Finite Group Databasebeta

← #13140040

Subgroups — #13140040

Order 1728.

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

normal subgroups only

This group has 743 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, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3C2×#88432C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10591168—31nomaximal
7#10591162—31nomaximal
8C23×(C3⋊D3)⋊C3—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)—24no—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
15C2×#6180C2212yescomplemented
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
18#88432—24no—
19C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
20C22×C3×D12—31no—
21#64139D311yescharacteristic, complemented
22C3×#930—31no—
23C2×#3714—32no—
24C2×#3714—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—21no—
34C22×(C3⋊D3)⋊C3—42no—
35C22×(C3⋊D3)⋊C3—42no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#638⋊C3—42no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C4⋊((C3⋊D3)⋊C3)—24no—
42C4⋊((C3⋊D3)⋊C3)—22no—
43C4⋊((C3⋊D3)⋊C3)—22no—
44C4⋊((C3⋊D3)⋊C3)D414yescomplemented
45#6180—22no—
46#5217—91no—
47#5036—91no—
48#4732—91no—
49C23×C3⋊D3—21no—
50C23×C3×D3—61no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C2×#638—21no—
55C2×C4⋊(C3⋊D3)—21no—
56C2×C4⋊(C3⋊D3)—24no—
57C2×C4⋊(C3⋊D3)C2×D312yes—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
59C2×C3×#67—61no—
60C2×C3×D12—32no—
61C2×C3×D12—31no—
62C2×C3×D12—64no—
63C2×C3×D12—61no—
64#3752—21no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
66C3×(C2×D3)⋊C4—61no—
67C3×C4⋊Dic3—31no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—62no—
71#3714—64no—
72#3714—64no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)D812yescomplemented
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×(C3⋊D3)⋊C3—24no—
78C2×(C3⋊D3)⋊C3—48no—
79C2×(C3⋊D3)⋊C3—42no—
80C2×(C3⋊D3)⋊C3—42no—
81C2×(C3⋊D3)⋊C3—41no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C4×H(3,𝔽3)—21no—
85C4×H(3,𝔽3)C2×D412yes—
86(C3⋊Dic3)⋊C3—41no—
87C22×C3×D4—91no—
88C22×D12—31no—
89C22×D12—91no—
90C3×#121—91no—
91C2×#213—92no—
92C2×#213—92no—
93#930—31no—
94#930—91no—
95C2×D24—92no—
96C3×C22⋊C8—91no—
97#832—92no—
98#830—31no—
99#830—31no—
100#830—61no—
101#821—92no—
102#809—92no—
103C23×C32—21no—
104C23×C32#6711yescharacteristic
105C23×C32#6711yescharacteristic
106C22×C3⋊D3—24no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—21no—
109C22×C3⋊D3—42no—
110C22×C3⋊D3—42no—
111C22×C3×D3—64no—
112C22×C3×D3—62no—
113C22×C3×D3—61no—
114C22×C3×D3—122no—
115C22×C3×D3—122no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—42no—
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125#638—42no—
126C2×C3⋊Dic3—21no—
127C4⋊(C3⋊D3)—24no—
128C4⋊(C3⋊D3)—22no—
129C4⋊(C3⋊D3)—22no—
130C4⋊(C3⋊D3)#6714yescomplemented
131C3×#67—122no—
132C2×C3×Dic3—61no—
133C3×D12—34no—
134C3×D12—64no—
135C3×D12—62no—
136C3×D12—62no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—62no—
139C3×C3⋊C8—122no—
140#447—271noSylow
141C2×H(3,𝔽3)—22no—
142C2×H(3,𝔽3)#12011yescharacteristic
143C2×H(3,𝔽3)C2×D811yescharacteristic
144C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
145(C3⋊D3)⋊C3—44no—
146(C3⋊D3)⋊C3—81no—
147C24×C3—181no—
148C23×D3—61no—
149C23×D3—181no—
150C2×C3×D4—92no—
151C2×C3×D4—91no—
152C2×C3×D4—181no—
153C2×C3×D4—184no—
154C2×C3×D4—181no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—61no—
158C22×C3×C4D3211yescharacteristic
159C2×#67—61no—
160C2×#67—181no—
161C2×D12—32no—
162C2×D12—31no—
163C2×D12—64no—
164C2×D12—61no—
165C2×D12—92no—
166C2×D12—91no—
167C2×D12—184no—
168C2×D12—181no—
169C2×C3×C8—92no—
170C3×C4⋊C4—91no—
171C3×C22⋊C4—181no—
172#213—184no—
173#213—184no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—31no—
177C4⋊Dic3—91no—
178C2×C3⋊C8—32no—
179C2×C3⋊C8—32no—
180C2×C3⋊C8—62no—
181D24—184no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—22no—
187C22×C32—21no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32#21312yescomplemented
191C22×C32#21312yescomplemented
192C22×C32C2×#6711yescharacteristic
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—24no—
195C2×C3⋊D3—48no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3×D3—64no—
202C2×C3×D3—128no—
203C2×C3×D3—122no—
204C2×C3×D3—122no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—22no—
211C32×C4—41no—
212C32×C4D3×D412yes—
213C32×C4C2×#6712yes—
