Finite Group Databasebeta

← #13161031

Subgroups — #13161031

Order 1728.

904 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723977C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5#11722042C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592464—31nomaximal
9#10591496—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×#6293—21no—
13#88864—21no—
14#88864—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64145—31no—
22#64145—34no—
23#64145—64no—
24#64097—31no—
25#63919—34no—
26C32×#102—31no—
27#63659D311yescharacteristic
28#63643—31no—
29#63642—31no—
30#63637—31no—
31C3×#832—34no—
32C3×#832D314yes—
33#63567—34no—
34#63567—34no—
35#63555—34no—
36C23×C33—21no—
37C33×D4—21no—
38C33×D4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47#6293—42no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8—21no—
50#4765—94no—
51#4765—91no—
52C2×C32×D4—24no—
53C2×C32×D4C2×D314yes—
54C2×C32×D4C2×D311yescharacteristic
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—61no—
58C2×#634—64no—
59C2×#627—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—31no—
62#3757—61no—
63#3757—61no—
64#3757—64no—
65#3757—64no—
66#3757—68no—
67#3757—68no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8C2×D314yes—
78#3724—64no—
79C22×C33—41no—
80C22×C33—41no—
81C22×C33—41no—
82C22×C33C2×D411yescharacteristic
83C33×C4#4111yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—21no—
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—41no—
90#936—94no—
91#936—91no—
92#936—184no—
93#888—94no—
94#888—91no—
95C3×#102—34no—
96C3×#102—91no—
97#832—34no—
98#832C3⋊D311yescharacteristic
99#816—94no—
100#816—91no—
101#815—94no—
102#815—91no—
103#810—94no—
104#810—91no—
105C23×C32—28no—
106C23×C32—24no—
107C23×C32—21no—
108C32×D4—21no—
109C32×D4—21no—
110C32×D4—24no—
111C32×D4—24no—
112C32×D4—44no—
113C32×D4—44no—
114C2×C32×C4—24no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—68no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—61no—
128#634—61no—
129#634—61no—
130#634—64no—
131#634—64no—
132#634—124no—
133#634—124no—
134#634—122no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—64no—
137#627—128no—
138C32×C8—61no—
139C3⋊(C3⋊C8)—21no—
140C3×C3⋊C8—24no—
141C3×C3⋊C8—64no—
142#489—271noSylow
143C2×C33—81no—
144C2×C33#12411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33#13711yescharacteristic
147C2×C3×D4—24no—
148C2×C3×D4D3214yes—
149C2×C3×D4C2×C3⋊D311yescharacteristic
150C2×Dic6—94no—
151C2×Dic6—91no—
152C2×Dic6—184no—
153C2×Dic6—181no—
154C2×Dic6—184no—
155C2×C3×C8—34no—
156C2×C3×C8—91no—
157C3×C4⋊C4—34no—
158C3×C4⋊C4—91no—
159#217—184no—
160#217—184no—
161#217—181no—
162#217—181no—
163#217—188no—
164#217—188no—
165C4⋊Dic3—34no—
166C4⋊Dic3C2×C3⋊D311yescharacteristic
167Dic3⋊C4—181no—
168Dic3⋊C4—184no—
169C4×Dic3—94no—
170C4×Dic3—91no—
171C4×Dic3—184no—
172C2×C3⋊C8—34no—
173C2×C3⋊C8C2×C3⋊D311yescharacteristic
174C22×C32—24no—
175C22×C32—48no—
176C22×C32—48no—
177C22×C32—48no—
178C22×C32—44no—
179C22×C32—41no—
180C22×C32—44no—
181C22×C32—41no—
182C22×C32—44no—
183C22×C32—41no—
184C22×C32D3×D414yes—
185C22×C32D3×D411yescharacteristic
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—121no—
189C32×C4#23514yes—
190C32×C4#23511yescharacteristic
191C32×C4#23714yes—
192C32×C4#23711yescharacteristic
193C3⋊Dic3—41no—
194C3⋊Dic3—61no—
