Finite Group Databasebeta

← #13177695

Subgroups — #13177695

Order 1728.

1488 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723976C211yesmaximal, characteristic
2C3×#64144C211yesmaximal, characteristic
3C32×#936C211yesmaximal, characteristic
4#11723642C212yesmaximal, complemented
5#11723641C212yesmaximal, complemented
6#11723597C212yesmaximal, complemented
7#11723596C212yesmaximal, complemented
8#11723444C212yesmaximal, complemented
9#11723393C212yesmaximal, complemented
10#10596143—31nomaximal
11#10595276—34nomaximal
12C2×C33×D4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C2×#6293C2212yes—
17#88864C2212yes—
18#88864C2211yescharacteristic
19#88857C2212yes—
20C3×#3757C2211yescharacteristic
21C3×#3757C2212yes—
22C3×#3750C2212yes—
23C32×#217C2212yes—
24C32×#217C2212yes—
25C32×C4×Dic3C2211yescharacteristic
26#88807C2212yes—
27#88807C2212yes—
28#88806C2212yes—
29#88806C2212yes—
30Dic3⋊(C3⋊Dic3)C2212yes—
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3×C3⋊Dic3C2212yes—
33C32×#124—31no—
34#64153—32no—
35#64144—31no—
36#64144—34no—
37#64144—64no—
38#64144D311yescharacteristic
39#64088—32no—
40#64088—32no—
41#64087—32no—
42#64087—32no—
43#64079—32no—
44C3×#936D314yes—
45C3×#935—34no—
46#63982—38no—
47#63964—38no—
48#63956—38no—
49#63956—38no—
50#63867—38no—
51#63839—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C33×D4—41no—
55C2×C33×C4C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)—41no—
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3×C3⋊Dic3—41no—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C32×Dic3C2312yes—
63C2×C32×Dic3C2312yes—
64C32×Dic6—41no—
65#6293—42no—
66#5672—94no—
67#5651—91no—
68C2×C32×Q8—31no—
69C2×C32×D4—24no—
70C2×C32×D4C2×D314yes—
71C2×C32×D4C2×D311yescharacteristic
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—34no—
74C22×C3⋊Dic3—64no—
75C22×C3⋊Dic3C2×D311yescharacteristic
76C2×#634—32no—
77C22×C3×Dic3—34no—
78C2×C3×Dic6C2×D314yes—
79C2×#627—38no—
80C32×C22⋊C4—32no—
81C32×C22⋊C4—32no—
82C32×C42—31no—
83#3757—32no—
84#3757—31no—
85#3757—34no—
86#3757—38no—
87#3757—64no—
88#3757—68no—
89#3757C2×D312yes—
90#3757C2×D311yescharacteristic
91C4⋊(C3⋊Dic3)—32no—
92C4⋊(C3⋊Dic3)—32no—
93#3750—32no—
94#3750—38no—
95#3750—32no—
96#3750—32no—
97#3750—32no—
98#3750—32no—
99#3750—68no—
100#3750C2×D312yes—
101C4×C3⋊Dic3—32no—
102C3×#217—34no—
103C3×#217—38no—
104C3×#217C2×D318yes—
105C3×#217C2×D318yes—
106C3×Dic3⋊C4—38no—
107C3×C4×Dic3C2×D314yes—
108#3724—38no—
109#3724—38no—
110Dic3⋊Dic3—38no—
111Dic3⋊Dic3—38no—
112Dic3⋊Dic3—38no—
113Dic3⋊Dic3—38no—
114Dic32—38no—
115C22×C33—21no—
116C22×C33—21no—
117C22×C33—22no—
118C22×C33—22no—
119C22×C33C2411yescharacteristic
120C33×C4—21no—
121C3⋊(C3⋊Dic3)—22no—
122C3⋊(C3⋊Dic3)—22no—
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3—22no—
125C32×Dic3—22no—
126C32×Dic3—22no—
127C3×#124—34no—
128C3×#123—91no—
129#944—98no—
130#936C3⋊D311yescharacteristic
131#935—34no—
132#935—94no—
133#935—91no—
