Finite Group Databasebeta

← #13138772

Subgroups — #13138772

Order 1728.

423 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
2#11720547C211yesmaximal, characteristic, complemented
3#63647⋊C3C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic, complemented
5#11720317C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10590665—31nomaximal
9#10590661—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88447C411yescharacteristic
20#88447C411yescharacteristic
21C4×#624—31no—
22C4×D3×Dic3—31no—
23#63647D311yescharacteristic
24#63642D311yescharacteristic, complemented
25C3×(C2×D3)⋊C8—31no—
26C3×#815—31no—
27C3×Dic3⋊C8—31no—
28C3×Dic3⋊C8—31no—
29C3×Dic3⋊C8—61no—
30Dic3⋊(C3⋊C8)—31no—
31Dic3⋊(C3⋊C8)—31no—
32#63557—31no—
33(C2×D3)⋊(C3⋊C8)—31no—
34#63555—31no—
35#63554—31no—
36C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C2×#1084C2×C411yescharacteristic
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45((C3⋊D3)⋊C3)⋊C4—22no—
46((C3⋊D3)⋊C3)⋊C4—22no—
47C8×H(3,𝔽3)—21no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4835—91no—
52#4719—91no—
53#4705—91no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×C4×D3—31no—
56C2×#624—31no—
57C2×#624—31no—
58C2×D3×Dic3—31no—
59C2×D3×Dic3—31no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D311yescharacteristic
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×C4×Dic3—31no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—31no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75Dic32—31no—
76Dic32—31no—
77Dic32—31no—
78Dic32—31no—
79C22×H(3,𝔽3)C22×C411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3M1612yes—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85#1084—21no—
86#1084M1612yes—
87(C3⋊Dic3)⋊C3—21no—
88(C3⋊Dic3)⋊C3—21no—
89(C3⋊Dic3)⋊C3—21no—
90C2×C4×Dic3—91no—
91C42×D3—91no—
92C42×D3—91no—
93C3×C4⋊C8—91no—
94C3×C22⋊C8—91no—
95C3×#97—91no—
96#830—91no—
97(C2×D3)⋊C8—31no—
98(C2×D3)⋊C8—91no—
99(C2×D3)⋊C8—91no—
100(C2×D3)⋊C8—91no—
101#815—31no—
102#815—91no—
103#815—91no—
104#815—91no—
105Dic3⋊C8—31no—
106Dic3⋊C8—31no—
107Dic3⋊C8—61no—
108Dic3⋊C8—91no—
109Dic3⋊C8—91no—
110C4⋊(C3⋊C8)—91no—
111#803—91no—
112C22×C3⋊D3C4×D311yescharacteristic
113C22×C3×D3—31no—
114C2×C32×C4—21no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3C4×D311yescharacteristic
118C2×C3⋊Dic3C4×D311yescharacteristic
119C2×C3⋊Dic3C4×D311yescharacteristic
120C4×C3⋊D3—21no—
121C4×C3⋊D3—21no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133#624—62no—
134#624—62no—
135D3×Dic3—62no—
136D3×Dic3—62no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—41no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—121no—
147#432—271noSylow
148C2×H(3,𝔽3)#11711yescharacteristic
149C2×H(3,𝔽3)C2×M1611yescharacteristic
150C2×H(3,𝔽3)C2×M1611yescharacteristic
151(C3⋊D3)⋊C3—22no—
152C22×C3×C4—91no—
153C22×Dic3—91no—
154C22×Dic3—91no—
155C2×C4×D3—31no—
156C2×C4×D3—91no—
157C2×C4×D3—91no—
158C2×C4×D3—91no—
159C2×C4×D3—91no—
160C2×C4×D3—91no—
161C2×C3×C8—31no—
162C2×C3×C8—31no—
163C2×C3×C8—61no—
164C2×C3×C8—91no—
165C2×C3×C8—91no—
166C2×C3×C8—91no—
167C2×C3×C8D3211yescharacteristic
168C3×C42—91no—
169C3×C42—91no—
170C4×Dic3—31no—
171C4×Dic3—31no—
172C4×Dic3—31no—
173C4×Dic3—61no—
174C4×Dic3—91no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C4×Dic3—91no—
178C4×Dic3—91no—
179C4×Dic3—91no—
180C4×Dic3—91no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—61no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—91no—
188C22×C32—21no—
189C22×C32C2×C4×D311yescharacteristic
190C22×C32C2×C4×D311yescharacteristic
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3ΓL(1,𝔽25)12yes—
