Finite Group Databasebeta

← #13154279

Subgroups — #13154279

Order 1728.

1350 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88746C211yesmaximal, characteristic
2C2×Q8×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×C9×C4⋊Q8C211yesmaximal, characteristic, complemented
4#11721857C211yesmaximal, characteristic, complemented
5#11721854C211yesmaximal, characteristic, complemented
6#11721752C214yesmaximal, complemented
7#11721750C214yesmaximal, complemented
8#11721712C211yesmaximal, characteristic, complemented
9#11721709C211yesmaximal, characteristic, complemented
10#10596195—31nomaximal
11#10589826—33nomaximal
12C2×C3×Q8×C9C2211yescharacteristic
13C2×C3×Q8×C9C2211yescharacteristic
14#88746—24no—
15Q8×C3⋊D9—24no—
16C2×C4⋊(C3⋊D9)C2212yes—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C4×C3⋊D9C2212yes—
19C2×C4×C3⋊D9C2212yes—
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×#6226C2212yes—
22C2×#6226C2211yescharacteristic
23C3×C9×C4⋊C4C2214yes—
24C3×C42×C9C2211yescharacteristic
25#88536C2212yes—
26#88536C2214yes—
27#88536C2214yes—
28#88534C2214yes—
29#88534C2212yes—
30C4×C3⋊Dic9C2211yescharacteristic
31C2×#3832—31no—
32C2×Q8×C3⋊D3—31no—
33C32×C4⋊Q8D311yescharacteristic
34#64156—31no—
35#64153—31no—
36#64105—34no—
37#64103—34no—
38#64084—31no—
39#64081—31no—
40C2×#3701—33no—
41C2×Q8×D9—33no—
42C9×C4⋊Q8D313yescomplemented
43#63509—33no—
44#63506—33no—
45#63458—312no—
46#63456—312no—
47#63437—33no—
48#63434—33no—
49C22×C3⋊D9C2312yes—
50C22×C3⋊D9C2311yescharacteristic
51C3×Q8×C9—22no—
52C3×Q8×C9—22no—
53C2×C3×C4×C9C2312yes—
54C2×C3×C4×C9C2312yes—
55C2×C3×C4×C9C2312yes—
56C2×C3×C4×C9C2311yescharacteristic
57C2×C3⋊Dic9C2312yes—
58C2×C3⋊Dic9C2312yes—
59C2×C3⋊Dic9C2311yescharacteristic
60C4⋊(C3⋊D9)—24no—
61C4⋊(C3⋊D9)—22no—
62C4×C3⋊D9—24no—
63C4×C3⋊D9—24no—
64C4×C3⋊D9D414yes—
65#6226—22no—
66#6226—24no—
67#5724—93no—
68#5724—91no—
69C2×C32×Q8C2×D311yescharacteristic
70C2×C32×Q8C2×D311yescharacteristic
71#3832—64no—
72Q8×C3⋊D3—64no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—32no—
75C2×C4×C3⋊D3—31no—
76C2×C4×C3⋊D3—32no—
77C2×C4×C3⋊D3—32no—
78C2×#634—31no—
79C2×#634—32no—
80C32×C4⋊C4C2×D314yes—
81C32×C42C2×D311yescharacteristic
82#3752—32no—
83#3752—34no—
84#3752—34no—
85#3750—32no—
86#3750—34no—
87C4×C3⋊Dic3—31no—
88C2×Q8×C9C2×D313yes—
89C2×Q8×C9C2×D313yes—
90#3701—612no—
91Q8×D9—612no—
92C2×D36—33no—
93C2×D36—36no—
94C2×C4×D9—33no—
95C2×C4×D9—36no—
96C2×C4×D9—36no—
97C2×Dic18—33no—
98C2×Dic18—36no—
99C9×C4⋊C4C2×D3112yes—
100C42×C9C2×D313yes—
101(C2×D9)⋊C4—312no—
102(C2×D9)⋊C4—36no—
103(C2×D9)⋊C4—312no—
104Dic9⋊C4—312no—
105Dic9⋊C4—36no—
106C4×Dic9—33no—
107C22×C3×C9C2411yescharacteristic
108C2×C3⋊D9—22no—
109C2×C3⋊D9—22no—
110C2×C3⋊D9—24no—
111C2×C3⋊D9C2×D412yes—
112C3×C4×C9—22no—
113C3×C4×C9—22no—
114C3×C4×C9—22no—
115C3×C4×C9C2×D412yes—
116C3⋊Dic9—22no—
117C3⋊Dic9—22no—
