Finite Group Databasebeta

← #13140193

Subgroups — #13140193

Order 1728.

709 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×#6229C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3C2×#88438C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720326C212yesmaximal, complemented
6#10591174—31nomaximal
7#10591171—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C2×#6229—21no—
11C2×#6229—24no—
12C2×#6229C2212yes—
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88438—24no—
19C22×C3×Dic6—31no—
20C22×C3×Dic6—31no—
21C22×C3×Dic6—61no—
22#64139D311yescharacteristic
23C3×#930—31no—
24C2×#3717—32no—
25C2×#3715—32no—
26#63657D311yescharacteristic, complemented
27C3×#830—31no—
28#63566—32no—
29#63565—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—21no—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084—24no—
36C2×#1084—22no—
37#6229—24no—
38#6229—22no—
39#6229D414yescomplemented
40#638⋊C3—42no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C4⋊((C3⋊D3)⋊C3)—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44#5227—91no—
45#5040—91no—
46#4738—91no—
47C22×C32×C4—21no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C2×#638—21no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C3×#67—61no—
53C22×C3×Dic3—62no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3—61no—
56C2×C3×D12—31no—
57C2×C3×Dic6—32no—
58C2×C3×Dic6—32no—
59C2×C3×Dic6—64no—
60C2×C3×Dic6—64no—
61C2×C3×Dic6—62no—
62C2×C3×Dic6—68no—
63C2×C3×Dic6—62no—
64C2×C3×Dic6—61no—
65C2×C3×Dic6—61no—
66#3752—21no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D312yes—
69C3×(C2×D3)⋊C4—61no—
70C3×C4⋊Dic3—31no—
71C2×C3×C3⋊C8—32no—
72#3717—64no—
73#3715—64no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)SD1612yescomplemented
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)C2×D412yes—
83#1084—24no—
84(C3⋊Dic3)⋊C3—41no—
85C22×C3×Q8—91no—
86C22×Dic6—31no—
87C22×Dic6—31no—
88C22×Dic6—61no—
89C3×#121—91no—
90C2×#215—92no—
91C2×#214—92no—
92#930—31no—
93#930—91no—
94C2×#204—92no—
95C3×C22⋊C8—91no—
96#835—92no—
97#830—31no—
98#830—91no—
99#816—92no—
100#810—92no—
101C23×C32—21no—
102C23×C32D1211yescharacteristic
103C23×C32#6711yescharacteristic
104C22×C3⋊D3—21no—
105C22×C3×D3—61no—
106C2×C32×C4—24no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4D1211yescharacteristic
112C2×C32×C4#6711yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115#638—42no—
116C2×C3⋊Dic3—21no—
117C4⋊(C3⋊D3)—22no—
118C3×#67—122no—
119C2×C3×Dic3—68no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—61no—
126C3×D12—62no—
127C3×Dic6—34no—
128C3×Dic6—34no—
129C3×Dic6—64no—
130C3×Dic6—68no—
131C3×Dic6—64no—
132C3×Dic6—64no—
133C3×Dic6—64no—
134C3×Dic6—62no—
135C3×Dic6—62no—
136C3⋊(C3⋊C8)—22no—
137C3×C3⋊C8—62no—
138#448—271noSylow
139C2×H(3,𝔽3)—22no—
140C2×H(3,𝔽3)#12011yescharacteristic
141C2×H(3,𝔽3)C2×SD1611yescharacteristic
142C2×H(3,𝔽3)#13711yescharacteristic
143(C3⋊D3)⋊C3—81no—
144C2×C3×Q8—92no—
145C2×C3×Q8—184no—
146C2×C3×Q8—181no—
147C2×C3×D4—91no—
148C2×C3×D4—181no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—181no—
153C22×C3×C4D3211yescharacteristic
154C2×#67—61no—
155C2×#67—181no—
156C22×Dic3—61no—
157C22×Dic3—61no—
158C22×Dic3—62no—
159C2×D12—31no—
160C2×D12—91no—
161C2×Dic6—32no—
162C2×Dic6—32no—
163C2×Dic6—62no—
164C2×Dic6—64no—
165C2×Dic6—64no—
166C2×Dic6—61no—
167C2×Dic6—61no—
168C2×Dic6—68no—
169C2×Dic6—62no—
170C2×C3×C8—92no—
171C3×C4⋊C4—91no—
172C3×C22⋊C4—181no—
173#215—184no—
174#214—184no—
175(C2×D3)⋊C4—61no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—31no—
178C4⋊Dic3—91no—
179C2×C3⋊C8—32no—
180C2×C3⋊C8—92no—
181#204—184no—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32#20412yescomplemented
191C22×C32#21412yescomplemented
192C22×C32C2×D1211yescharacteristic
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—22no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—22no—
