Finite Group Databasebeta

← #13139958

Subgroups — #13139958

Order 1728.

386 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#64135⋊C3C211yesmaximal, characteristic
2#11720573C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#11720312C211yesmaximal, characteristic, complemented
5#11720311C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10591062—31nomaximal
9#10591056—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#3757⋊C3C411yescharacteristic
14#3757⋊C3C411yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64135D311yescharacteristic
22C3×#926—31no—
23#63657D311yescharacteristic
24C3×#830—31no—
25C3×#830—31no—
26C3×#830—31no—
27C3×#830—61no—
28#63572—31no—
29#63572—31no—
30Dic3⋊(C3⋊C8)—31no—
31Dic3⋊(C3⋊C8)—31no—
32#63555—31no—
33#63554—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C23×H(3,𝔽3)C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45#6180—21no—
46#6180—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5000—91no—
50#4720—91no—
51#4720—91no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55#3757Dic311yescharacteristic
56#3757Dic311yescharacteristic
57C4⋊(C3⋊Dic3)Dic311yescharacteristic
58C4⋊(C3⋊Dic3)Dic311yescharacteristic
59C4×C3⋊Dic3C2×D311yescharacteristic
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#217—31no—
64C3×#217—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76C2×C3×C3⋊C8—61no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)—21no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)C22×C411yescharacteristic
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—21no—
88C3×#117—91no—
89#926—31no—
90#926—91no—
91C3×C22⋊C8—91no—
92C3×C22⋊C8—91no—
93#830—31no—
94#830—31no—
95#830—31no—
96#830—61no—
97#830—91no—
98#815—91no—
99#815—91no—
100Dic3⋊C8—91no—
101Dic3⋊C8—91no—
102Dic3⋊C8—91no—
103Dic3⋊C8—91no—
104C8×Dic3—91no—
105C8×Dic3—91no—
106C4⋊(C3⋊C8)—91no—
107C4⋊(C3⋊C8)—91no—
108#803—91no—
109C4×C3⋊C8—91no—
110C23×C32—21no—
111C23×C32C4×D311yescharacteristic
112C23×C32C2×Dic311yescharacteristic
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4C4×D311yescharacteristic
122C2×C32×C4C2×Dic311yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3C2×Dic311yescharacteristic
126C2×C3⋊Dic3C2×Dic311yescharacteristic
127C2×C3⋊Dic3C2×Dic311yescharacteristic
128C2×C3⋊Dic3C2×Dic311yescharacteristic
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—121no—
142C3×C3⋊C8—121no—
143#433—271noSylow
144C2×H(3,𝔽3)—41no—
145C2×H(3,𝔽3)#11711yescharacteristic
146C2×H(3,𝔽3)#13111yescharacteristic
147C2×H(3,𝔽3)#13111yescharacteristic
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C22×C3×C4D3211yescharacteristic
152C2×C3×C8—91no—
153C2×C3×C8—91no—
154C2×C3×C8—91no—
155C2×C3×C8—91no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C22⋊C4—91no—
159C3×C22⋊C4—91no—
160C3×C42—91no—
161C3×C42—91no—
162#217—31no—
163#217—31no—
164#217—91no—
165#217—91no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—91no—
169C4⋊Dic3—91no—
170C4×Dic3—31no—
171C4×Dic3—31no—
172C4×Dic3—91no—
173C4×Dic3—91no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—91no—
184C22×C32—22no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32C2×C4×D311yescharacteristic
195C22×C32C22×Dic311yescharacteristic
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—41no—
201C32×C4#23511yescharacteristic
202C32×C4#23511yescharacteristic
203C32×C4#23711yescharacteristic
204C32×C4#23711yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—21no—
208C3⋊Dic3—21no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213#117—91no—
214C4⋊C8—271no—
215C4⋊C8—271no—
216C22⋊C8—91no—
217C22⋊C8—91no—
218#97—271no—
219C4×C8—271no—
220H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
221C23×C3—31no—
222C23×C3—31no—
223C23×C3—61no—
224C23×C3D3×Dic311yescharacteristic
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—121no—
242C2×C3×C4—121no—
243C2×C3×C4D3×Dic311yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—91no—
253C3×C8—181no—
254C3×C8—181no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C3⋊C8—61no—
260C3⋊C8—61no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—121no—
264C3⋊C8—121no—
265C3⋊C8—181no—
266C3⋊C8—181no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—42no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32#87711yescharacteristic
274C2×C32#90111yescharacteristic
275C2×C32#90711yescharacteristic
276C2×C32#92611yescharacteristic
277C2×C32#94811yescharacteristic
278C2×C32#94811yescharacteristic
279C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C2×C8—91no—
281C2×C8—91no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C22⋊C4—91no—
287C22⋊C4—91no—
288C42—91no—
289C42—91no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—62no—
302C22×C3—61no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3C2×D3×Dic311yescharacteristic
306C3×C4—21no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—61no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—121no—
316C3×C4—181no—
317C3×C4—181no—
318C3×C4—181no—
319C3×C4—181no—
320C3×C4#380411yescharacteristic
321C3×C4#380411yescharacteristic
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325Dic3—61no—
326Dic3—181no—
327Dic3—181no—
328Dic3—181no—
329Dic3—181no—
330C32—21no—
331C32#472011yescharacteristic, complemented
332C32#500011yescharacteristic, complemented
333C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
341C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C8—181no—
343C8—181no—
344C8—181no—
345C8—181no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—41no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—121no—
357C2×C3—121no—
358C2×C3—122no—
359C2×C3#6381411yescharacteristic
360C2×C3#6381511yescharacteristic
361C2×C3#6395511yescharacteristic
362C22—21no—
363C22—21no—
364C22—21no—
365C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
366C4—21no—
367C4—181no—
368C4—181no—
369C4—181no—
370C4—181no—
371C4#8867111yescharacteristic
372C4#8867111yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3#1059105611yescharacteristic
377C2—41no—
378C2#1172105411yescharacteristic
379C2#1172105511yescharacteristic
380C2#1172129611yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#433
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720312C2semidirect
5. #11720311C2semidirect
6. #11720311C2semidirect
7. #11720310C2semidirect
220. H(3,𝔽3)#433semidirect
331. C32#4720semidirect
332. C32#5000semidirect

Maximal subgroups

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

subgroupindex
#64135⋊C32
#117205732
#63657⋊C32
#117203122
#117203112
#117203102
#105910623
#105910563