Finite Group Databasebeta

← #13140035

Subgroups — #13140035

Order 1728.

708 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 93 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic, complemented
3C2×#88431C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10591183—31nomaximal
7#10591172—31nomaximal
8C23×(C3⋊D3)⋊C3—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—24no—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×#634⋊C3C2211yescharacteristic, complemented
14C2×#6180C2212yescomplemented
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
17#3750⋊C3—21no—
18#88431—24no—
19C22×C4⋊(C3⋊D3)D311yescharacteristic
20C22×C3×D12—31no—
21#64133D311yescharacteristic, complemented
22C3×#924—31no—
23C2×#3717—32no—
24C2×#3716—32no—
25C3×#830—31no—
26C3×#830—31no—
27C3×#830—61no—
28#63564—32no—
29#63563—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—24no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—42no—
34C22×(C3⋊D3)⋊C3—42no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×(C3⋊Dic3)⋊C3—21no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C4⋊((C3⋊D3)⋊C3)—24no—
41C4⋊((C3⋊D3)⋊C3)—22no—
42C4⋊((C3⋊D3)⋊C3)D414yes—
43#634⋊C3—22no—
44#6180—22no—
45#5218—91no—
46#5046—91no—
47#4731—91no—
48C23×C3⋊D3—21no—
49C23×C3×D3—61no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D311yescharacteristic
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)—24no—
55C2×C4⋊(C3⋊D3)C2×D312yes—
56C2×#634C2×D311yescharacteristic, complemented
57C2×C3×D12—32no—
58C2×C3×D12—64no—
59C2×C3×D12—61no—
60C2×C3×Dic6—31no—
61#3757—21no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
63#3750—21no—
64C3×#217—61no—
65C3×C4⋊Dic3—31no—
66C3×Dic3⋊C4—61no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3717—64no—
71#3716—64no—
72C22×H(3,𝔽3)—21no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)SD1612yescomplemented
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×(C3⋊D3)⋊C3—24no—
77C2×(C3⋊D3)⋊C3—48no—
78C2×(C3⋊D3)⋊C3—42no—
79C2×(C3⋊D3)⋊C3—42no—
80C4×H(3,𝔽3)—21no—
81C4×H(3,𝔽3)C2×D412yes—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84C22×C3×D4—91no—
85C22×D12—31no—
86C22×D12—91no—
87C3×C22⋊Q8—91no—
88C2×#215—92no—
89C2×#214—92no—
90#924—31no—
91#924—91no—
92C2×#204—92no—
93C3×C22⋊C8—91no—
94#832—92no—
95#830—31no—
96#830—31no—
97#830—61no—
98#821—92no—
99#809—92no—
100C23×C32—21no—
101C23×C32#6711yescharacteristic
102C23×C32#6711yescharacteristic
103C22×C3⋊D3—24no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—42no—
106C22×C3⋊D3—42no—
107C22×C3×D3—64no—
108C22×C3×D3—62no—
109C22×C3×D3—122no—
110C22×C3×D3—122no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—42no—
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C4⋊(C3⋊D3)—24no—
123C4⋊(C3⋊D3)—22no—
124C4⋊(C3⋊D3)#6714yes—
125#634—22no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—34no—
129C3×D12—64no—
130C3×D12—62no—
131C3×Dic6—62no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—122no—
135#450—271noSylow
136C2×H(3,𝔽3)—22no—
137C2×H(3,𝔽3)#12011yescharacteristic
138C2×H(3,𝔽3)C2×SD1611yescharacteristic
139C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
140(C3⋊D3)⋊C3—44no—
141C24×C3—181no—
142C23×D3—61no—
143C23×D3—181no—
144C2×C3×Q8—91no—
145C2×C3×D4—92no—
146C2×C3×D4—181no—
147C2×C3×D4—184no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C22×C3×C4D3211yescharacteristic
152C2×D12—32no—
153C2×D12—64no—
154C2×D12—61no—
155C2×D12—92no—
156C2×D12—184no—
157C2×D12—181no—
158C2×Dic6—31no—
159C2×Dic6—91no—
160C2×C3×C8—92no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—181no—
163C3×C22⋊C4—181no—
164#217—61no—
165#217—181no—
166#215—184no—
167#214—184no—
168C4⋊Dic3—31no—
169C4⋊Dic3—91no—
170Dic3⋊C4—61no—
171Dic3⋊C4—181no—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—62no—
175#204—184no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—22no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32#21412yescomplemented
185C22×C32#21512yescomplemented
186C22×C32C2×#6711yescharacteristic
187C22×C32C2×#6711yescharacteristic
188C2×C3⋊D3—24no—
189C2×C3⋊D3—48no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—42no—