204C32×C4D3×D412yes—
205C32×C4D3×D412yes—
206C3⋊Dic3—41no—
207C3×Dic3—68no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C3×Dic3—121no—
211C22×Q8—91no—
212C2×SD16—272no—
213#121—91no—
214#103—272no—
215C22⋊C8—91no—
216H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—61no—
220C23×C3—181no—
221C23×C3#62611yescharacteristic
222C22×D3—61no—
223C22×D3—181no—
224C3×Q8—94no—
225C3×Q8—184no—
226C3×Q8—182no—
227C3×D4—182no—
228C3×D4—362no—
229C2×C3×C4—22no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—64no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—184no—
240C2×C3×C4—182no—
241C2×C3×C4—181no—
242C2×C3×C4#62611yescharacteristic
243C2×C3×C4C2×D3211yescharacteristic
244#67—122no—
245#67—362no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—64no—
249C2×Dic3—64no—
250C2×Dic3—68no—
251C2×Dic3—64no—
252C2×Dic3—61no—
253C2×Dic3—181no—
254D12—62no—
255D12—182no—
256Dic6—34no—
257Dic6—34no—
258Dic6—64no—
259Dic6—62no—
260Dic6—62no—
261Dic6—64no—
262Dic6—64no—
263Dic6—68no—
264Dic6—64no—
265C3×C8—182no—
266C3⋊C8—62no—
267C3⋊C8—182no—
268C2×C32—24no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32#88211yescharacteristic
275C2×C32C2×#20411yescharacteristic
276C2×C32#90911yescharacteristic
277C2×C32C2×#21411yescharacteristic
278C2×C32#93711yescharacteristic
279C2×C32#95111yescharacteristic
280C3⋊D3—81no—
281C3×D3—241no—
282C2×Q8—92no—
283C2×Q8—184no—
284C2×Q8—181no—
285C2×D4—91no—
286C2×D4—181no—
287C22×C4—181no—
288C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
289SD16—544no—
290C2×C8—92no—
291C4⋊C4—91no—
292C22⋊C4—181no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—32no—
298C22×C3—32no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—62no—
304C22×C3—62no—
305C22×C3—61no—
306C22×C3—62no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3—361no—
310C22×C3#371612yes—
311C22×C3C2×#62611yescharacteristic
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—121no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—361no—
318C3×C4—21no—
319C3×C4—32no—
320C3×C4—32no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—62no—
324C3×C4—62no—
325C3×C4—184no—
326C3×C4—361no—
327C3×C4#381112yes—
328Dic3—64no—
329Dic3—64no—
330Dic3—68no—
331Dic3—121no—
332Dic3—361no—
333C32—21no—
334C32#473811yescharacteristic, complemented
335C32#504011yescharacteristic, complemented
336C23—181no—
337C23#620011yescharacteristic
338Q8—94no—
339Q8—184no—
340Q8—182no—
341D4—182no—
342D4—362no—
343C2×C4—22no—
344C2×C4—184no—
345C2×C4—182no—
346C2×C4—181no—
347C2×C4#620011yescharacteristic
348C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C8—182no—
350C2×C3—22no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—62no—
358C2×C3—62no—
359C2×C3—64no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—721no—
364C2×C3C2×#371611yescharacteristic
365C2×C3#6383111yescharacteristic
366C2×C3#6398111yescharacteristic
367D3—241no—
368D3—721no—
369C22—21no—
370C22—21no—
371C22—361no—
372C22—361no—
373C22—361no—
374C22#8843812yescomplemented
375C22C2×#620011yescharacteristic
376C4—21no—
377C4—184no—
378C4—361no—
379C4#8867712yes—
380C3—31no—
381C3—31no—
382C3—61no—
383C3#1059118111yescharacteristic
384C2—22no—
385C2—721no—
386C2C2×#8843811yescharacteristic
387C2#1172107411yescharacteristic
388C2#1172132311yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#448
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6229C2semidirect
3. C2×#88438C2semidirect
4. #63657⋊C3C2semidirect
5. #11720326C2semidirect
26. #63657D3semidirect
39. #6229D4semidirect
76. C22×H(3,𝔽3)SD16semidirect
190. C22×C32#204semidirect
191. C22×C32#214semidirect
216. H(3,𝔽3)#448semidirect
334. C32#4738semidirect
335. C32#5040semidirect
374. C22#88438semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
#64139⋊C32
C2×#884382
#63657⋊C32
#117203262
#105911743
#105911713