Finite Group Databasebeta

← #13162034

Subgroups — #13162034

Order 1728.

842 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 91 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#11724619C211yesmaximal, characteristic
2C3×#64156C211yesmaximal, characteristic, complemented
3C3×#64151C212yesmaximal, complemented
4#11722077C212yesmaximal, complemented
5#11722073C211yesmaximal, characteristic, complemented
6#10591659—32nomaximal
7#10591658—31nomaximal
8C2×C33×Q8C2211yescharacteristic, complemented
9C3×#3828C2212yes—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11#89049—22no—
12C4⋊#1113—22no—
13#89041—22no—
14C3×#3756—22no—
15C3×#3755—22no—
16C3×#3752—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
18C3×#3748C2212yescomplemented
19#64308—31no—
20#64307—32no—
21#64156D311yescharacteristic, complemented
22#64151D312yescomplemented
23C3×#947—32no—
24C3×#947—61no—
25C3×#942—32no—
26C3×#942—32no—
27C3×#942—62no—
28#63570—32no—
29#63569—32no—
30#63568—32no—
31#63559—31no—
32#63558—32no—
33C22×C3×C3⋊D3D412yes—
34C2×#1113—22no—
35C33×Q8—22no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C3×#638—22no—
39C2×C3×C3⋊Dic3—21no—
40C3×C4⋊(C3⋊D3)—22no—
41C3×C4⋊(C3⋊D3)D412yes—
42C3×C4×C3⋊D3—22no—
43C3×#634D412yes—
44#6295—24no—
45#6294—22no—
46C3×C3⋊(C3⋊C8)—22no—
47#5202—92no—
48#5061—91no—
49C2×C32×Q8—21no—
50C2×C32×Q8—22no—
51C2×C32×Q8—41no—
52C2×C32×Q8C2×D311yescharacteristic, complemented
53C2×C32×Q8C2×D312yescomplemented
54#3828C2×D312yes—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C3×#235—32no—
57C3×#235—32no—
58C3×#235—62no—
59C2×C3×D12—32no—
60C2×C3×D12—61no—
61#3811—62no—
62D3×#67—64no—
63#3802—62no—
64D3×D12—62no—
65D3×D12—62no—
66#3799—62no—
67#3796—62no—
68#3795—62no—
69#3756—22no—
70#3755—22no—
71#3752—22no—
72C4×C3⋊Dic3C2×D311yescharacteristic, complemented
73#3748C2×D312yescomplemented
74C3×#216—62no—
75C3×#216—62no—
76C3×#216—64no—
77C3×#215—62no—
78C3×#215—62no—
79C3×#215—64no—
80C3×(C2×D3)⋊C4—64no—
81C3×(C2×D3)⋊C4—64no—
82C3×C4×Dic3—32no—
83C3×C4×Dic3—61no—
84C3×#208—32no—
85C3×#208—32no—
86C3×#208—62no—
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—42no—
91#1113—82no—
92C33×C4—41no—
93C33×C4C2×D412yes—
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3—41no—
96#1009—92no—
97#1004—91no—
98C3×#137—92no—
99C3×#124—91no—
100#947—32no—
101#947—61no—
102#942—32no—
103#942—32no—
104#942—62no—
105#837—92no—
106#833—91no—
107#825—92no—
108#823—92no—
109#805—92no—
110C22×C3⋊D3#6712yes—
111C22×C3×D3—34no—
112C22×C3×D3—62no—
113C2×D32—62no—
114C2×D32—64no—
115C32×Q8—22no—
116C32×Q8—24no—
117C32×Q8—22no—
118C32×Q8—22no—
119C32×Q8—42no—
120C32×Q8—42no—
121C32×Q8—44no—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4—42no—
129C2×C32×C4—42no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D312yes—
132#638—22no—
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—22no—
135C4⋊(C3⋊D3)#6712yes—