192C2×C3×D3—64no—
193C2×C3×D3—128no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—22no—
199C32×C4—41no—
200C32×C4D3×D412yes—
201C32×C4C2×#6712yes—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C22×D4—91no—
207C2×SD16—272no—
208C22⋊Q8—91no—
209#102—272no—
210C22⋊C8—91no—
211H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
212C23×C3—31no—
213C23×C3—31no—
214C23×C3—61no—
215C23×C3—182no—
216C23×C3—184no—
217C23×C3—362no—
218C23×C3—362no—
219C23×C3#62511yescharacteristic
220C22×D3—64no—
221C22×D3—62no—
222C22×D3—122no—
223C22×D3—122no—
224C22×D3—182no—
225C22×D3—184no—
226C22×D3—362no—
227C22×D3—362no—
228C3×Q8—182no—
229C3×D4—94no—
230C3×D4—182no—
231C3×D4—184no—
232C2×C3×C4—22no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—122no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4#62511yescharacteristic
245C2×C3×C4C2×D3211yescharacteristic
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—181no—
249C2×Dic3—181no—
250D12—34no—
251D12—64no—
252D12—62no—
253D12—94no—
254D12—182no—
255D12—184no—
256Dic6—62no—
257Dic6—182no—
258C3×C8—182no—
259C3⋊C8—62no—
260C3⋊C8—62no—
261C3⋊C8—122no—
262C2×C32—22no—
263C2×C32—22no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—42no—
268C2×C32#93211yescharacteristic
269C2×C32C2×#21411yescharacteristic
270C2×C32#93711yescharacteristic
271C2×C32C2×#21511yescharacteristic
272C2×C32#94911yescharacteristic
273C2×C32#95311yescharacteristic
274C3⋊D3—44no—
275C3×D3—124no—
276C24—181no—
277C2×Q8—91no—
278C2×D4—92no—
279C2×D4—184no—
280C2×D4—181no—
281C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
282SD16—544no—
283C2×C8—92no—
284C4⋊C4—91no—
285C4⋊C4—181no—
286C22⋊C4—181no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—61no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—62no—
299C22×C3—121no—
300C22×C3—121no—
301C22×C3—184no—
302C22×C3—362no—
303C22×C3—362no—
304C22×C3—368no—
305C22×C3#371512yes—
306C22×C3C2×#62511yescharacteristic
307C2×D3—64no—
308C2×D3—128no—
309C2×D3—122no—
310C2×D3—122no—
311C2×D3—184no—
312C2×D3—362no—
313C2×D3—362no—
314C2×D3—368no—
315C3×C4—21no—
316C3×C4—32no—
317C3×C4—32no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—62no—
321C3×C4—121no—
322C3×C4—361no—
323C3×C4—361no—
324C3×C4D3×#6712yes—
325Dic3—121no—
326Dic3—121no—
327Dic3—361no—
328Dic3—361no—
329C32—21no—
330C32#504611yescharacteristic, complemented
331C32#521811yescharacteristic, complemented
332C23—184no—
333C23—182no—
334C23—362no—
335C23—362no—
336C23#619811yescharacteristic
337Q8—182no—
338D4—94no—
339D4—184no—
340D4—182no—
341C2×C4—22no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4#619811yescharacteristic
345C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C8—182no—
347C2×C3—22no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—62no—
355C2×C3—62no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—122no—
360C2×C3—364no—
361C2×C3#6382311yescharacteristic
362C2×C3C2×#371511yescharacteristic
363C2×C3#6397611yescharacteristic
364D3—124no—
365D3—364no—
366C22—21no—
367C22—21no—
368C22—184no—
369C22—368no—
370C22—362no—
371C22—362no—
372C22#8843112yescomplemented
373C22C2×#619811yescharacteristic
374C4—21no—
375C4—361no—
376C4—361no—
377C4#8867612yes—
378C3—31no—
379C3—31no—
380C3—61no—
381C3#1059116911yescharacteristic
382C2—22no—
383C2—364no—
384C2C2×#8843111yescharacteristic
385C2#1172106111yescharacteristic
386C2#1172131611yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#450
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
2. #64133⋊C3C2semidirect
3. C2×#88431C2semidirect
4. #11720573C2semidirect
5. #11720318C2semidirect
13. C2×#634⋊C3C22semidirect
14. C2×#6180C22semidirect
16. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. #64133D3semidirect
56. C2×#634C2×D3semidirect
62. C4⋊(C3⋊Dic3)C2×D3semidirect
74. C22×H(3,𝔽3)SD16semidirect
184. C22×C32#214semidirect
185. C22×C32#215semidirect
211. H(3,𝔽3)#450semidirect
330. C32#5046semidirect
331. C32#5218semidirect
372. C22#88431semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
#64133⋊C32
C2×#884312
#117205732
#117203182
#105911833
#105911723