194C2×C3×D3—32no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4#23511yescharacteristic
200C32×C4#23511yescharacteristic
201C32×C4#23711yescharacteristic
202C32×C4#23711yescharacteristic
203C3⋊Dic3—21no—
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3×Dic3—32no—
207C3×Dic3—32no—
208C3×Dic3—62no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—121no—
215C2×C42—271no—
216C4⋊C8—91no—
217C4⋊C8—271no—
218C22⋊C8—91no—
219C22⋊C8—271no—
220#97—91no—
221#97—271no—
222H(3,𝔽3)#43211yescharacteristic, complemented, Sylow
223C23×C3—91no—
224C22×D3—31no—
225C22×D3—91no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—61no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—31no—
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—91no—
247C2×Dic3—91no—
248C2×Dic3—91no—
249C2×Dic3—182no—
250C2×Dic3—182no—
251C4×D3—61no—
252C4×D3—61no—
253C4×D3—181no—
254C4×D3—181no—
255C4×D3—182no—
256C4×D3—182no—
257C4×D3—182no—
258C4×D3—182no—
259C3×C8—21no—
260C3×C8—61no—
261C3×C8—61no—
262C3×C8—121no—
263C3×C8—181no—
264C3×C8—181no—
265C3×C8—181no—
266C3⋊C8—61no—
267C3⋊C8—61no—
268C3⋊C8—61no—
269C3⋊C8—61no—
270C3⋊C8—121no—
271C3⋊C8—181no—
272C3⋊C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32#87211yescharacteristic
277C2×C32#87711yescharacteristic
278C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
279C2×C32D3×M1611yescharacteristic
280C2×C32D3×M1611yescharacteristic
281C2×C32D3×M1611yescharacteristic
282C3⋊D3—22no—
283C3×D3—62no—
284C22×C4—91no—
285C22×C4—271no—
286C22×C4—271no—
287C2×C8—91no—
288C2×C8—91no—
289C2×C8—91no—
290C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
291C42—91no—
292C42—91no—
293C42—271no—
294C42—271no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—61no—
298C22×C3—92no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3C4×D3211yescharacteristic
302C2×D3—32no—
303C2×D3—61no—
304C2×D3—61no—
305C2×D3—92no—
306C2×D3—181no—
307C2×D3—181no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—92no—
315C3×C4—181no—
316C3×C4—181no—
317C3×C4—181no—
318C3×C4—181no—
319C3×C4#379511yescharacteristic
320C3×C4#380411yescharacteristic
321Dic3—32no—
322Dic3—32no—
323Dic3—62no—
324Dic3—61no—
325Dic3—61no—
326Dic3—61no—
327Dic3—61no—
328Dic3—61no—
329Dic3—121no—
330Dic3—181no—
331Dic3—181no—
332Dic3—181no—
333C32—21no—
334C32#470511yescharacteristic, complemented
335C32#471911yescharacteristic, complemented
336C23—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—542no—
345C2×C4—542no—
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C8—21no—
348C8—181no—
349C8—181no—
350C8—181no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—182no—
361C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
362C2×C3#6379111yescharacteristic
363C2×C3#6387711yescharacteristic
364D3—62no—
365D3—182no—
366C22—92no—
367C22—181no—
368C22—181no—
369C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
370C4—92no—
371C4—181no—
372C4—181no—
373C4—181no—
374C4—181no—
375C4#8864811yescharacteristic
376C4#8867111yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3#1059066011yescharacteristic
381C2—182no—
382C2#1172083511yescharacteristic
383C2#1172083611yescharacteristic
384C2#1172112011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#432
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720547C2semidirect
4. #63642⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720304C2semidirect
24. #63642D3semidirect
222. H(3,𝔽3)#432semidirect
334. C32#4705semidirect
335. C32#4719semidirect

Maximal subgroups

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

subgroupindex
C4×((C3⋊D3)⋊C3)⋊C42
#117205472
#63647⋊C32
#63642⋊C32
#117203172
#117203122
#117203042
#105906653
#105906613