134#935—184no—
135#924—92no—
136#889—92no—
137#888—92no—
138#879—98no—
139#879—98no—
140#879—92no—
141#879—92no—
142#878—98no—
143#878—98no—
144#878—92no—
145#870—98no—
146C23×C32—24no—
147C23×C32—24no—
148C23×C32C22×D314yes—
149C23×C32C22×D314yes—
150C23×C32C22×D311yescharacteristic
151C23×C32C22×D311yescharacteristic
152C32×Q8—121no—
153C32×D4—41no—
154C32×D4—44no—
155C32×D4—48no—
156C2×C32×C4—24no—
157C2×C32×C4—32no—
158C2×C32×C4—32no—
159C2×C32×C4C22×D314yes—
160C2×C32×C4C22×D311yescharacteristic
161C2×C3⋊Dic3—32no—
162C2×C3⋊Dic3—32no—
163C2×C3⋊Dic3—38no—
164C2×C3⋊Dic3—38no—
165C2×C3⋊Dic3—41no—
166C2×C3⋊Dic3—68no—
167C2×C3⋊Dic3—68no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—124no—
170C2×C3⋊Dic3—128no—
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D312yes—
173#634—122no—
174C2×C3×Dic3—38no—
175C2×C3×Dic3—38no—
176C2×C3×Dic3—124no—
177C2×C3×Dic3C22×D318yes—
178C2×C3×Dic3C22×D318yes—
179C3×Dic6—44no—
180#627—128no—
181#563—271noSylow
182C2×C33—41no—
183C2×C33—41no—
184C2×C33H(5,𝔽2)11yescharacteristic
185C2×C332−1+412yes—
186C2×C3×Q8—34no—
187C2×C3×D4—24no—
188C2×C3×D4D3214yes—
189C2×C3×D4C2×C3⋊D311yescharacteristic
190C22×C3×C4—91no—
191C22×Dic3—34no—
192C22×Dic3—94no—
193C22×Dic3—91no—
194C22×Dic3—184no—
195C2×Dic6—98no—
196C2×Dic6—92no—
197C2×Dic6C2×C3⋊D311yescharacteristic
198C3×C4⋊C4—92no—
199C3×C22⋊C4—38no—
200C3×C22⋊C4—38no—
201C3×C22⋊C4—92no—
202C3×C22⋊C4—91no—
203C3×C42—34no—
204#217—38no—
205#217—34no—
206#217—98no—
207#217—94no—
208#217—91no—
209#217—92no—
210#217—188no—
211#217—184no—
212#217C2×C3⋊D312yes—
213#217C2×C3⋊D312yes—
214C4⋊Dic3—98no—
215C4⋊Dic3—98no—
216C4⋊Dic3—92no—
217C4⋊Dic3—92no—
218Dic3⋊C4—38no—
219Dic3⋊C4—98no—
220Dic3⋊C4—98no—
221Dic3⋊C4—98no—
222Dic3⋊C4—98no—
223Dic3⋊C4—98no—
224Dic3⋊C4—92no—
225Dic3⋊C4—92no—
226Dic3⋊C4—92no—
227Dic3⋊C4—92no—
228Dic3⋊C4—92no—
229Dic3⋊C4—188no—
230C4×Dic3—98no—
231C4×Dic3—92no—
232C4×Dic3C2×C3⋊D311yescharacteristic
233C22×C32—28no—
234C22×C32—21no—
235C22×C32—24no—
236C22×C32—21no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—28no—
242C22×C32—28no—
243C22×C32—48no—
244C22×C32—44no—
245C22×C32—48no—
246C22×C32C23×D314yes—
247C22×C32C23×D311yescharacteristic
248C32×C4—21no—
249C32×C4—24no—
250C32×C4—44no—
251C32×C4—62no—
252C32×C4—62no—
253C3⋊Dic3—22no—
254C3⋊Dic3—22no—
255C3⋊Dic3—62no—
256C3⋊Dic3—62no—
257C3⋊Dic3—68no—
258C3⋊Dic3—68no—
259C3⋊Dic3—128no—
260C3⋊Dic3—128no—
261C3×Dic3—28no—
262C3×Dic3—28no—
263C3×Dic3—68no—
264C3×Dic3—68no—
265C4⋊Q8—272no—
266#126—272no—
267#126—272no—
268#125—272no—
269#124—31no—
270#123—91no—
271#123—271no—
272C22⋊Q8—272no—
273C22⋊Q8—272no—
274C33#56311yescharacteristic, complemented, Sylow
275C23×C3—24no—
276C23×C3—24no—
277C23×C3C2×D3214yes—
278C23×C3C2×D3214yes—
279C23×C3C22×C3⋊D311yescharacteristic
280C23×C3C22×C3⋊D311yescharacteristic
281C3×Q8—124no—
282C3×D4—44no—
283C3×D4—41no—
284C3×D4—48no—
285C2×C3×C4—24no—
286C2×C3×C4—38no—
287C2×C3×C4—38no—
288C2×C3×C4—92no—