136C4×C3⋊D3—22no—
137#634#6712yes—
138C3×#67—62no—
139C3×#67—62no—
140C3×#67—122no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C3×D12—32no—
144C3×D12—32no—
145C3×D12—62no—
146C3×D12—62no—
147C3×D12—62no—
148C3×D12—122no—
149C3×C4×D3—62no—
150C3×C4×D3—62no—
151C3×C4×D3—122no—
152C3×Dic6—32no—
153C3×Dic6—32no—
154C3×Dic6—62no—
155#626—64no—
156#626—64no—
157#625—64no—
158#624—62no—
159D3×Dic3—64no—
160C3⋊(C3⋊C8)—22no—
161C3×C3⋊C8—64no—
162C3×C3⋊C8—62no—
163C3×C3⋊C8—62no—
164#455—271noSylow
165C2×C33—21no—
166C2×C33#12011yescharacteristic
167C3×C3⋊D3—42no—
168C3×C3⋊D3—42no—
169C3×#41—92no—
170C2×C3×Q8—21no—
171C2×C3×Q8—22no—
172C2×C3×Q8—41no—
173C2×C3×Q8D3211yescharacteristic, complemented
174C2×C3×Q8D3212yescomplemented
175C2×C3×D4—91no—
176C2×#67—182no—
177#239—182no—
178#237—182no—
179D3×D4—182no—
180D3×D4—182no—
181D3×D4—184no—
182#235—32no—
183#235—32no—
184#235—62no—
185#235—182no—
186C2×D12—32no—
187C2×D12—61no—
188C2×D12—182no—
189C3×Q16—182no—
190C3×SD16—182no—
191C3×M16—92no—
192C3×C22⋊C4—182no—
193C3×C42—91no—
194#216—62no—
195#216—62no—
196#216—64no—
197#215—62no—
198#215—62no—
199#215—64no—
200(C2×D3)⋊C4—64no—
201(C2×D3)⋊C4—64no—
202C4×Dic3—32no—
203C4×Dic3—61no—
204#208—32no—
205#208—32no—
206#208—62no—
207C22×C32—22no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32D3×D412yes—
211C22×C32C2×#6711yescharacteristic
212C2×C3⋊D3—22no—
213C2×C3⋊D3—22no—
214C2×C3⋊D3—42no—
215C2×C3×D3—64no—
216C2×C3×D3—62no—
217C2×C3×D3—62no—
218C2×C3×D3—122no—
219C2×C3×D3—122no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222D32—244no—
223D32—242no—
224C32×C4—22no—
225C32×C4—22no—
226C32×C4—22no—
227C32×C4—42no—
228C32×C4—42no—
229C32×C4—44no—
230C32×C4—44no—
231C32×C4—41no—
232C32×C4—42no—
233C32×C4D3×D412yes—
234C32×C4D3×D412yes—
235C32×C4C2×#6712yes—
236C3⋊Dic3—22no—
237C3⋊Dic3—41no—
238C3×Dic3—62no—
239C3×Dic3—62no—
240C3×Dic3—122no—
241C3×Dic3—122no—
242C3×Dic3—122no—
243H(5,𝔽2)—271no—
244#137—92no—
245#124—91no—
246#104—272no—
247#100—271no—
248C33#45511yescharacteristic, complemented, Sylow
249C23×C3—92no—
250C22×D3—34no—
251C22×D3—62no—
252C22×D3—184no—
253C22×D3—182no—
254C3×Q8—24no—
255C3×Q8—22no—
256C3×Q8—22no—
257C3×Q8—22no—
258C3×Q8—42no—
259C3×Q8—42no—
260C3×Q8—44no—
261C3×Q8—92no—
262C3×D4—92no—
263C3×D4—182no—
264C3×D4—182no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—21no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—182no—
274C2×C3×C4—181no—
275C2×C3×C4C2×D3211yescharacteristic
276C2×C3×C4C2×D3212yes—
277#67—62no—
278#67—62no—
279#67—122no—
280#67—184no—
281#67—184no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—182no—
285D12—32no—
286D12—32no—
287D12—62no—
288D12—62no—
289D12—62no—
290D12—122no—
291D12—184no—
292C4×D3—62no—
293C4×D3—62no—
294C4×D3—122no—
295C4×D3—184no—
296Dic6—32no—
297Dic6—32no—
298Dic6—62no—
299C3×C8—182no—
300C3⋊C8—64no—
301C3⋊C8—62no—
302C3⋊C8—62no—
