Finite Group Databasebeta

← #13160035

Subgroups — #13160035

Order 1728.

1412 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×#89026C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic, complemented
3#11723413C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592469—31nomaximal
9#10590941—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15#89026—22no—
16#89026C414yescomplemented
17C2×C33×C8C2211yescharacteristic, complemented
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
19#88850—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21C32×C4⋊Dic3C2211yescharacteristic, complemented
22#88804—21no—
23Dic3×C3⋊Dic3—21no—
24C2×D4×C3⋊D3—31no—
25C2×#3802—34no—
26C2×#3743—31no—
27C2×#3743—34no—
28C2×#3743—64no—
29#64101—31no—
30#63849—34no—
31C32×#102—31no—
32#63659—31no—
33#63648D311yescharacteristic, complemented
34#63636—31no—
35C3×#821—34no—
36C3×#821D314yescomplemented
37#63562—34no—
38#63562—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C22×C32×D3—21no—
42C2×D3×C3⋊D3—24no—
43C2×D3×C3⋊D3—42no—
44C2×D3×C3⋊D3—42no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C4×C3⋊(C3⋊D3)C2×C412yes—
48C4×C3⋊(C3⋊D3)D412yes—
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)C2×C412yes—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12C2×C412yes—
55#6290—24no—
56C33×C8—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58#4770—94no—
59#4770—91no—
60C23×C3⋊D3—61no—
61C22×D32—64no—
62C2×C32×D4—31no—
63C2×#638—61no—
64D4×C3⋊D3—34no—
65D4×C3⋊D3—62no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—34no—
69C2×C4×C3⋊D3—64no—
70C2×C3×D12—34no—
71C2×C3×D12C2×D314yes—
72C2×#625—64no—
73#3802—316no—
74#3802—68no—
75C2×C32×C8—24no—
76C2×C32×C8C2×D314yescomplemented
77C2×C32×C8C2×D311yescharacteristic, complemented
78C32×C4⋊C4—31no—
79#3752—61no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
81C4×C3⋊Dic3—61no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85#3743—62no—
86#3743—68no—
87#3743—616no—
88C3×C4⋊Dic3—34no—
89C3×C4⋊Dic3C2×D314yescomplemented
90#3722—64no—
91Dic32—64no—
92C22×C33C2×D411yescharacteristic
93C2×C3⋊(C3⋊D3)—21no—
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)C22⋊C412yes—
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—41no—
99C2×C32×D3—22no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102D3×C3⋊D3—48no—
103C33×C4C22×C411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)C22⋊C412yes—
106C3×C3⋊Dic3—41no—
107C32×Dic3—41no—
108C2×D3×D4—94no—
109C2×D3×D4—91no—
110C2×ΓL(1,𝔽25)—94no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—184no—
113#892—94no—
114#892—91no—
115C3×#102—34no—
116C3×#102—91no—
117#832—94no—
118#832—91no—
119#821—34no—
120#821C3⋊D311yescharacteristic, complemented
121#809—94no—
122#809—91no—
123C23×C32—61no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—31no—
126C22×C3⋊D3—34no—
127C22×C3⋊D3—64no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—122no—
130C22×C3⋊D3—122no—
131C22×C3×D3—24no—
132C22×C3×D3—64no—
133C2×D32—616no—
134C2×D32—128no—
135C2×D32—128no—
136C32×D4—32no—
137C32×D4—61no—
138C2×C32×C4—24no—
139C2×C32×C4—61no—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D311yescharacteristic
142#638—64no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—31no—
