Finite Group Databasebeta

← #13160077

Subgroups — #13160077

Order 1728.

1244 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1244 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×#89017C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic, complemented
3#11723503C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592502—31nomaximal
9#10590957—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6292—21no—
14C2×#6289—21no—
15C2×Dic3×C3⋊D3—21no—
16#89017—22no—
17#89017C414yescomplemented
18C2×C33×C8C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20#88850—22no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C32×C4⋊Dic3C2211yescharacteristic
23#88806—21no—
24C2×#3832—31no—
25C2×#3796—34no—
26C2×#3743—31no—
27C2×#3743—34no—
28C2×#3743—64no—
29C4⋊C4×C3⋊D3—31no—
30#63900—34no—
31C32×#103—31no—
32#63662—31no—
33#63648D311yescharacteristic, complemented
34#63636—31no—
35C3×#821—34no—
36C3×#816D314yescomplemented
37#63565—34no—
38#63562—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4×C3⋊(C3⋊D3)C2×C412yes—
44C4×C3⋊(C3⋊D3)D412yes—
45C2×C3×C3⋊Dic3—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)C2×C412yescomplemented
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6C2×C412yescomplemented
52#6292—24no—
53#6289—42no—
54Dic3×C3⋊D3—24no—
55C33×C8—21no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57#4803—94no—
58#4771—91no—
59C2×C32×Q8—31no—
60#3832—34no—
61#3832—62no—
62C2×C4⋊(C3⋊D3)—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—34no—
66C2×C4×C3⋊D3—61no—
67C2×C4×C3⋊D3—64no—
68C2×C4×C3⋊D3—61no—
69C2×C3×D12—34no—
70C2×C3×Dic6C2×D314yes—
71C2×#626—64no—
72C2×#624—64no—
73C2×D3×Dic3—64no—
74#3796—316no—
75#3796—68no—
76C2×C32×C8—24no—
77C2×C32×C8C2×D314yes—
78C2×C32×C8C2×D311yescharacteristic
79C32×C4⋊C4—31no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—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×D314yes—
90#3724—64no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)—21no—
94C2×C3⋊(C3⋊D3)C22⋊C412yes—
95C2×C3×C3⋊D3—22no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)C22⋊C412yes—
101C3×C3⋊Dic3—41no—
102C32×Dic3—22no—
103C32×Dic3—41no—
104C2×#239—94no—
105C2×#237—91no—
106C2×ΓL(1,𝔽25)—94no—
107C2×ΓL(1,𝔽25)—91no—
108C2×ΓL(1,𝔽25)—184no—
109D3×C4⋊C4—94no—
110D3×C4⋊C4—91no—
111C3×#103—34no—
112C3×#102—91no—
113#835—94no—
114#832—91no—
115#821—34no—
116#816C3⋊D311yescharacteristic, complemented
117#810—91no—
118#809—94no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3—31no—
121C22×C3⋊D3—34no—
122C22×C3⋊D3—64no—
123C22×C3×D3—64no—
124C32×Q8—32no—
125C32×Q8—61no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—64no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)C4×D312yescomplemented
138C4×C3⋊D3—32no—
139C4×C3⋊D3—32no—
140C4×C3⋊D3—38no—
141C4×C3⋊D3—38no—
142C4×C3⋊D3—68no—
143C4×C3⋊D3—68no—
144C4×C3⋊D3—64no—
145C4×C3⋊D3—122no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—64no—
149C3×D12—38no—
150C3×D12—64no—
151C3×Dic6—24no—
152C3×Dic6C4×D318yescomplemented
153#626—616no—
154#624—128no—
155D3×Dic3—616no—
156C32×C8—28no—
157C32×C8—24no—
158C32×C8—21no—
159C3⋊(C3⋊C8)—68no—
160C3⋊(C3⋊C8)—64no—
161C3⋊(C3⋊C8)—61no—
162#417—271noSylow
163C2×C33C2×C22⋊C411yescharacteristic
164C2×C33AGL(1,ℤ/8)11yescharacteristic
165C2×C33#13711yescharacteristic
166C3⋊(C3⋊D3)—22no—
167C3×C3⋊D3—42no—
168C2×C3×Q8—34no—
169C2×C3×D4—91no—
170C2×#67—181no—
171C22×Dic3—181no—
172#239—916no—
173#239—188no—
174#237—94no—
175#237—182no—
176C2×D12—34no—
177C2×D12—184no—
178C2×C4×D3—94no—
179C2×C4×D3—91no—
180C2×C4×D3—184no—
181C2×C4×D3—184no—
182C2×C4×D3—181no—
183C2×C4×D3—184no—
184C2×Dic6C2×C3⋊D311yescharacteristic
185C2×C3×C8—24no—
186C2×C3×C8D3214yes—
187C2×C3×C8C2×C3⋊D311yescharacteristic
188C3×C4⋊C4—34no—
189C3×C4⋊C4—91no—
190C4⋊Dic3—34no—