303C2×C32—21no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—21no—
308C2×C32—41no—
309C2×C32—42no—
310C2×C32—42no—
311C2×C32#93712yes—
312C2×C32#93711yescharacteristic
313C3⋊D3—42no—
314C3⋊D3—42no—
315C3×D3—122no—
316C3×D3—124no—
317C3×D3—122no—
318C3×D3—242no—
319C3×D3—242no—
320#41—92no—
321#41—542no—
322C2×Q8#111311yescharacteristic, complemented
323C2×D4—91no—
324C2×D4—542no—
325C2×D4—542no—
326Q16—182no—
327SD16—182no—
328M16—92no—
329C22⋊C4—182no—
330C42—91no—
331C22×C3—21no—
332C22×C3—22no—
333C22×C3—41no—
334C22×C3—182no—
335C22×C3—182no—
336C22×C3—362no—
337C22×C3D3×#6712yes—
338C22×C3#381111yescharacteristic
339C2×D3—62no—
340C2×D3—62no—
341C2×D3—64no—
342C2×D3—122no—
343C2×D3—122no—
344C2×D3—124no—
345C2×D3—124no—
346C2×D3—722no—
347C2×D3—724no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—44no—
353C3×C4—44no—
354C3×C4—41no—
355C3×C4—42no—
356C3×C4—42no—
357C3×C4—182no—
358C3×C4—361no—
359C3×C4D3×#6712yes—
360C3×C4D3×#6712yes—
361C3×C4#381112yes—
362Dic3—62no—
363Dic3—62no—
364Dic3—122no—
365Dic3—122no—
366Dic3—122no—
367C32—21no—
368C32—22no—
369C32—41no—
370C32#506111yescharacteristic, complemented
371C32#520212yescomplemented
372C23—92no—
373C23—542no—
374Q8—22no—
375Q8—92no—
376D4—92no—
377D4—182no—
378D4—182no—
379D4—544no—
380C2×C4—21no—
381C2×C4—182no—
382C2×C4—181no—
383C2×C4—542no—
384C2×C4C2×#111311yescharacteristic
385C8—182no—
386C2×C3—22no—
387C2×C3—21no—
388C2×C3—22no—
389C2×C3—22no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—42no—
393C2×C3—41no—
394C2×C3—362no—
395C2×C3—362no—
396C2×C3#6398311yescharacteristic
397C2×C3#6398312yes—
398D3—122no—
399D3—124no—
400D3—122no—
401D3—242no—
402D3—242no—
403C22—182no—
404C22—182no—
405C22—362no—
406C22—2162no—
407C22#8904911yescharacteristic
408C4—41no—
409C4—182no—
410C4—361no—
411C4#8904912yes—
412C3—21no—
413C3—22no—
414C3—41no—
415C3#1059165811yescharacteristic, complemented
416C3#1059165912yescomplemented
417C2—21no—
418C2—362no—
419C2—362no—
420C2#1172372911yescharacteristic
subgroupG/H
centerC2#11723729
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC2×C33#120
prime radical#11724619C2
hypercenterC2×Q8#1113
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64156C2semidirect
3. C3×#64151C2semidirect
4. #11722077C2semidirect
5. #11722073C2semidirect
8. C2×C33×Q8C22semidirect
17. C3×C4×C3⋊Dic3C22semidirect
18. C3×#3748C22semidirect
21. #64156D3semidirect
22. #64151D3semidirect
52. C2×C32×Q8C2×D3semidirect
53. C2×C32×Q8C2×D3semidirect
72. C4×C3⋊Dic3C2×D3semidirect
73. #3748C2×D3semidirect
173. C2×C3×Q8D32semidirect
174. C2×C3×Q8D32semidirect
248. C33#455semidirect
322. C2×Q8#1113semidirect
370. C32#5061semidirect
371. C32#5202semidirect
415. C3#10591658semidirect
416. C3#10591659semidirect

Maximal subgroups

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

subgroupindex
#117246192
C3×#641562
C3×#641512
#117220772
#117220732
#105916593
#105916583