Finite Group Databasebeta

← #13139014

Subgroups — #13139014

Order 1728.

378 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 69 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#11721110C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720549C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic
8#10590872—31nomaximal
9#10590868—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#88450—21no—
18#88450—21no—
19#88449—21no—
20#88448—21no—
21#63872—31no—
22#63869—31no—
23#63864—31no—
24#63863—31no—
25#63648D311yescharacteristic
26C8×C3⋊Dic3D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#816—31no—
29C3×#816—31no—
30C3×#816—61no—
31C3×C8×Dic3—31no—
32#63566—31no—
33#63565—31no—
34#63564—31no—
35#63562—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40#6229—21no—
41#6229—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#4800—91no—
50#4768—91no—
51#4697—91no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×D12—31no—
54C2×C3×Dic6—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—61no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4×C3⋊Dic3C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×C4×Dic3—31no—
68C2×C3×C3⋊C8—31no—
69#3724—61no—
70Dic3⋊Dic3—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73#3722—61no—
74#3722—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—41no—
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88#947—91no—
89#936—91no—
90#890—91no—
91#890—91no—
92#875—91no—
93#870—91no—
94C3×#103—91no—
95C3×#102—91no—
96C3×C4×C8—91no—
97#835—91no—
98#832—91no—
99#821—31no—
100#821—91no—
101#821—91no—
102#816—31no—
103#816—31no—
104#816—61no—
105#816—91no—
106C8×Dic3—31no—
107C8×Dic3—91no—
108#810—91no—
109#809—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—61no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3D1211yescharacteristic
117C2×C3⋊Dic3D1211yescharacteristic
118C4⋊(C3⋊D3)—21no—
119C4⋊(C3⋊D3)—21no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—121no—
128C2×C3×Dic3—121no—
129C3×D12—61no—
130C3×D12—61no—
131C3×Dic6—62no—
132C3×Dic6—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C3×Dic6—121no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—41no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142#488—271noSylow
143C2×H(3,𝔽3)#12411yescharacteristic
144C2×H(3,𝔽3)#13511yescharacteristic
145C2×H(3,𝔽3)#13511yescharacteristic
146(C3⋊D3)⋊C3—81no—
147C2×C3×Q8—91no—
148C2×C3×D4—91no—
149C2×D12—31no—
150C2×D12—91no—
151C2×Dic6—31no—
152C2×Dic6—31no—
153C2×Dic6—61no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—91no—
158C2×C3×C8D3211yescharacteristic
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C42—91no—
162#217—181no—
163#217—181no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167(C2×D3)⋊C4—181no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—61no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174C4⋊Dic3—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C4×Dic3—31no—
180C4×Dic3—91no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32C2×D1211yescharacteristic
185C22×C32D3×D411yescharacteristic
186C2×C3⋊D3—41no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4#23511yescharacteristic
195C32×C4#23511yescharacteristic
196C32×C4#23511yescharacteristic
197C32×C4#23711yescharacteristic
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—122no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210#125—271no—
211#124—271no—
212#103—91no—
213#103—271no—
214#102—91no—
215#102—271no—
216C4×C8—91no—
217H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
218C23×C3—181no—
219C22×D3—61no—
220C22×D3—181no—
221C3×Q8—181no—
222C3×Q8—181no—
223C3×D4—181no—
224C3×D4—181no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—61no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4C2×D3211yescharacteristic
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C2×Dic3—121no—
244C2×Dic3—121no—
245C2×Dic3—181no—
246D12—61no—
247D12—61no—
248D12—181no—
249D12—181no—
250Dic6—61no—
251Dic6—61no—
252Dic6—61no—
253Dic6—61no—
254Dic6—62no—
255Dic6—121no—
256C3×C8—21no—
257C3×C8—61no—
258C3×C8—61no—
259C3×C8—121no—
260C3×C8—181no—
261C3⋊C8—61no—
262C3⋊C8—181no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32#87511yescharacteristic
267C2×C32#88511yescharacteristic
268C2×C32#90411yescharacteristic
269C2×C32#90411yescharacteristic
270C2×C32#91211yescharacteristic
271C2×C32#91611yescharacteristic
272C3⋊D3—81no—
273C3×D3—241no—
274C2×Q8—91no—
275C2×D4—91no—
276C2×C8—91no—
277C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—541no—
281C4⋊C4—541no—
282C22⋊C4—541no—
283C22⋊C4—541no—
284C42—91no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—361no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3D3×D1211yescharacteristic
292C2×D3—121no—
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—361no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—181no—
305C3×C4—181no—
306C3×C4—361no—
307C3×C4—361no—
308C3×C4—361no—
309C3×C4#379811yescharacteristic
310C3×C4#380411yescharacteristic
311Dic3—61no—
312Dic3—61no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—121no—
317Dic3—122no—
318Dic3—121no—
319Dic3—181no—
320Dic3—181no—
321Dic3—241no—
322Dic3—361no—
323C32—21no—
324C32#469711yescharacteristic, complemented
325C32#476811yescharacteristic, complemented
326C23—181no—
327Q8—181no—
328Q8—181no—
329D4—181no—
330D4—181no—
331C2×C4—91no—
332C2×C4—91no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C8—21no—
338C8—181no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—61no—
348C2×C3—721no—
349C2×C3#6380611yescharacteristic
350C2×C3#6380711yescharacteristic
351C2×C3#6385211yescharacteristic
352D3—241no—
353D3—721no—
354C22—361no—
355C22—361no—
356C22—361no—
357C22#8865411yescharacteristic
358C4—181no—
359C4—181no—
360C4—361no—
361C4—361no—
362C4—361no—
363C4#8865011yescharacteristic
364C4#8867111yescharacteristic
365C3—31no—
366C3—31no—
367C3—61no—
368C3#1059086311yescharacteristic
369C2—721no—
370C2#1172085411yescharacteristic
371C2#1172085511yescharacteristic
372C2#1172110311yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721093C2semidirect
3. #11720549C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720326C2semidirect
26. C8×C3⋊Dic3D3semidirect
217. H(3,𝔽3)#488semidirect
324. C32#4697semidirect
325. C32#4768semidirect

Maximal subgroups

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

subgroupindex
#117211102
#117210932
#117205492
#63648⋊C32
C8×(C3⋊Dic3)⋊C32
#117203262
#117203082
#105908723
#105908683