118C3⋊Dic9C2×D412yes—
119C2×#239—93no—
120C2×#239—91no—
121C2×D3×Q8—93no—
122C2×D3×Q8—91no—
123C3×C4⋊Q8D913yescomplemented
124C3×C4⋊Q8C3⋊D311yescharacteristic
125#947—93no—
126#947—91no—
127#944—93no—
128#944—91no—
129#896—912no—
130#896—94no—
131#894—912no—
132#894—94no—
133#875—93no—
134#875—91no—
135#872—93no—
136#872—91no—
137C22×C3⋊D3—31no—
138C22×C3⋊D3—32no—
139C32×Q8—22no—
140C32×Q8—22no—
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3—31no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—62no—
150C4×C3⋊D3—34no—
151C4×C3⋊D3—64no—
152C4×C3⋊D3—64no—
153#634—64no—
154#634—62no—
155C22×D9—33no—
156C22×D9—36no—
157Q8×C9—26no—
158Q8×C9—26no—
159C2×C4×C9C22×D316yes—
160C2×C4×C9C22×D316yes—
161C2×C4×C9C22×D316yes—
162C2×C4×C9C22×D313yes—
163C2×Dic9—33no—
164C2×Dic9—36no—
165C2×Dic9—36no—
166D36—612no—
167D36—66no—
168C4×D9—312no—
169C4×D9—612no—
170C4×D9—612no—
171Dic18—612no—
172Dic18—66no—
173#536—271noSylow
174C2×C3×C9C22×D411yescharacteristic
175C2×C3×C92−1+412yes—
176C3⋊D9—22no—
177C3⋊D9—42no—
178C2×C3×Q8C2×D913yes—
179C2×C3×Q8C2×D913yes—
180C2×C3×Q8C2×C3⋊D311yescharacteristic
181C2×C3×Q8C2×C3⋊D311yescharacteristic
182#239—184no—
183#239—1812no—
184D3×Q8—184no—
185D3×Q8—1812no—
186C2×D12—93no—
187C2×D12—96no—
188C2×D12—91no—
189C2×D12—92no—
190C2×C4×D3—93no—
191C2×C4×D3—91no—
192C2×C4×D3—96no—
193C2×C4×D3—96no—
194C2×C4×D3—92no—
195C2×C4×D3—92no—
196C2×Dic6—93no—
197C2×Dic6—96no—
198C2×Dic6—91no—
199C2×Dic6—92no—
200C3×C4⋊C4C2×D9112yes—
201C3×C4⋊C4C2×C3⋊D314yes—
202C3×C42C2×D913yes—
203C3×C42C2×C3⋊D311yescharacteristic
204(C2×D3)⋊C4—96no—
205(C2×D3)⋊C4—912no—
206(C2×D3)⋊C4—912no—
207(C2×D3)⋊C4—92no—
208(C2×D3)⋊C4—94no—
209(C2×D3)⋊C4—94no—
210Dic3⋊C4—96no—
211Dic3⋊C4—912no—
212Dic3⋊C4—92no—
213Dic3⋊C4—94no—
214C4×Dic3—93no—
215C4×Dic3—91no—
216C22×C32C23×D311yescharacteristic
217C2×C3⋊D3—32no—
218C2×C3⋊D3—62no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—64no—
221C32×C4—22no—
222C32×C4—22no—
223C32×C4—22no—
224C32×C4D3×D412yes—
225C3⋊Dic3—32no—
226C3⋊Dic3—62no—
227C3⋊Dic3—62no—
228C22×C9C23×D313yes—
229C2×D9—36no—
230C2×D9—66no—
231C2×D9—66no—
232C2×D9—612no—
233C4×C9—26no—
234C4×C9—26no—
235C4×C9—26no—
236C4×C9D3×D416yes—
237Dic9—36no—
238Dic9—66no—
239Dic9—66no—
240C2×#41—271no—
241C22×Q8—271no—
242C4⋊Q8—271no—
243C4⋊Q8C3⋊D911yescharacteristic, complemented
244#124—271no—
245#124—271no—
246#123—274no—
247C22⋊Q8—274no—
248#117—271no—
249C3×C9#53611yescharacteristic, complemented, Sylow
250C22×D3—91no—
251C22×D3—93no—
252C22×D3—92no—
253C22×D3—96no—
254C3×Q8—26no—
255C3×Q8—26no—
256C3×Q8—22no—
257C3×Q8—22no—
258C2×C3×C4C22×D916yes—
259C2×C3×C4C22×D916yes—
260C2×C3×C4C22×D916yes—
261C2×C3×C4C22×D913yes—
262C2×C3×C4C22×C3⋊D312yes—
263C2×C3×C4C22×C3⋊D312yes—
264C2×C3×C4C22×C3⋊D312yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266C2×Dic3—91no—
267C2×Dic3—93no—
