Finite Group Databasebeta

← #13140624

Subgroups — #13140624

Order 1728.

488 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#88546C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
3#11721083C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720312C211yesmaximal, characteristic, complemented
8#10591446—31nomaximal
9#10591440—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3—21no—
13#88546C414yescomplemented
14C2×#6180C2211yescharacteristic
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88449—21no—
21#88447—21no—
22D4×C3⋊Dic3D311yescharacteristic
23C3×D4×Dic3—31no—
24C2×C3×#214—31no—
25C2×C3×#214—31no—
26C2×C3×#214—61no—
27#63843—31no—
28Dic3×Dic6—31no—
29#63659D311yescharacteristic
30C3×#832—31no—
31#63575—31no—
32#63575—31no—
33#63566—31no—
34#63565—31no—
35#63555—31no—
36#63554—31no—
37C23×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2×C412yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42#6229—21no—
43#6229C2×C412yescomplemented
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—42no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#6180C2×C412yescomplemented
50(C3⋊(C3⋊C8))⋊C3—21no—
51#5164—91no—
52#5018—91no—
53#4757—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C2×D311yescharacteristic
57C22×C3⋊Dic3—21no—
58C22×C3×Dic3—61no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—61no—
62#3757—21no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×#217—61no—
68C3×#214—34no—
69C3×#214—34no—
70C3×#214—64no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78Dic3⋊Dic3—61no—
79Dic3⋊Dic3—61no—
80Dic32—61no—
81Dic32—61no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—22no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92(C3⋊Dic3)⋊C3—41no—
93C2×C3×SD16—91no—
94C3×C4×D4—91no—
95Q8×Dic3—91no—
96D4×Dic3—31no—
97D4×Dic3—91no—
98C2×#214—31no—
99C2×#214—31no—
100C2×#214—61no—
101#887—91no—
102C4×Dic6—91no—
103C3×#102—91no—
104#835—91no—
105#832—31no—
106#832—91no—
107#818—91no—
108#816—91no—
109#815—91no—
110#815—91no—
111#810—91no—
112#807—91no—
113#807—91no—
114#803—91no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—41no—
118C32×D4—21no—
119C32×D4—21no—
120C32×D4—22no—
121C32×D4—41no—
122C32×D4C4×D312yes—
123C32×D4C4×D312yes—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—42no—
130C2×C3⋊Dic3#6711yescharacteristic
131C2×C3⋊Dic3#6711yescharacteristic
132C2×C3×Dic3—31no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—121no—
139C2×C3×Dic3—122no—
140C3×Dic6—32no—
141C3×Dic6—32no—
142C3×Dic6—62no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C3×Dic6—121no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—32no—
148C3×C3⋊C8—32no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—61no—
151#440—271noSylow
152C2×H(3,𝔽3)—42no—
153C2×H(3,𝔽3)C4×D411yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155C2×H(3,𝔽3)#13711yescharacteristic
156C2×C3×Q8—91no—
157C2×C3×D4—31no—
158C2×C3×D4—31no—
159C2×C3×D4—61no—
160C2×C3×D4D3211yescharacteristic
161C22×C3×C4—181no—
162C22×Dic3—61no—
163C22×Dic3—181no—
164C2×Dic6—31no—
165C2×Dic6—31no—
166C2×Dic6—61no—
167C3×SD16—94no—
168C2×C3×C8—91no—
169C2×C3×C8—91no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4—91no—
172C3×C22⋊C4—181no—
173C3×C42—91no—
174#217—61no—
175#217—181no—
176#214—34no—
177#214—34no—
178#214—64no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—91no—
182C4⋊Dic3—91no—
183C4⋊Dic3—181no—
184Dic3⋊C4—181no—
185Dic3⋊C4—181no—
186C4×Dic3—31no—
187C4×Dic3—91no—
188C4×Dic3—181no—
189C4×Dic3—181no—
190C4×Dic3—181no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—61no—
195C2×C3⋊C8—91no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32—42no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—81no—
205C22×C32—81no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×#6711yescharacteristic
208C32×C4—21no—
209C32×C4—21no—
