Finite Group Databasebeta

← #13140538

Subgroups — #13140538

Order 1728.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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
1D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#11720314C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591495—31nomaximal
9#10591408—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C2×#6196—21no—
13C4⋊#1084C2211yescharacteristic
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#88448—21no—
21D4×C3⋊Dic3D311yescharacteristic
22C3×D4×Dic3—31no—
23#63919—31no—
24#63918—31no—
25#63659D311yescharacteristic
26C3×#832—31no—
27C3×#832—31no—
28C3×#832—31no—
29C3×#832—61no—
30#63574—31no—
31#63573—31no—
32#63573—31no—
33#63573—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36C23×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)Q812yescomplemented
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—42no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46#6196—42no—
47#6180—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5008—91no—
50#4761—91no—
51#4761—91no—
52C2×C32×D4—21no—
53C2×C32×D4C2×D311yescharacteristic
54C2×C32×D4C2×D311yescharacteristic
55C22×C3⋊Dic3—21no—
56C22×C3×Dic3—61no—
57C2×#627—61no—
58C2×#627—61no—
59#3757—21no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×#217—61no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4×Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77C22×H(3,𝔽3)—22no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)C2×Q811yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—41no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3SD1612yes—
88C3×C4×D4—91no—
89D4×Dic3—31no—
90D4×Dic3—91no—
91#888—91no—
92#888—91no—
93#869—91no—
94C3×#102—91no—
95C3×#102—91no—
96#832—31no—
97#832—31no—
98#832—31no—
99#832—61no—
100#832—91no—
101#817—91no—
102#817—91no—
103Dic3⋊C8—91no—
104Dic3⋊C8—91no—
105#808—91no—
106#808—91no—
107#808—91no—
108#808—91no—
109C4⋊(C3⋊C8)—91no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—41no—
113C32×D4—21no—
114C32×D4—21no—
115C32×D4—22no—
116C32×D4—41no—
117C32×D4Dic612yescomplemented
118C32×D4Dic612yescomplemented
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—42no—
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3⋊Dic3#6711yescharacteristic
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—121no—
134C2×C3×Dic3—122no—
135#627—122no—
136#627—122no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—121no—
142#476—271noSylow
143C2×H(3,𝔽3)—42no—
144C2×H(3,𝔽3)C22⋊Q811yescharacteristic
145C2×H(3,𝔽3)C2×SD1611yescharacteristic
146C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
147C2×C3×D4—31no—
148C2×C3×D4—31no—
149C2×C3×D4—61no—
150C2×C3×D4D3211yescharacteristic
151C22×C3×C4—181no—
152C22×Dic3—61no—
153C22×Dic3—181no—
154C2×Dic6—181no—
155C2×Dic6—181no—
156C2×Dic6—181no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C22⋊C4—181no—
163C3×C42—91no—
164#217—61no—
165#217—181no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32—21no—
186C22×C32—42no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—81no—
192C22×C32—81no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×#6711yescharacteristic
195C32×C4—21no—
196C32×C4—21no—
197C32×C4C2×Dic611yescharacteristic
198C32×C4C2×Dic611yescharacteristic
199C32×C4#23511yescharacteristic
200C32×C4#23711yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3⋊Dic3#21412yes—
205C3×Dic3—32no—
206C3×Dic3—61no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—241no—
212C4⋊Q8—271no—
213C4×D4—91no—
214#106—271no—
215#106—271no—
216C4⋊C8—271no—
217#102—91no—
218#102—91no—
219H(3,𝔽3)#47611yescharacteristic, complemented, Sylow
220C23×C3—21no—
221C23×C3—61no—
222C23×C3—61no—
223C23×C3—121no—
224C3×D4—21no—
225C3×D4—32no—
226C3×D4—32no—
227C3×D4—61no—
228C3×D4—61no—
229C3×D4—62no—
230C3×D4—121no—
231C3×D4#62712yes—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—362no—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—91no—
249C2×Dic3—91no—
250C2×Dic3—121no—
251C2×Dic3—122no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—362no—
255Dic6—362no—
256Dic6—362no—
257Dic6—362no—
258C3×C8—181no—
259C3×C8—181no—
260C3⋊C8—61no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—121no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—42no—
269C2×C32—42no—
270C2×C32—82no—
271C2×C32#87811yescharacteristic
272C2×C32#91111yescharacteristic
273C2×C32D3×SD1611yescharacteristic
274C2×C32#92411yescharacteristic
275C2×C32C2×#21411yescharacteristic
276C2×C32#94911yescharacteristic
277C2×Q8—541no—
278C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
279C22×C4—181no—
280C2×C8—91no—
281C2×C8—91no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—541no—
286C22⋊C4—181no—
287C42—91no—
288C22×C3—22no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—62no—
294C22×C3—62no—
295C22×C3—61no—
296C22×C3—122no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—121no—
300C22×C3—121no—
301C22×C3—241no—
302C22×C3—241no—
303C22×C3D3×#6711yescharacteristic
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—61no—
309C3×C4—61no—
310C3×C4—92no—
311C3×C4—181no—
312C3×C4—361no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4#380411yescharacteristic
316C3×C4C2×#62711yescharacteristic
317Dic3—32no—
318Dic3—61no—
319Dic3—92no—
320Dic3—121no—
321Dic3—121no—
322Dic3—121no—
323Dic3—121no—
324Dic3—181no—
325Dic3—241no—
326Dic3—361no—
327Dic3—361no—
328C32—21no—
329C32#476111yescharacteristic, complemented
330C32#500811yescharacteristic, complemented
331C23—21no—
332Q8—1082no—
333D4—21no—
334D4#619612yescomplemented
335C2×C4—91no—
336C2×C4—91no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—362no—
341C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C8—181no—
343C8—181no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—42no—
351C2×C3—61no—
352C2×C3—61no—
353C2×C3—61no—
354C2×C3—122no—
355C2×C3—122no—
356C2×C3—242no—
357C2×C3#6392511yescharacteristic
358C2×C3D3×#21411yescharacteristic
359C2×C3#6396411yescharacteristic
360C22—22no—
361C22—41no—
362C22—41no—
363C22#8867611yescharacteristic
364C4—92no—
365C4—181no—
366C4—361no—
367C4—361no—
368C4—361no—
369C4C2×#619611yescharacteristic
370C4#8867111yescharacteristic
371C3—31no—
372C3—31no—
373C3—61no—
374C3#1059140811yescharacteristic
375C2—42no—
376C2#1172119511yescharacteristic
377C2#1172119811yescharacteristic
378C2#1172129911yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#476
LCSG ▷ C2×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
2. #11721116C2semidirect
5. #11720314C2semidirect
6. #11720311C2semidirect
7. #11720307C2semidirect
38. D4×H(3,𝔽3)Q8semidirect
117. C32×D4Dic6semidirect
118. C32×D4Dic6semidirect
219. H(3,𝔽3)#476semidirect
329. C32#4761semidirect
330. C32#5008semidirect
334. D4#6196semidirect

Maximal subgroups

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

subgroupindex
D4×(C3⋊Dic3)⋊C32
#117211162
#117205782
#63659⋊C32
#117203142
#117203112
#117203072
#105914953
#105914083