214C3⋊Dic3—41no—
215C3×Dic3—121no—
216C22×D4—91no—
217C2×D8—272no—
218#121—91no—
219#102—272no—
220C22⋊C8—91no—
221H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
222C23×C3—31no—
223C23×C3—31no—
224C23×C3—61no—
225C23×C3—182no—
226C23×C3—181no—
227C23×C3—184no—
228C23×C3—362no—
229C23×C3—362no—
230C23×C3#62511yescharacteristic
231C22×D3—64no—
232C22×D3—62no—
233C22×D3—61no—
234C22×D3—122no—
235C22×D3—122no—
236C22×D3—182no—
237C22×D3—184no—
238C22×D3—181no—
239C22×D3—362no—
240C22×D3—362no—
241C3×D4—94no—
242C3×D4—182no—
243C3×D4—182no—
244C3×D4—184no—
245C3×D4—362no—
246C2×C3×C4—22no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—122no—
256C2×C3×C4—181no—
257C2×C3×C4#62511yescharacteristic
258C2×C3×C4C2×D3211yescharacteristic
259#67—122no—
260#67—362no—
261C2×Dic3—61no—
262C2×Dic3—181no—
263D12—34no—
264D12—64no—
265D12—62no—
266D12—62no—
267D12—94no—
268D12—182no—
269D12—182no—
270D12—184no—
271C3×C8—182no—
272C3⋊C8—62no—
273C3⋊C8—62no—
274C3⋊C8—122no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—42no—
281C2×C32C2×#21311yescharacteristic
282C2×C32C2×#21311yescharacteristic
283C2×C32#93211yescharacteristic
284C2×C32#93711yescharacteristic
285C2×C32#94911yescharacteristic
286C2×C32#95311yescharacteristic
287C3⋊D3—44no—
288C3⋊D3—81no—
289C3×D3—124no—
290C3×D3—241no—
291C24—181no—
292C2×D4—92no—
293C2×D4—91no—
294C2×D4—184no—
295C2×D4—181no—
296C2×D4—181no—
297C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
298D8—544no—
299C2×C8—92no—
300C4⋊C4—91no—
301C22⋊C4—181no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—32no—
307C22×C3—32no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—62no—
314C22×C3—121no—
315C22×C3—121no—
316C22×C3—184no—
317C22×C3—362no—
318C22×C3—362no—
319C22×C3—361no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3—368no—
323C22×C3#371312yes—
324C22×C3C2×#62511yescharacteristic
325C2×D3—64no—
326C2×D3—128no—
327C2×D3—122no—
328C2×D3—122no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—184no—
333C2×D3—362no—
334C2×D3—362no—
335C2×D3—368no—
336C2×D3—361no—
337C2×D3—361no—
338C2×D3—361no—
339C3×C4—21no—
340C3×C4—32no—
341C3×C4—32no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—62no—
345C3×C4—121no—
346C3×C4—361no—
347C3×C4D3×#6712yes—
348Dic3—121no—
349Dic3—361no—
350C32—21no—
351C32#503611yescharacteristic, complemented
352C32#521711yescharacteristic, complemented
353C23—184no—
354C23—182no—
355C23—181no—
356C23—362no—
357C23—362no—
358C23#619811yescharacteristic
359D4—94no—
360D4—184no—
361D4—182no—
362D4—182no—
363D4—362no—
364C2×C4—22no—
365C2×C4—181no—
366C2×C4#619811yescharacteristic
367C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C8—182no—
369C2×C3—22no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—62no—
377C2×C3—62no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—122no—
382C2×C3—364no—
383C2×C3—721no—
384C2×C3C2×#371311yescharacteristic
385C2×C3#6382311yescharacteristic
386C2×C3#6397611yescharacteristic
387D3—124no—
388D3—241no—
389D3—364no—
390D3—721no—
391C22—21no—
392C22—21no—
393C22—184no—
394C22—368no—
395C22—362no—
396C22—362no—
397C22—361no—
398C22—361no—
399C22—361no—
400C22#8843212yescomplemented
401C22C2×#619811yescharacteristic
402C4—21no—
403C4—361no—
404C4#8867612yes—
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059116011yescharacteristic
409C2—22no—
410C2—364no—
411C2—721no—
412C2C2×#8843211yescharacteristic
413C2#1172106111yescharacteristic
414C2#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
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#447
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×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#88432C2semidirect
4. #11720573C2semidirect
5. #11720318C2semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×#6180C22semidirect
17. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
19. C22×C4⋊(C3⋊D3)D3semidirect
21. #64139D3semidirect
44. C4⋊((C3⋊D3)⋊C3)D4semidirect
58. C2×C4⋊(C3⋊D3)C2×D3semidirect
65. C4⋊(C3⋊Dic3)C2×D3semidirect
75. C22×H(3,𝔽3)D8semidirect
130. C4⋊(C3⋊D3)#67semidirect
190. C22×C32#213semidirect
191. C22×C32#213semidirect
221. H(3,𝔽3)#447semidirect
351. C32#5036semidirect
352. C32#5217semidirect
400. C22#88432semidirect

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
#64139⋊C32
C2×#884322
#117205732
#117203182
#105911683
#105911623