210C32×C4C2×C4×D311yescharacteristic
211C32×C4C2×C4×D311yescharacteristic
212C32×C4#23511yescharacteristic
213C32×C4#23711yescharacteristic
214C3⋊Dic3—21no—
215C3⋊Dic3—21no—
216C3⋊Dic3—41no—
217C3⋊Dic3—41no—
218C3×Dic3—62no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—62no—
222C3×Dic3—122no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C2×SD16—91no—
226C4×Q8—271no—
227C4×D4—91no—
228#107—271no—
229#103—271no—
230#102—91no—
231#97—271no—
232H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C3×Q8—92no—
238C3×Q8—181no—
239C3×D4—21no—
240C3×D4—32no—
241C3×D4—32no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—62no—
245C3×D4—121no—
246C3×D4#62412yes—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—61no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4—362no—
256C2×C3×C4C2×D3211yescharacteristic
257C2×Dic3—31no—
258C2×Dic3—31no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—91no—
264C2×Dic3—91no—
265C2×Dic3—121no—
266C2×Dic3—122no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269C2×Dic3—362no—
270Dic6—32no—
271Dic6—32no—
272Dic6—62no—
273Dic6—61no—
274Dic6—61no—
275Dic6—121no—
276C3×C8—92no—
277C3×C8—181no—
278C3⋊C8—32no—
279C3⋊C8—32no—
280C3⋊C8—62no—
281C3⋊C8—61no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—42no—
287C2×C32—42no—
288C2×C32—82no—
289C2×C32#67⋊C411yescharacteristic
290C2×C32#91111yescharacteristic
291C2×C32#91511yescharacteristic
292C2×C32C4×#6711yescharacteristic
293C2×C32#93211yescharacteristic
294C2×C32#95111yescharacteristic
295C2×Q8—91no—
296C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C22×C4—181no—
298SD16—94no—
299C2×C8—91no—
300C2×C8—91no—
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—541no—
304C22⋊C4—181no—
305C42—91no—
306C42—541no—
307C22×C3—22no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—61no—
315C22×C3—122no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—241no—
321C22×C3—241no—
322C22×C3D3×#6711yescharacteristic
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—61no—
328C3×C4—61no—
329C3×C4—181no—
330C3×C4—181no—
331C3×C4—182no—
332C3×C4—361no—
333C3×C4—361no—
334C3×C4#380411yescharacteristic
335C3×C4C2×#62411yescharacteristic
336Dic3—62no—
337Dic3—62no—
338Dic3—61no—
339Dic3—61no—
340Dic3—122no—
341Dic3—121no—
342Dic3—121no—
343Dic3—181no—
344Dic3—181no—
345Dic3—361no—
346Dic3—361no—
347C32—21no—
348C32#475711yescharacteristic, complemented
349C32#501811yescharacteristic, complemented
350C23—21no—
351Q8—92no—
352Q8—181no—
353D4—21no—
354D4#619912yes—
355C2×C4—91no—
356C2×C4—91no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—362no—
361C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C8—92no—
363C8—181no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—42no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—122no—
375C2×C3—122no—
376C2×C3—242no—
377C2×C3#6392911yescharacteristic
378C2×C3#6393211yescharacteristic
379C2×C3#6395211yescharacteristic
380C22—22no—
381C22—41no—
382C22—41no—
383C22#8867611yescharacteristic
384C4—182no—
385C4—181no—
386C4—181no—
387C4—361no—
388C4—361no—
389C4#8867111yescharacteristic
390C4C2×#619911yescharacteristic
391C3—31no—
392C3—31no—
393C3—61no—
394C3#1059140511yescharacteristic
395C2—42no—
396C2#1172119911yescharacteristic
397C2#1172120011yescharacteristic
398C2#1172129311yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#440
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11721083C2semidirect
5. #11720326C2semidirect
6. #11720315C2semidirect
7. #11720312C2semidirect
13. #88546C4semidirect
43. #6229C2×C4semidirect
49. #6180C2×C4semidirect
232. H(3,𝔽3)#440semidirect
348. C32#4757semidirect
349. C32#5018semidirect

Maximal subgroups

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

subgroupindex
C2×#885462
D4×(C3⋊Dic3)⋊C32
#117210832
#63659⋊C32
#117203262
#117203152
#117203122
#105914463
#105914403