Finite Group Databasebeta

← #13138757

Subgroups — #13138757

Order 1728.

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

normal subgroups only

This group has 390 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#11721117C211yesmaximal, characteristic, complemented
2#11721113C211yesmaximal, characteristic
3#11720548C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic
7#11720313C211yesmaximal, characteristic, complemented
8#10590895—31nomaximal
9#10590893—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#6196—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#88450—21no—
19#88450—21no—
20#88449—21no—
21#63918—31no—
22#63918—31no—
23#63864—31no—
24#63860—31no—
25#63648D311yescharacteristic
26#63643D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#816—31no—
29C3×#815—31no—
30C3×#815—31no—
31C3×#815—61no—
32#63567—31no—
33#63566—31no—
34#63564—31no—
35#63563—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47#6196—42no—
48C8×H(3,𝔽3)—21no—
49#6180—21no—
50#4794—91no—
51#4765—91no—
52#4714—91no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×Dic6—31no—
57C2×#627—61no—
58C2×#627—61no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4×Dic3—31no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74#3722—61no—
75#3722—61no—
76(C2×D3)⋊Dic3—61no—
77(C2×D3)⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C2×(C3⋊D3)⋊C3—41no—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—21no—
85#1084—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89#947—91no—
90#936—91no—
91#888—91no—
92#888—91no—
93#875—91no—
94#869—91no—
95C3×#103—91no—
96C3×#102—91no—
97C3×#97—91no—
98#835—91no—
99#832—91no—
100#821—31no—
101#821—91no—
102#821—91no—
103#816—31no—
104#816—91no—
105#816—91no—
106#815—31no—
107#815—31no—
108#815—61no—
109#810—91no—
110#809—91no—
111C22×C3⋊D3—21no—
112C22×C3×D3—61no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119C4⋊(C3⋊D3)—21no—
120C4⋊(C3⋊D3)—21no—
121#634—21no—
122#634—21no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×D12—61no—
133C3×D12—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136#627—122no—
137#627—122no—
138C32×C8—22no—
139C32×C8—21no—
140C32×C8—21no—
141C3×C3⋊C8—62no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—61no—
144#489—271noSylow
145C2×H(3,𝔽3)#12411yescharacteristic
146C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
147C2×H(3,𝔽3)#13711yescharacteristic
148(C3⋊D3)⋊C3—81no—
149C2×C3×Q8—91no—
150C2×C3×D4—91no—
151C2×D12—31no—
152C2×D12—91no—
153C2×Dic6—31no—
154C2×Dic6—91no—
155C2×Dic6—181no—
156C2×Dic6—181no—
157C2×Dic6—181no—
158C2×C3×C8—31no—
159C2×C3×C8—31no—
160C2×C3×C8—61no—
161C2×C3×C8—91no—
162C2×C3×C8D3211yescharacteristic
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C42—91no—
166#217—181no—
167#217—181no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—91no—
175C4⋊Dic3—91no—
176C4⋊Dic3—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—61no—
185C22×C32—21no—
186C22×C32D3×D411yescharacteristic
187C22×C32D3×D411yescharacteristic
188C2×C3⋊D3—41no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4#23511yescharacteristic
197C32×C4#23511yescharacteristic
198C32×C4#23711yescharacteristic
199C32×C4#23711yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3⋊Dic3—41no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C4⋊Q8—271no—
213#124—271no—
214#103—91no—
215#103—271no—
216#102—91no—
217#102—271no—
218#97—91no—
219H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
220C23×C3—181no—
221C22×D3—61no—
222C22×D3—181no—
223C3×Q8—181no—
224C3×Q8—181no—
225C3×D4—181no—
226C3×D4—181no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—31no—
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—181no—
248D12—61no—
249D12—61no—
250D12—181no—
251D12—181no—
252Dic6—61no—
253Dic6—61no—
254Dic6—181no—
255Dic6—181no—
256Dic6—362no—
257Dic6—362no—
258Dic6—362no—
259C3×C8—21no—
260C3×C8—61no—
261C3×C8—61no—
262C3×C8—62no—
263C3×C8—181no—
264C3⋊C8—61no—
265C3⋊C8—61no—
266C3⋊C8—62no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32#88511yescharacteristic
271C2×C32#88511yescharacteristic
272C2×C32#91111yescharacteristic
273C2×C32#91411yescharacteristic
274C2×C32#91511yescharacteristic
275C2×C32#91811yescharacteristic
276C3⋊D3—81no—
277C3×D3—241no—
278C2×Q8—91no—
279C2×Q8—541no—
280C2×D4—91no—
281C2×C8—91no—
282C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—541no—
286C22⋊C4—541no—
287C22⋊C4—541no—
288C42—91no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—61no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3#380211yescharacteristic
296C2×D3—121no—
297C2×D3—121no—
298C2×D3—121no—
299C2×D3—361no—
300C2×D3—361no—
301C2×D3—361no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—181no—
309C3×C4—181no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4—361no—
313C3×C4#379811yescharacteristic
314C3×C4#380511yescharacteristic
315Dic3—61no—
316Dic3—61no—
317Dic3—61no—
318Dic3—61no—
319Dic3—121no—
320Dic3—121no—
321Dic3—121no—
322Dic3—121no—
323Dic3—121no—
324Dic3—361no—
325Dic3—361no—
326Dic3—361no—
327C32—21no—
328C32#476511yescharacteristic, complemented
329C32#479411yescharacteristic, complemented
330C23—181no—
331Q8—181no—
332Q8—181no—
333Q8—1082no—
334D4—181no—
335D4—181no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C8—21no—
343C8—181no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—61no—
351C2×C3—61no—
352C2×C3—61no—
353C2×C3—721no—
354C2×C3#6379811yescharacteristic
355C2×C3#6381111yescharacteristic
356C2×C3#6391511yescharacteristic
357D3—241no—
358D3—721no—
359C22—361no—
360C22—361no—
361C22—361no—
362C22#8865511yescharacteristic
363C4—181no—
364C4—181no—
365C4—361no—
366C4—361no—
367C4—361no—
368C4#8865011yescharacteristic
369C4#8867211yescharacteristic
370C3—31no—
371C3—31no—
372C3—61no—
373C3#1059086611yescharacteristic
374C2—721no—
375C2#1172083911yescharacteristic
376C2#1172084511yescharacteristic
377C2#1172114511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#489
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
1. #11721117C2semidirect
3. #11720548C2semidirect
5. #63643⋊C3C2semidirect
7. #11720313C2semidirect
26. #63643D3semidirect
219. H(3,𝔽3)#489semidirect
328. C32#4765semidirect
329. C32#4794semidirect

Maximal subgroups

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

subgroupindex
#117211172
#117211132
#117205482
#63648⋊C32
#63643⋊C32
#117203182
#117203132
#105908953
#105908933