Finite Group Databasebeta

← #13137971

Subgroups — #13137971

Order 1728.

412 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721760C211yesmaximal, characteristic
2#11721094C211yesmaximal, characteristic
3#63637⋊C3C211yesmaximal, characteristic, complemented
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic
6#11720309C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590521—31nomaximal
9#10590429—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13#88542—21no—
14C4⋊#1084C2211yescharacteristic
15C4×#1084—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88448—21no—
21C3×#892—31no—
22C3×#892—31no—
23C3×#892—61no—
24#63857—31no—
25#63856—31no—
26#63637D311yescharacteristic, complemented
27#63634D311yescharacteristic, complemented
28C3×#810—31no—
29C3×#807—31no—
30#63574—31no—
31#63573—31no—
32#63567—31no—
33#63565—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C4×#272—21no—
42C4×#272—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#4996—91no—
50#4888—91no—
51#4815—91no—
52C2×#634C2×D311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—61no—
56C2×C3×Dic6—31no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—62no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4×Dic3—61no—
71C3×C4×Dic3—61no—
72C3×C4×Dic3—62no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75#3724—61no—
76Dic3⋊Dic3—61no—
77(C2×D3)⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×#272—21no—
81C2×#272—21no—
82C2×#272—21no—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)#4111yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86#1084—21no—
87#1084—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90C3×#117—91no—
91#946—91no—
92#924—91no—
93#897—91no—
94#892—31no—
95#892—31no—
96#892—61no—
97C3×#107—91no—
98C3×#103—91no—
99#835—91no—
100#830—91no—
101(C2×D3)⋊C8—91no—
102(C2×D3)⋊C8—91no—
103#817—91no—
104#816—91no—
105#810—31no—
106#810—91no—
107#810—91no—
108#808—91no—
109#808—91no—
110#807—31no—
111#807—91no—
112C22×C3×D3—31no—
113C22×C3×D3—31no—
114C22×C3×D3—61no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123#634—21no—
124#634—21no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×Dic6—61no—
140C3×Dic6—61no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—21no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145#485—271noSylow
146C2×H(3,𝔽3)#12311yescharacteristic
147C2×H(3,𝔽3)#13511yescharacteristic
148C2×H(3,𝔽3)#13711yescharacteristic
149#272—41no—
150C2×C3×Q8—91no—
151C22×C3×C4—91no—
152C2×C4×D3—31no—
153C2×C4×D3—31no—
154C2×C4×D3—61no—
155C2×Dic6—31no—
156C2×Dic6—91no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4D3211yescharacteristic
165C3×C22⋊C4—181no—
166C3×C42—181no—
167#217—181no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—61no—
170(C2×D3)⋊C4—62no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—61no—
177C4⋊Dic3—91no—
178C4⋊Dic3—181no—
179Dic3⋊C4—181no—
180Dic3⋊C4—181no—
181C4×Dic3—61no—
182C4×Dic3—61no—
183C4×Dic3—62no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—91no—
188C22×C32—21no—
189C22×C32D3×D411yescharacteristic
190C22×C32D3×D411yescharacteristic
191C2×C3×D3—62no—
192C2×C3×D3—61no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—42no—
203C32×C4—41no—
204C32×C4—41no—
205C32×C4#23511yescharacteristic
206C32×C4#23511yescharacteristic
207C32×C4#23711yescharacteristic
208C32×C4#23911yescharacteristic
209C3⋊Dic3—41no—
210C3⋊Dic3—41no—
211C3×Dic3—62no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—122no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C22⋊Q8—271no—
220#117—91no—
221#107—91no—
222#106—271no—
223#103—91no—
224#103—271no—
225C22⋊C8—271no—
226H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C22×D3—31no—
229C22×D3—31no—
230C22×D3—61no—
231C3×Q8—181no—
232C3×Q8—181no—
233C2×C3×C4—21no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C2×D3211yescharacteristic
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—62no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257C4×D3—61no—
258C4×D3—61no—
259C4×D3—61no—
260C4×D3—61no—
261C4×D3—62no—
262C4×D3—62no—
263Dic6—61no—
264Dic6—61no—
265Dic6—181no—
266Dic6—181no—
267C3×C8—181no—
268C3×C8—181no—
269C3⋊C8—61no—
270C3⋊C8—61no—
271C3⋊C8—181no—
272C3⋊C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32#88311yescharacteristic
277C2×C32#89411yescharacteristic
278C2×C32#91211yescharacteristic
279C2×C32#91611yescharacteristic
280C2×C32#91811yescharacteristic
281C2×C32#91811yescharacteristic
282C3×D3—122no—
283C3×D3—121no—
284C3×D3—121no—
285C2×Q8—91no—
286C22×C4—91no—
287C2×C8—91no—
288C2×C8—91no—
289C4⋊C4—91no—
290C4⋊C4—91no—
291C4⋊C4—541no—
292C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
293C22⋊C4—181no—
294C22⋊C4—541no—
295C42—181no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3—181no—
302C22×C3#381111yescharacteristic
303C2×D3—61no—
304C2×D3—61no—
305C2×D3—61no—
306C2×D3—61no—
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—62no—
310C2×D3—121no—
311C2×D3—121no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—41no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—121no—
320C3×C4—121no—
321C3×C4—122no—
322C3×C4—181no—
323C3×C4—361no—
324C3×C4—361no—
325C3×C4—361no—
326C3×C4#379611yescharacteristic
327C3×C4#379811yescharacteristic
328Dic3—61no—
329Dic3—61no—
330Dic3—62no—
331Dic3—121no—
332Dic3—121no—
333Dic3—122no—
334Dic3—121no—
335Dic3—121no—
336Dic3—361no—
337Dic3—361no—
338C32—21no—
339C32#481511yescharacteristic, complemented
340C32#488811yescharacteristic, complemented
341C23—91no—
342Q8—181no—
343Q8—181no—
344C2×C4—21no—
345C2×C4—91no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C8—181no—
353C8—181no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—361no—
364C2×C3#6385311yescharacteristic
365C2×C3#6393611yescharacteristic
366C2×C3#6394511yescharacteristic
367D3—121no—
368D3—121no—
369D3—122no—
370C22—181no—
371C22—181no—
372C22—181no—
373C22#8867711yescharacteristic
374C4—41no—
375C4—181no—
376C4—361no—
377C4—361no—
378C4—361no—
379C4#8864911yescharacteristic
380C4#8865011yescharacteristic
381C3—31no—
382C3—31no—
383C3—61no—
384C3#1059052611yescharacteristic
385C2—361no—
386C2#1172110411yescharacteristic
387C2#1172119211yescharacteristic
388C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#485
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #63637⋊C3C2semidirect
4. #63634⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720307C2semidirect
26. #63637D3semidirect
27. #63634D3semidirect
226. H(3,𝔽3)#485semidirect
339. C32#4815semidirect
340. C32#4888semidirect

Maximal subgroups

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

subgroupindex
#117217602
#117210942
#63637⋊C32
#63634⋊C32
#117203232
#117203092
#117203072
#105905213
#105904293