Finite Group Databasebeta

← #10592926⋊C3 #13143600

Subgroups — #10592926⋊C3 #13143600

Order 1728.

709 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 122 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×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C212yesmaximal, complemented
3#64139⋊C3C211yesmaximal, characteristic
4#63662⋊C3C212yesmaximal, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6#10592926C311yesmaximal, characteristic, complemented
7C3×#5227—31nomaximal
8C2×Q8×H(3,𝔽3)—24no—
9C2×Q8×H(3,𝔽3)—21no—
10C2×Q8×H(3,𝔽3)C2212yes—
11C22×C4×H(3,𝔽3)—21no—
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15#3755⋊C3—24no—
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
19C22×C32×Q8—21no—
20C22×C32×Q8D311yescharacteristic, complemented
21C22×C32×Q8C2×C311yescharacteristic, complemented
22C2×#3755C2×C312yescomplemented
23#64139C2×C311yescharacteristic
24C2×C3×#215—32no—
25C3×#930—31no—
26#63662C2×C312yescomplemented
27#63657C2×C311yescharacteristic, complemented
28C3×#835—32no—
29C3×#830—31no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—21no—
32Q8×H(3,𝔽3)—24no—
33Q8×H(3,𝔽3)—22no—
34Q8×H(3,𝔽3)D414yescomplemented
35C2×C4×H(3,𝔽3)—24no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40#638⋊C3—42no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C4⋊((C3⋊D3)⋊C3)—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C3×#448—91no—
45#5227—31no—
46#5227—91no—
47C2×C32×Q8—28no—
48C2×C32×Q8—22no—
49C2×C32×Q8—22no—
50C2×C32×Q8—24no—
51C2×C32×Q8—24no—
52C2×C32×Q8—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D312yes—
55C2×C32×Q8C22×C312yes—
56C22×C32×C4—22no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C22×C311yescharacteristic
62C2×#638—21no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C3×#67—61no—
65C2×C3×D12—31no—
66#3755—24no—
67#3752—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C312yes—
70C3×#215—64no—
71C3×(C2×D3)⋊C4—61no—
72C3×C4⋊Dic3—31no—
73C2×C3×C3⋊C8—32no—
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)—24no—
82C4×H(3,𝔽3)—21no—
83C4×H(3,𝔽3)C2×D412yes—
84(C3⋊Dic3)⋊C3—41no—
85C22×C3×Q8—31no—
86C22×C3×Q8—31no—
87C22×C3×Q8—61no—
88C22×C3×Q8C3×D311yescharacteristic
89C2×C3×SD16—92no—
90C3×#121—91no—
91C2×#215—32no—
92C2×#215—92no—
93#930—31no—
94#930—91no—
95C3×#103—92no—
96C3×C22⋊C8—91no—
97#835—32no—
98#835—92no—
99#830—31no—
100#830—91no—
101C23×C32—21no—
102C23×C32#6711yescharacteristic
103C23×C32C3×D411yescharacteristic
104C22×C3⋊D3—21no—
105C22×C3×D3—61no—
106C32×Q8—28no—
107C32×Q8—24no—
108C32×Q8—24no—
109C32×Q8—24no—
110C32×Q8—22no—
111C32×Q8—22no—
112C32×Q8—24no—
113C32×Q8#6714yescomplemented
114C32×Q8C3×D414yescomplemented
115C2×C32×C4—28no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4C3×D411yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C23×C311yescharacteristic
130#638—42no—
131C2×C3⋊Dic3—21no—
132C4⋊(C3⋊D3)—22no—
133C3×#67—122no—
134C2×C3×Dic3—61no—
135C3×D12—62no—
136C3⋊(C3⋊C8)—22no—
137C3×C3⋊C8—62no—
138#448—91noSylow
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—24no—
145C2×C3×Q8—21no—
146C2×C3×Q8—32no—
147C2×C3×Q8—32no—
148C2×C3×Q8—61no—
149C2×C3×Q8—61no—
150C2×C3×Q8—64no—
151C2×C3×Q8—64no—
152C2×C3×Q8—68no—
153C2×C3×Q8—62no—
154C2×C3×Q8—62no—
155C2×C3×Q8C2×C3×D312yes—
156C2×C3×D4—91no—
157C2×C3×D4—181no—
158C22×C3×C4—21no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—61no—
162C22×C3×C4—61no—
163C22×C3×C4—62no—
164C22×C3×C4—61no—
165C22×C3×C4C2×C3×D311yescharacteristic
166C2×#67—61no—
167C2×#67—181no—
168C2×D12—31no—
169C2×D12—91no—
170C3×SD16—184no—
171C2×C3×C8—92no—
172C3×C4⋊C4—91no—
173C3×C22⋊C4—181no—
174#215—64no—
175#215—184no—
176(C2×D3)⋊C4—61no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—31no—
179C4⋊Dic3—91no—
180C2×C3⋊C8—32no—
181C2×C3⋊C8—92no—
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#21512yescomplemented
191C22×C32C3×SD1612yescomplemented