145C2×C3⋊Dic3—34no—
146C2×C3⋊Dic3—64no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)C4×D312yes—
149C4×C3⋊D3—32no—
150C4×C3⋊D3—32no—
151C4×C3⋊D3—38no—
152C4×C3⋊D3—38no—
153C4×C3⋊D3—68no—
154C4×C3⋊D3—68no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—64no—
157C3×D12—24no—
158C3×D12—38no—
159C3×D12—64no—
160C3×D12C4×D318yes—
161#625—616no—
162C32×C8—28no—
163C32×C8—24no—
164C32×C8—21no—
165C3⋊(C3⋊C8)—68no—
166C3⋊(C3⋊C8)—64no—
167C3⋊(C3⋊C8)—61no—
168#418—271noSylow
169C2×C33C2×C22⋊C411yescharacteristic
170C2×C33AGL(1,ℤ/8)11yescharacteristic
171C2×C33AGL(1,ℤ/8)11yescharacteristic
172C3⋊(C3⋊D3)—22no—
173C3×C3⋊D3—42no—
174C32×D3—42no—
175C23×D3—181no—
176C23×D3—184no—
177C2×C3×D4—34no—
178C2×C3×D4—91no—
179C2×#67—181no—
180C2×#67—184no—
181D3×D4—916no—
182D3×D4—94no—
183D3×D4—182no—
184D3×D4—188no—
185C2×D12—34no—
186C2×D12C2×C3⋊D311yescharacteristic
187C2×C4×D3—94no—
188C2×C4×D3—91no—
189C2×C4×D3—184no—
190C2×C3×C8—24no—
191C2×C3×C8D3214yescomplemented
192C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
193C3×C4⋊C4—34no—
194C3×C4⋊C4—91no—
195(C2×D3)⋊C4—181no—
196(C2×D3)⋊C4—184no—
197C4⋊Dic3—34no—
198C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
199C4×Dic3—181no—
200C4×Dic3—184no—
201C2×C3⋊C8—94no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—184no—
204ΓL(1,𝔽25)—188no—
205ΓL(1,𝔽25)—182no—
206ΓL(1,𝔽25)—1816no—
207C22×C32—24no—
208C22×C32—62no—
209C22×C32—121no—
210C22×C32—121no—
211C22×C32D3×D414yes—
212C22×C32D3×D411yescharacteristic
213C2×C3⋊D3—22no—
214C2×C3⋊D3—32no—
215C2×C3⋊D3—38no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—68no—
219C2×C3⋊D3—61no—
220C2×C3⋊D3—61no—
221C2×C3⋊D3—64no—
222C2×C3⋊D3—64no—
223C2×C3⋊D3—124no—
224C2×C3⋊D3—124no—
225C2×C3⋊D3—128no—
226C2×C3×D3—28no—
227C2×C3×D3—44no—
228C2×C3×D3—44no—
229C2×C3×D3—68no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232D32—1232no—
233C32×C4—24no—
234C32×C4—24no—
235C32×C4—121no—
236C32×C4C2×C4×D314yes—
237C32×C4C2×C4×D311yescharacteristic
238C32×C4D3×D414yes—
239C32×C4D3×D411yescharacteristic
240C3⋊Dic3—32no—
241C3⋊Dic3—38no—
242C3⋊Dic3—41no—
243C3⋊Dic3—68no—
244C3×Dic3—44no—
245C3×Dic3—124no—
246C22×D4—271no—
247C2×M16—271no—
248#117—271no—
249#102—31no—
250#102—91no—
251#102—271no—
252#102—271no—
253C33#41811yescharacteristic, complemented, Sylow
254C23×C3—64no—
255C23×C3—181no—
256C22×D3—21no—
257C22×D3—64no—
258C22×D3—94no—
259C22×D3—91no—
260C22×D3—184no—
261C22×D3—184no—
262C22×D3—1816no—
263C22×D3—362no—
264C22×D3—362no—
265C22×D3—368no—
266C22×D3—368no—
267C3×D4—38no—
268C3×D4—64no—
269C3×D4—92no—
270C3×D4—181no—
271C2×C3×C4—24no—
272C2×C3×C4—64no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3214yes—
275C2×C3×C4C22×C3⋊D311yescharacteristic
276#67—184no—
277#67—1816no—
278C2×Dic3—21no—
279C2×Dic3—64no—
280C2×Dic3—94no—
281C2×Dic3—91no—
282C2×Dic3—184no—
283D12—21no—
284D12—38no—
285D12—64no—
286D12C4×C3⋊D312yes—
287C4×D3—98no—
288C4×D3—98no—
289C4×D3—92no—
290C4×D3—92no—
291C4×D3—188no—
292C4×D3—188no—
293C3×C8—28no—
294C3×C8—24no—
295C3×C8—21no—
296C3⋊C8—188no—
297C3⋊C8—184no—
298C3⋊C8—181no—
299C2×C32—24no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—122no—
303C2×C32D3×C22⋊C414yes—
304C2×C32D3×C22⋊C411yescharacteristic
305C2×C32#91114yes—
306C2×C32#91111yescharacteristic
