⋊C418yes—
225C2×#41—271no—
226C22×C8—31no—
227#117—271no—
228#103—271no—
229#103—271no—
230#102—91no—
231#102—271no—
232C33#41611yescharacteristic, complemented, Sylow
233C23×C3—34no—
234C23×C3—181no—
235C22×D3—64no—
236C22×D3C4⋊(C3⋊D3)11yescharacteristic
237C3×D4—92no—
238C3×D4—181no—
239C2×C3×C4—24no—
240C2×C3×C4—34no—
241C2×C3×C4—38no—
242C2×C3×C4—38no—
243C2×C3×C4—181no—
244C2×C3×C4C2×D3214yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246#67—1816no—
247#67—184no—
248C2×Dic3—64no—
249C2×Dic3—184no—
250C2×Dic3—181no—
251C2×Dic3—184no—
252C2×Dic3—184no—
253C2×Dic3—181no—
254C2×Dic3—364no—
255C2×Dic3—364no—
256C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
257D12—38no—
258D12—64no—
259C4×D3—1816no—
260C4×D3C4×C3⋊D312yes—
261C4×D3#63812yes—
262Dic6—98no—
263Dic6—92no—
264Dic6—188no—
265Dic6—184no—
266Dic6—181no—
267Dic6—364no—
268C3×C8—24no—
269C3×C8—21no—
270C3×C8—44no—
271C3×C8—64no—
272C3⋊C8—21no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—62no—
277C2×C32D3×C22⋊C411yescharacteristic
278C2×C32#90414yes—
279C2×C32#90414yes—
280C2×C32#91211yescharacteristic
281C2×C32#91611yescharacteristic
282C2×C32C2×(C2×D3)⋊C414yes—
283C3⋊D3—42no—
284C3×D3—28no—
285C3×D3—128no—
286#41—274no—
287#41—542no—
288C2×Q8—271no—
289C2×D4—91no—
290C2×D4—541no—
291C22×C4—31no—
292C22×C4—541no—
293C2×C8—31no—
294C2×C8—62no—
295C2×C8D3×C3⋊D311yescharacteristic
296C4⋊C4—91no—
297C4⋊C4—271no—
298C22⋊C4—541no—
299C42—541no—
300C22×C3—24no—
301C22×C3—38no—
302C22×C3—64no—
303C22×C3—64no—
304C22×C3—182no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3D3×D1214yes—
308C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—68no—
312C2×D3—124no—
313C2×D3—124no—
314C2×D3#375212yes—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—38no—
318C3×C4—361no—
319C3×C4C4×D3214yes—
320C3×C4D3×#6714yes—
321C3×C4C2×C4×C3⋊D311yescharacteristic
322C3×C4C2×#63811yescharacteristic
323Dic3—124no—
324Dic3—182no—
325Dic3—188no—
326Dic3—368no—
327Dic3—364no—
328Dic3—361no—
329Dic3—724no—
330Dic3#375212yes—
331C32—24no—
332C32#477211yescharacteristic, complemented
333C32#511314yescomplemented
334C23—31no—
335C23—181no—
336Q8—272no—
337Q8—541no—
338D4—92no—
339D4—181no—
340D4—544no—
341C2×C4—31no—
342C2×C4—32no—
343C2×C4—32no—
344C2×C4—181no—
345C2×C4—544no—
346C2×C4—541no—
347C2×C4—541no—
348C2×C4C2×D3×C3⋊D311yescharacteristic
349C8—21no—
350C8—61no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—24no—
354C2×C3—68no—
355C2×C3—362no—
356C2×C3#6380614yes—
357C2×C3#6380714yes—
358C2×C3D3×(C2×D3)⋊C414yes—
359C2×C3#6411311yescharacteristic
360C2×C3#6411311yescharacteristic
361C2×C3C2×#375211yescharacteristic
362D3—22no—
363D3—128no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—182no—
368C22—361no—
369C22—361no—
370C22D3×C4⋊(C3⋊D3)11yescharacteristic
371C4—32no—
372C4—361no—
373C4—542no—
374C4—1081no—
375C4C4×D3×C3⋊D311yescharacteristic
376C4D3×#63811yescharacteristic
377C3—24no—
378C3#1059095414yescomplemented
379C3#1059281211yescharacteristic, complemented
380C2—62no—
381C2—362no—
382C2#1172333511yescharacteristic
383C2#1172333611yescharacteristic
384C2D3×#375211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89021C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#416
LCSG ▷ 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
2. C2×C32×D3×C8C2semidirect
3. #11723424C2semidirect
4. #11722186C2semidirect
5. C3×#63648C2semidirect
6. #11722042C2semidirect
7. #11722034C2semidirect
16. #89021C4semidirect
28. C2×C3×D3×C8D3semidirect
31. #63648D3semidirect
106. C2×D3×C8C3⋊D3semidirect
232. C33#416semidirect
332. C32#4772semidirect
333. C32#5113semidirect
378. C3#10590954semidirect
379. C3#10592812semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C2×C32×D3×C82
#117234242
#117221862
C3×#636482
#117220422
#117220342
#105928123
#105909543