289C2×C3×C4—92no—
290C2×C3×C4—361no—
291C2×C3×C4C2×D3214yes—
292C2×C3×C4C22×C3⋊D311yescharacteristic
293C2×Dic3—38no—
294C2×Dic3—38no—
295C2×Dic3—98no—
296C2×Dic3—98no—
297C2×Dic3—92no—
298C2×Dic3—92no—
299C2×Dic3—124no—
300C2×Dic3—188no—
301C2×Dic3—188no—
302C2×Dic3—364no—
303C2×Dic3—361no—
304C2×Dic3—368no—
305C2×Dic3C22×C3⋊D312yes—
306C2×Dic3C22×C3⋊D312yes—
307Dic6—41no—
308Dic6—362no—
309Dic6—368no—
310C2×C32—28no—
311C2×C32—24no—
312C2×C32—48no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—44no—
316C2×C32—44no—
317C2×C32—84no—
318C2×C32#100414yes—
319C2×C32#100411yescharacteristic
320C2×C32#101018yes—
321C2×C32#101012yes—
322C2×Q8—31no—
323C2×Q8—272no—
324C2×D4D3×C3⋊D311yescharacteristic
325C22×C4—91no—
326C22×C4—271no—
327C4⋊C4—92no—
328C4⋊C4—272no—
329C4⋊C4—272no—
330C4⋊C4—272no—
331C4⋊C4—272no—
332C4⋊C4—272no—
333C4⋊C4—272no—
334C4⋊C4—272no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—92no—
338C22⋊C4—91no—
339C22⋊C4—272no—
340C22⋊C4—271no—
341C42—31no—
342C42—272no—
343C22×C3—28no—
344C22×C3—28no—
345C22×C3—28no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—24no—
353C22×C3—48no—
354C22×C3—44no—
355C22×C3—48no—
356C22×C3C22×D3214yes—
357C22×C3C23×C3⋊D311yescharacteristic
358C3×C4—24no—
359C3×C4—21no—
360C3×C4—44no—
361C3×C4—68no—
362C3×C4—68no—
363C3×C4—182no—
364C3×C4—182no—
365Dic3—22no—
366Dic3—22no—
367Dic3—68no—
368Dic3—68no—
369Dic3—188no—
370Dic3—188no—
371Dic3—182no—
372Dic3—182no—
373Dic3—368no—
374Dic3—368no—
375C32—24no—
376C32#565111yescharacteristic, complemented
377C32#567214yescomplemented
378C23C2×D3×C3⋊D311yescharacteristic
379C23C2×D3×C3⋊D311yescharacteristic
380Q8—121no—
381Q8—1082no—
382D4—41no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—92no—
386C2×C4—92no—
387C2×C4—272no—
388C2×C4—272no—
389C2×C4—361no—
390C2×C4—1081no—
391C2×C4C2×D3×C3⋊D311yescharacteristic
392C2×C3—28no—
393C2×C3—24no—
394C2×C3—48no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—44no—
398C2×C3—44no—
399C2×C3—84no—
400C2×C3#6430918yes—
401C2×C3#6433014yes—
402C2×C3#6436111yescharacteristic
403C2×C3#6436712yes—
404C22—22no—
405C22—22no—
406C22—21no—
407C22—21no—
408C22C22×D3×C3⋊D311yescharacteristic
409C4—21no—
410C4—62no—
411C4—62no—
412C4—182no—
413C4—182no—
414C4—542no—
415C4—542no—
416C3—24no—
417C3#1059527614yescomplemented
418C3#1059614311yescharacteristic, complemented
419C2—41no—
420C2—41no—
421C2#1172458212yes—
422C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11723642C2semidirect
5. #11723641C2semidirect
6. #11723597C2semidirect
7. #11723596C2semidirect
8. #11723444C2semidirect
9. #11723393C2semidirect
274. C33#563semidirect
376. C32#5651semidirect
377. C32#5672semidirect
417. C3#10595276semidirect
418. C3#10596143semidirect

Maximal subgroups

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

subgroupindex
#117239762
C3×#641442
C32×#9362
#117236422
#117236412
#117235972
#117235962
#117234442
#117233932
#105961433
#105952763