268C2×Dic3—92no—
269C2×Dic3—92no—
270C2×Dic3—96no—
271C2×Dic3—96no—
272D12—184no—
273D12—1812no—
274D12—186no—
275D12—182no—
276C4×D3—912no—
277C4×D3—94no—
278C4×D3—184no—
279C4×D3—1812no—
280C4×D3—184no—
281C4×D3—1812no—
282Dic6—184no—
283Dic6—1812no—
284Dic6—186no—
285Dic6—182no—
286C2×C32C2×D3×D411yescharacteristic
287C2×C32#100712yes—
288C3⋊D3—62no—
289C3⋊D3—122no—
290C2×C9C2×D3×D413yes—
291C2×C9#100716yes—
292D9—66no—
293D9—126no—
294#41—544no—
295C2×Q8—272no—
296C2×Q8—271no—
297C2×Q8—544no—
298C2×Q8C2×C3⋊D911yescharacteristic
299C2×Q8C2×C3⋊D911yescharacteristic
300C2×D4—272no—
301C2×D4—271no—
302C22×C4—271no—
303C22×C4—272no—
304C22×C4—272no—
305C4⋊C4—272no—
306C4⋊C4—274no—
307C4⋊C4C2×C3⋊D914yes—
308C22⋊C4—272no—
309C22⋊C4—274no—
310C22⋊C4—274no—
311C42—271no—
312C42C2×C3⋊D911yescharacteristic
313C22×C3C23×D913yes—
314C22×C3C23×C3⋊D311yescharacteristic
315C2×D3—92no—
316C2×D3—96no—
317C2×D3—182no—
318C2×D3—186no—
319C2×D3—182no—
320C2×D3—184no—
321C2×D3—186no—
322C2×D3—1812no—
323C3×C4—22no—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—26no—
327C3×C4—26no—
328C3×C4—26no—
329C3×C4D4×D916yes—
330C3×C4D4×C3⋊D312yes—
331Dic3—92no—
332Dic3—96no—
333Dic3—182no—
334Dic3—182no—
335Dic3—186no—
336Dic3—186no—
337C32#572411yescharacteristic
338C9#572413yescomplemented
339C23—271no—
340C23—272no—
341Q8—22no—
342Q8—22no—
343Q8—544no—
344Q8—542no—
345D4—544no—
346D4—542no—
347C2×C4—271no—
348C2×C4—274no—
349C2×C4—272no—
350C2×C4—272no—
351C2×C4—544no—
352C2×C4—544no—
353C2×C4C22×C3⋊D912yes—
354C2×C4C22×C3⋊D912yes—
355C2×C4C22×C3⋊D912yes—
356C2×C4C22×C3⋊D911yescharacteristic
357C2×C3C2×D4×D913yes—
358C2×C3#6371316yes—
359C2×C3C2×D4×C3⋊D311yescharacteristic
360C2×C3#6436412yes—
361D3—182no—
362D3—186no—
363D3—362no—
364D3—366no—
365C22—272no—
366C22—542no—
367C22—542no—
368C22—544no—
369C22C23×C3⋊D911yescharacteristic
370C4—22no—
371C4—22no—
372C4—22no—
373C4—272no—
374C4—542no—
375C4—542no—
376C4D4×C3⋊D912yes—
377C3#1058982613yescomplemented
378C3#1059619511yescharacteristic
379C2—542no—
380C2—1082no—
381C2C2×D4×C3⋊D911yescharacteristic
382C2#1172266712yes—
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×C4⋊Q8C2
socleC22×C32C23×D3
prime radicalC2×#88746C2
hypercenterC4⋊Q8C3⋊D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C3⋊D9C2semidirect
3. C3×C9×C4⋊Q8C2semidirect
4. #11721857C2semidirect
5. #11721854C2semidirect
6. #11721752C2semidirect
7. #11721750C2semidirect
8. #11721712C2semidirect
9. #11721709C2semidirect
42. C9×C4⋊Q8D3semidirect
123. C3×C4⋊Q8D9semidirect
243. C4⋊Q8C3⋊D9semidirect
249. C3×C9#536semidirect
338. C9#5724semidirect
377. C3#10589826semidirect

Maximal subgroups

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

subgroupindex
C2×#887462
C2×Q8×C3⋊D92
C3×C9×C4⋊Q82
#117218572
#117218542
#117217522
#117217502
#117217122
#117217092
#105961953
#105898263