Finite Group Databasebeta

← #10592745⋊C3 #13142884

Subgroups — #10592745⋊C3 #13142884

Order 1728.

732 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 110 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
2H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#63636⋊C3C212yesmaximal, complemented
6#10592745C311yesmaximal, characteristic, complemented
7C3×#5046—31nomaximal
8C23×(C3⋊D3)⋊C3—21no—
9C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)—24no—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C4⋊C4×H(3,𝔽3)—21no—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
16C22⋊C4×H(3,𝔽3)—21no—
17#3755⋊C3—24no—
18C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
19C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
20C22×C3×D12—31no—
21C32×C22⋊Q8—21no—
22C32×C22⋊Q8D311yescharacteristic, complemented
23C32×C22⋊Q8C2×C311yescharacteristic, complemented
24C2×#3755C2×C312yescomplemented
25C2×C3×#215—32no—
26#63657C2×C311yescharacteristic, complemented
27#63636C2×C312yescomplemented
28C3×#830—31no—
29C3×#809—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—
35Q8×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C4⋊((C3⋊D3)⋊C3)—24no—
42C4⋊((C3⋊D3)⋊C3)—22no—
43C4⋊((C3⋊D3)⋊C3)D414yes—
44(C3⋊(C3⋊C8))⋊C3—22no—
45C3×#450—91no—
46#5046—31no—
47#5046—91no—
48C23×C3⋊D3—21no—
49C23×C3×D3—61no—
50C2×C32×Q8—21no—
51C2×C32×Q8C2×D311yescharacteristic, complemented
52C2×C32×Q8C22×C311yescharacteristic, complemented
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×C4⋊(C3⋊D3)—21no—
57C2×C4⋊(C3⋊D3)—24no—
58C2×C4⋊(C3⋊D3)C22×C312yes—
59C2×C3×D12—32no—
60C2×C3×D12—64no—
61C2×C3×D12—61no—
62C32×C4⋊C4—22no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2×D311yescharacteristic, complemented
67C32×C4⋊C4C22×C311yescharacteristic, complemented
68C32×C22⋊C4—22no—
69C32×C22⋊C4—21no—
70C32×C22⋊C4—21no—
71#3755—24no—
72C2×C3⋊(C3⋊C8)C22×C312yescomplemented
73C3×#215—64no—
74C2×C3×C3⋊C8—32no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)SD1612yescomplemented
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—24no—
80C2×(C3⋊D3)⋊C3—48no—
81C2×(C3⋊D3)⋊C3—42no—
82C2×(C3⋊D3)⋊C3—42no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)C2×D412yes—
87C22×C3×D4—91no—
88C22×D12—31no—
89C22×D12—91no—
90C2×C3×SD16—92no—
91C3×C22⋊Q8—31no—
92C3×C22⋊Q8—31no—
93C3×C22⋊Q8—61no—
94C3×C22⋊Q8C3×D311yescharacteristic
95C2×#215—32no—
96C2×#215—92no—
97C3×#102—92no—
98C3×C22⋊C8—91no—
99#830—31no—
100#830—91no—
101#809—32no—
102#809—92no—
103C23×C32—21no—
104C23×C32#6711yescharacteristic
105C23×C32C3×D411yescharacteristic
106C22×C3⋊D3—24no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—42no—
109C22×C3⋊D3—42no—
110C22×C3×D3—64no—
111C22×C3×D3—62no—
112C22×C3×D3—122no—
113C22×C3×D3—122no—
114C32×Q8—22no—
115C32×Q8—22no—
116C32×Q8—42no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—42no—
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4C3×D411yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C23×C311yescharacteristic
132C4⋊(C3⋊D3)—24no—
133C4⋊(C3⋊D3)—22no—
134C4⋊(C3⋊D3)C3×D414yes—
135C3×D12—34no—
136C3×D12—64no—
137C3×D12—62no—
138C3⋊(C3⋊C8)—22no—
139C3×C3⋊C8—62no—
140#450—91noSylow
141C2×H(3,𝔽3)—22no—
142C2×H(3,𝔽3)#12011yescharacteristic
143C2×H(3,𝔽3)C2×SD1611yescharacteristic
144C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
145(C3⋊D3)⋊C3—44no—
146C24×C3—181no—
147C23×D3—61no—
148C23×D3—181no—
149C2×C3×Q8—31no—
150C2×C3×Q8—31no—
151C2×C3×Q8—61no—
152C2×C3×Q8C2×C3×D311yescharacteristic
153C2×C3×D4—92no—
154C2×C3×D4—181no—
155C2×C3×D4—184no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—61no—
159C22×C3×C4C2×C3×D311yescharacteristic
160C2×D12—32no—
161C2×D12—64no—
162C2×D12—61no—
163C2×D12—92no—
164C2×D12—184no—
165C2×D12—181no—
166C3×SD16—184no—
167C2×C3×C8—92no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—62no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4C2×C3×D311yescharacteristic
176C3×C22⋊C4—21no—
177C3×C22⋊C4—61no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—62no—
180#215—64no—
181#215—184no—
182C2×C3⋊C8—32no—
183C2×C3⋊C8—92no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32#21512yescomplemented
193C22×C32C3×SD1612yescomplemented
194C22×C32C2×#6711yescharacteristic
195C22×C32C2×C3×D411yescharacteristic
196C2×C3⋊D3—24no—
197C2×C3⋊D3—48no—
198C2×C3⋊D3—42no—
199C2×C3⋊D3—42no—
200C2×C3×D3—64no—
201C2×C3×D3—128no—
202C2×C3×D3—122no—
203C2×C3×D3—122no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—22no—
207C32×C4—41no—
208C32×C4—42no—