307C2×C32#91414yes—
308C2×C32#91411yescharacteristic
309C3⋊D3—42no—
310C3⋊D3—62no—
311C3⋊D3—68no—
312C3⋊D3—128no—
313C3×D3—48no—
314C3×D3—128no—
315C24—541no—
316C2×D4—31no—
317C2×D4—91no—
318C2×D4—274no—
319C2×D4—542no—
320C2×D4—541no—
321C22×C4—271no—
322M16—542no—
323C2×C8—271no—
324C2×C8D3×C3⋊D311yescharacteristic, complemented
325C4⋊C4—31no—
326C4⋊C4—91no—
327C22⋊C4—541no—
328C42—541no—
329C22×C3—24no—
330C22×C3—68no—
331C22×C3—124no—
332C22×C3—124no—
333C22×C3—182no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3#380214yes—
337C22×C3D4×C3⋊D311yescharacteristic
338C2×D3—22no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—68no—
342C2×D3—92no—
343C2×D3—98no—
344C2×D3—124no—
345C2×D3—124no—
346C2×D3—188no—
347C2×D3—184no—
348C2×D3—184no—
349C2×D3—181no—
350C2×D3—181no—
351C2×D3—364no—
352C2×D3—364no—
353C2×D3—368no—
354C2×D3—3632no—
355C3×C4—24no—
356C3×C4—24no—
357C3×C4—124no—
358C3×C4—361no—
359C3×C4C4×D3214yes—
360C3×C4#381114yes—
361C3×C4C2×C4×C3⋊D311yescharacteristic
362C3×C4D4×C3⋊D311yescharacteristic
363Dic3—41no—
364Dic3—92no—
365Dic3—98no—
366Dic3—124no—
367Dic3—188no—
368C32—24no—
369C32#477014yescomplemented
370C32#477011yescharacteristic, complemented
371C23—61no—
372C23—181no—
373C23—271no—
374C23—544no—
375C23—1082no—
376C23—1082no—
377D4—32no—
378D4—61no—
379D4—92no—
380D4—181no—
381D4—544no—
382C2×C4—61no—
383C2×C4—181no—
384C2×C4—271no—
385C2×C4—272no—
386C2×C4—272no—
387C2×C4C2×D3×C3⋊D311yescharacteristic
388C8—21no—
389C8—541no—
390C2×C3—24no—
391C2×C3—24no—
392C2×C3—24no—
393C2×C3—128no—
394C2×C3—362no—
395C2×C3#6379814yes—
396C2×C3#6379814yes—
397C2×C3#6392114yes—
398C2×C3C22⋊C4×C3⋊D311yescharacteristic
399C2×C3#6412011yescharacteristic
400C2×C3#6412311yescharacteristic
401D3—42no—
402D3—128no—
403D3—182no—
404D3—188no—
405D3—368no—
406C22—62no—
407C22—121no—
408C22—121no—
409C22—182no—
410C22—272no—
411C22—361no—
412C22—361no—
413C22—541no—
414C22—541no—
415C22—1088no—
416C22#8902611yescharacteristic
417C4—121no—
418C4—272no—
419C4—361no—
420C4C4×D3×C3⋊D311yescharacteristic
421C4#8903911yescharacteristic
422C3—24no—
423C3#1059094114yescomplemented
424C3#1059246911yescharacteristic, complemented
425C2—122no—
426C2—362no—
427C2—542no—
428C2#1172332311yescharacteristic
429C2#1172332411yescharacteristic
430C2#1172353911yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88850C2semidirect
3. #11723413C2semidirect
4. C3×#63648C2semidirect
5. C32×#821C2semidirect
6. #11722033C2semidirect
7. #11722031C2semidirect
16. #89026C4semidirect
17. C2×C33×C8C22semidirect
18. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
20. C3×C4⋊(C3⋊Dic3)C22semidirect
21. C32×C4⋊Dic3C22semidirect
33. #63648D3semidirect
36. C3×#821D3semidirect
76. C2×C32×C8C2×D3semidirect
77. C2×C32×C8C2×D3semidirect
80. C4⋊(C3⋊Dic3)C2×D3semidirect
89. C3×C4⋊Dic3C2×D3semidirect
120. #821C3⋊D3semidirect
191. C2×C3×C8D32semidirect
192. C2×C3×C8C2×C3⋊D3semidirect
198. C4⋊Dic3C2×C3⋊D3semidirect
253. C33#418semidirect
324. C2×C8D3×C3⋊D3semidirect
369. C32#4770semidirect
370. C32#4770semidirect
423. C3#10590941semidirect
424. C3#10592469semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C2×#888502
#117234132
C3×#636482
C32×#8212
#117220332
#117220312
#105924693
#105909413