195C3⋊Dic3—61no—
196C3⋊Dic3—64no—
197C3⋊Dic3—64no—
198C3⋊Dic3—124no—
199C3⋊Dic3—124no—
200C3⋊Dic3—128no—
201C3⋊Dic3—124no—
202C3⋊Dic3—121no—
203C3×Dic3—44no—
204C3×Dic3—124no—
205C4⋊Q8—271no—
206#124—271no—
207#103—271no—
208#103—271no—
209#102—31no—
210#102—91no—
211#97—271no—
212C33#48911yescharacteristic, complemented, Sylow
213C23×C3—28no—
214C23×C3—24no—
215C23×C3—21no—
216C3×D4—24no—
217C3×D4—24no—
218C3×D4—21no—
219C3×D4—21no—
220C3×D4—44no—
221C3×D4—44no—
222C2×C3×C4—24no—
223C2×C3×C4—64no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—64no—
229C2×Dic3—94no—
230C2×Dic3—94no—
231C2×Dic3—91no—
232C2×Dic3—91no—
233C2×Dic3—184no—
234C2×Dic3—184no—
235C2×Dic3—188no—
236C2×Dic3—184no—
237C2×Dic3—181no—
238Dic6—184no—
239Dic6—184no—
240Dic6—181no—
241Dic6—181no—
242Dic6—364no—
243Dic6—364no—
244Dic6—362no—
245Dic6—368no—
246C3×C8—64no—
247C3×C8—181no—
248C3⋊C8—21no—
249C3⋊C8—64no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—88no—
254C2×C32—84no—
255C2×C32—81no—
256C2×C32#88514yes—
257C2×C32#88511yescharacteristic
258C2×C32#91114yes—
259C2×C32#91111yescharacteristic
260C2×C32#91514yes—
261C2×C32#91511yescharacteristic
262C2×Q8—271no—
263C2×Q8—541no—
264C2×D4D3×C3⋊D311yescharacteristic
265C2×C8—31no—
266C2×C8—91no—
267C4⋊C4—31no—
268C4⋊C4—91no—
269C4⋊C4—541no—
270C22⋊C4—541no—
271C22⋊C4—541no—
272C42—271no—
273C22×C3—24no—
274C22×C3—48no—
275C22×C3—48no—
276C22×C3—48no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—44no—
281C22×C3—44no—
282C22×C3—44no—
283C22×C3#381114yes—
284C22×C3D4×C3⋊D311yescharacteristic
285C3×C4—24no—
286C3×C4—24no—
287C3×C4—124no—
288C3×C4—361no—
289C3×C4#380414yes—
290C3×C4#380414yes—
291C3×C4#382811yescharacteristic
292C3×C4#383011yescharacteristic
293Dic3—41no—
294Dic3—124no—
295Dic3—181no—
296Dic3—181no—
297Dic3—184no—
298Dic3—184no—
299Dic3—364no—
300Dic3—364no—
301Dic3—368no—
302Dic3—361no—
303Dic3—364no—
304C32—24no—
305C32#476514yescomplemented
306C32#476511yescharacteristic, complemented
307C23—21no—
308Q8—541no—
309Q8—541no—
310Q8—1082no—
311D4—21no—
312D4—21no—
313C2×C4—61no—
314C2×C4—181no—
315C2×C4—271no—
316C2×C4—271no—
317C2×C4—541no—
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—24no—
324C2×C3—88no—
325C2×C3—84no—
326C2×C3—81no—
327C2×C3#6392714yes—
328C2×C3#6393114yes—
329C2×C3#6395314yes—
330C2×C3#6409411yescharacteristic
331C2×C3#6412011yescharacteristic
332C2×C3#6412411yescharacteristic
333C22—41no—
334C22—41no—
335C22—41no—
336C22#8903911yescharacteristic
337C4—121no—
338C4—361no—
339C4—541no—
340C4—541no—
341C4—1081no—
342C4#8902811yescharacteristic
343C4#8903111yescharacteristic
344C3—24no—
345C3#1059149614yescomplemented
346C3#1059246411yescharacteristic, complemented
347C2—81no—
348C2#1172354811yescharacteristic
349C2#1172355411yescharacteristic
350C2#1172359211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#489
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723536C2semidirect
5. #11722042C2semidirect
6. #11722041C2semidirect
7. #11722021C2semidirect
212. C33#489semidirect
305. C32#4765semidirect
306. C32#4765semidirect
345. C3#10591496semidirect
346. C3#10592464semidirect

Maximal subgroups

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

subgroupindex
#117239772
#117235362
C3×#636592
C32×#8322
#117220422
#117220412
#117220212
#105924643
#105914963