209C32×C4—42no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4D3×D412yes—
215C32×C4C2×C3×D412yes—
216C22×D4—91no—
217C2×SD16—92no—
218C22⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
219#102—92no—
220C22⋊C8—91no—
221H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
222C23×C3—31no—
223C23×C3—31no—
224C23×C3—61no—
225C23×C3—182no—
226C23×C3—184no—
227C23×C3—362no—
228C23×C3—362no—
229C23×C3C3×#6711yescharacteristic
230C22×D3—64no—
231C22×D3—62no—
232C22×D3—122no—
233C22×D3—122no—
234C22×D3—182no—
235C22×D3—184no—
236C22×D3—362no—
237C22×D3—362no—
238C3×Q8—22no—
239C3×Q8—62no—
240C3×Q8—62no—
241C3×Q8—122no—
242C3×D4—94no—
243C3×D4—182no—
244C3×D4—184no—
245C2×C3×C4—22no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—122no—
263C2×C3×C4C3×#6711yescharacteristic
264C2×C3×C4C22×C3×D311yescharacteristic
265D12—34no—
266D12—64no—
267D12—62no—
268D12—94no—
269D12—182no—
270D12—184no—
271C3×C8—182no—
272C3⋊C8—62no—
273C3⋊C8—182no—
274C2×C32—22no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—42no—
280C2×C32#93711yescharacteristic
281C2×C32C2×#21511yescharacteristic
282C2×C32#94911yescharacteristic
283C2×C32C3×#12011yescharacteristic
284C2×C32C2×C3×SD1611yescharacteristic
285C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
286C3⋊D3—44no—
287C3×D3—124no—
288C24—181no—
289C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
290C2×D4—92no—
291C2×D4—184no—
292C2×D4—181no—
293C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
294SD16—184no—
295C2×C8—92no—
296C4⋊C4—21no—
297C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
298C22⋊C4—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—31no—
302C22×C3—31no—
303C22×C3—32no—
304C22×C3—32no—
305C22×C3—61no—
306C22×C3—61no—
307C22×C3—61no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—62no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—184no—
314C22×C3—362no—
315C22×C3—362no—
316C22×C3—368no—
317C22×C3C3×#21512yes—
318C22×C3C2×C3×#6711yescharacteristic
319C2×D3—64no—
320C2×D3—128no—
321C2×D3—122no—
322C2×D3—122no—
323C2×D3—184no—
324C2×D3—362no—
325C2×D3—362no—
326C2×D3—368no—
327C3×C4—21no—
328C3×C4—32no—
329C3×C4—32no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—62no—
335C3×C4—121no—
336C3×C4—121no—
337C3×C4—121no—
338C3×C4—121no—
339C3×C4—121no—
340C3×C4—122no—
341C3×C4—122no—
342C3×C4C3×D3×D412yes—
343C32—21no—
344C32#504611yescharacteristic, complemented
345C32C3×#45011yescharacteristic, complemented
346C23—184no—
347C23—182no—
348C23—362no—
349C23—362no—
350C23#638⋊C311yescharacteristic
351Q8—22no—
352D4—94no—
353D4—184no—
354D4—182no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4#638⋊C311yescharacteristic
359C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
360C8—182no—
361C2×C3—22no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—62no—
369C2×C3—62no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—122no—
374C2×C3—364no—
375C2×C3C3×#93711yescharacteristic
376C2×C3C2×C3×#21511yescharacteristic
377C2×C3C3×#94911yescharacteristic
378D3—124no—
379D3—364no—
380C22—21no—
381C22—21no—
382C22—184no—
383C22—368no—
384C22—362no—
385C22—362no—
386C22#3755⋊C312yescomplemented
387C22C2×#638⋊C311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4D4×(C3⋊D3)⋊C312yes—
392C3—31no—
393C3—31no—
394C3—61no—
395C3C3×#504611yescharacteristic
396C2—22no—
397C2—364no—
398C2#64146⋊C311yescharacteristic
399C2C2×#3755⋊C311yescharacteristic
400C2#64158⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#450
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
2. H(3,𝔽3)×C22⋊Q8C2semidirect
3. C2×#3755⋊C3C2semidirect
4. #63657⋊C3C2semidirect
5. #63636⋊C3C2semidirect
6. #10592745C3semidirect
9. C2×Q8×H(3,𝔽3)C22semidirect
15. C4⋊C4×H(3,𝔽3)C22semidirect
18. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. C32×C22⋊Q8D3semidirect
23. C32×C22⋊Q8C2×C3semidirect
24. C2×#3755C2×C3semidirect
26. #63657C2×C3semidirect
27. #63636C2×C3semidirect
51. C2×C32×Q8C2×D3semidirect
52. C2×C32×Q8C22×C3semidirect
66. C32×C4⋊C4C2×D3semidirect
67. C32×C4⋊C4C22×C3semidirect
72. C2×C3⋊(C3⋊C8)C22×C3semidirect
77. C22×H(3,𝔽3)SD16semidirect
192. C22×C32#215semidirect
193. C22×C32C3×SD16semidirect
218. C22⋊Q8(C3⋊D3)⋊C3semidirect
221. H(3,𝔽3)#450semidirect
289. C2×Q8C2×(C3⋊D3)⋊C3semidirect
297. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
344. C32#5046semidirect
345. C32C3×#450semidirect
386. C22#3755⋊C3semidirect

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
H(3,𝔽3)×C22⋊Q82
C2×#3755⋊C32
#63657⋊C32
#63636⋊C32
#105927453
C3×#50463