192C22×C32C2×#6711yescharacteristic
193C22×C32C2×C3×D411yescharacteristic
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—28no—
201C32×C4—22no—
202C32×C4—24no—
203C32×C4—21no—
204C32×C4—24no—
205C32×C4—21no—
206C32×C4—22no—
207C32×C4C2×#6712yes—
208C32×C4C2×C3×D412yes—
209C3⋊Dic3—41no—
210C3×Dic3—121no—
211C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
212C2×SD16—92no—
213#121—91no—
214#103—92no—
215C22⋊C8—91no—
216H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—61no—
220C23×C3—181no—
221C23×C3C3×#6711yescharacteristic
222C22×D3—61no—
223C22×D3—181no—
224C3×Q8—24no—
225C3×Q8—22no—
226C3×Q8—34no—
227C3×Q8—34no—
228C3×Q8—62no—
229C3×Q8—62no—
230C3×Q8—64no—
231C3×Q8—64no—
232C3×Q8—68no—
233C3×Q8—64no—
234C3×Q8—64no—
235C3×Q8C3×#6714yes—
236C3×D4—182no—
237C3×D4—362no—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4—64no—
250C2×C3×C4—64no—
251C2×C3×C4—68no—
252C2×C3×C4—64no—
253C2×C3×C4—64no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—181no—
257C2×C3×C4C3×#6711yescharacteristic
258C2×C3×C4C22×C3×D311yescharacteristic
259#67—122no—
260#67—362no—
261C2×Dic3—61no—
262C2×Dic3—181no—
263D12—62no—
264D12—182no—
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×C32C2×#21511yescharacteristic
275C2×C32#94211yescharacteristic
276C2×C32#95311yescharacteristic
277C2×C32C3×#12011yescharacteristic
278C2×C32C2×C3×SD1611yescharacteristic
279C2×C32C3×#13711yescharacteristic
280C3⋊D3—81no—
281C3×D3—241no—
282C2×Q8—24no—
283C2×Q8—21no—
284C2×Q8C2×(C3⋊D3)⋊C312yes—
285C2×D4—91no—
286C2×D4—181no—
287C22×C4—21no—
288C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
289SD16—184no—
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×C3C3×#21512yes—
311C22×C3C2×C3×#6711yescharacteristic
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—121no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—361no—
318C3×C4—24no—
319C3×C4—21no—
320C3×C4—32no—
321C3×C4—32no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—64no—
325C3×C4—64no—
326C3×C4—68no—
327C3×C4—62no—
328C3×C4—62no—
329C3×C4—361no—
330C3×C4C2×C3×#6712yes—
331Dic3—121no—
332Dic3—361no—
333C32—21no—
334C32#522711yescharacteristic, complemented
335C32C3×#44811yescharacteristic, complemented
336C23—181no—
337C23#638⋊C311yescharacteristic
338Q8—24no—
339Q8—22no—
340Q8#638⋊C314yescomplemented
341D4—182no—
342D4—362no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—181no—
347C2×C4#638⋊C311yescharacteristic
348C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
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×C3×#21511yescharacteristic
365C2×C3C3×#94211yescharacteristic
366C2×C3C3×#95311yescharacteristic
367D3—241no—
368D3—721no—
369C22—21no—
370C22—21no—
371C22—361no—
372C22—361no—
373C22—361no—
374C22#3755⋊C312yescomplemented
375C22C2×#638⋊C311yescharacteristic
376C4—24no—
377C4—21no—
378C4—361no—
379C4C2×#638⋊C312yes—
380C3—31no—
381C3—31no—
382C3—61no—
383C3C3×#522711yescharacteristic
384C2—22no—
385C2—721no—
386C2C2×#3755⋊C311yescharacteristic
387C2#64151⋊C311yescharacteristic
388C2#1172170211yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radical#64139⋊C3C2
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#448
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
2. C2×#3755⋊C3C2semidirect
4. #63662⋊C3C2semidirect
5. #63657⋊C3C2semidirect
6. #10592926C3semidirect
20. C22×C32×Q8D3semidirect
21. C22×C32×Q8C2×C3semidirect
22. C2×#3755C2×C3semidirect
26. #63662C2×C3semidirect
27. #63657C2×C3semidirect
34. Q8×H(3,𝔽3)D4semidirect
76. C22×H(3,𝔽3)SD16semidirect
113. C32×Q8#67semidirect
114. C32×Q8C3×D4semidirect
190. C22×C32#215semidirect
191. C22×C32C3×SD16semidirect
211. C22×Q8(C3⋊D3)⋊C3semidirect
216. H(3,𝔽3)#448semidirect
334. C32#5227semidirect
335. C32C3×#448semidirect
340. Q8#638⋊C3semidirect
374. C22#3755⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×#3755⋊C32
#64139⋊C32
#63662⋊C32
#63657⋊C32
#105929263
C3×#52273