191C4⋊Dic3C2×C3⋊D311yescharacteristic
192Dic3⋊C4—181no—
193Dic3⋊C4—184no—
194C2×C3⋊C8—94no—
195C2×C3⋊C8—91no—
196C2×C3⋊C8—184no—
197ΓL(1,𝔽25)—188no—
198ΓL(1,𝔽25)—182no—
199ΓL(1,𝔽25)—1816no—
200C22×C32—24no—
201C22×C32D3×D414yes—
202C22×C32D3×D411yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—32no—
205C2×C3⋊D3—38no—
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—68no—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—61no—
211C2×C3⋊D3—64no—
212C2×C3⋊D3—64no—
213C2×C3⋊D3—124no—
214C2×C3⋊D3—124no—
215C2×C3×D3—68no—
216C2×C3×D3—124no—
217C2×C3×D3—124no—
218C32×C4—24no—
219C32×C4—24no—
220C32×C4—62no—
221C32×C4—121no—
222C32×C4C2×C4×D314yes—
223C32×C4C2×C4×D311yescharacteristic
224C32×C4D3×D414yes—
225C32×C4D3×D411yescharacteristic
226C3⋊Dic3—32no—
227C3⋊Dic3—38no—
228C3⋊Dic3—41no—
229C3⋊Dic3—68no—
230C3×Dic3—28no—
231C3×Dic3—44no—
232C3×Dic3—124no—
233C2×#41—271no—
234C2×M16—271no—
235C2×C4⋊C4—271no—
236#103—31no—
237#103—271no—
238#102—91no—
239#102—271no—
240C33#41711yescharacteristic, complemented, Sylow
241C23×C3—181no—
242C22×D3—64no—
243C22×D3—94no—
244C22×D3—91no—
245C22×D3—184no—
246C3×Q8—38no—
247C3×Q8—64no—
248C3×D4—92no—
249C3×D4—181no—
250C2×C3×C4—24no—
251C2×C3×C4—64no—
252C2×C3×C4—64no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256#67—184no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—64no—
260C2×Dic3—94no—
261C2×Dic3—91no—
262C2×Dic3—184no—
263C2×Dic3—184no—
264D12—38no—
265D12—64no—
266D12—1816no—
267C4×D3—98no—
268C4×D3—98no—
269C4×D3—92no—
270C4×D3—92no—
271C4×D3—188no—
272C4×D3—188no—
273C4×D3—1816no—
274C4×D3—362no—
275C4×D3—368no—
276Dic6—21no—
277Dic6C4×C3⋊D312yescomplemented
278C3×C8—28no—
279C3×C8—24no—
280C3×C8—21no—
281C3⋊C8—188no—
282C3⋊C8—184no—
283C3⋊C8—181no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32D3×C22⋊C414yes—
288C2×C32D3×C22⋊C411yescharacteristic
289C2×C32#91111yescharacteristic
290C2×C32#91414yes—
291C2×C32#91511yescharacteristic
292C2×C32#91814yes—
293C3⋊D3—42no—
294C3⋊D3—62no—
295C3⋊D3—68no—
296C3⋊D3—128no—
297C3×D3—128no—
298#41—274no—
299#41—542no—
300C2×Q8—31no—
301C2×D4—91no—
302C2×D4—541no—
303C22×C4—271no—
304C22×C4—541no—
305C22×C4—541no—
306M16—542no—
307C2×C8—271no—
308C2×C8D3×C3⋊D311yescharacteristic
309C4⋊C4—31no—
310C4⋊C4—91no—
311C4⋊C4—541no—
312C22×C3—24no—
313C22×C3—182no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3#380214yes—
317C22×C3D4×C3⋊D311yescharacteristic
318C2×D3—68no—
319C2×D3—92no—
320C2×D3—98no—
321C2×D3—124no—
322C2×D3—124no—
323C2×D3—188no—
324C2×D3—184no—
325C2×D3—184no—
326C2×D3—181no—
327C2×D3—181no—
328C2×D3—364no—
329C2×D3—364no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—68no—
333C3×C4—124no—
334C3×C4—361no—
335C3×C4C4×D3214yes—
336C3×C4#381114yes—
337C3×C4C2×C4×C3⋊D311yescharacteristic
338C3×C4D4×C3⋊D311yescharacteristic
339Dic3—22no—
340Dic3—41no—
341Dic3—92no—
342Dic3—98no—
343Dic3—124no—
344Dic3—188no—
345C32—24no—
346C32#477111yescharacteristic, complemented
347C32#480314yescomplemented
348C23—181no—
349C23—271no—
350Q8—32no—
351Q8—61no—
352D4—92no—
353D4—181no—
354D4—544no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—181no—
358C2×C4—271no—
359C2×C4—272no—
360C2×C4—272no—
361C2×C4—544no—
362C2×C4—1082no—
363C2×C4C2×D3×C3⋊D311yescharacteristic
364C8—21no—
365C8—541no—
366C2×C3—24no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—362no—
370C2×C3#6379814yes—
371C2×C3#6381114yes—
372C2×C3#6392114yes—
373C2×C3C22⋊C4×C3⋊D311yescharacteristic
374C2×C3#6412311yescharacteristic
375C2×C3#6412711yescharacteristic
376D3—128no—
377D3—188no—
378D3—182no—
379D3—368no—
380C22—182no—
381C22—272no—
382C22—361no—
383C22—361no—
384C22—541no—
385C22—541no—
386C22#8902611yescharacteristic
387C4—62no—
388C4—121no—
389C4—272no—
390C4—361no—
391C4C4×D3×C3⋊D311yescharacteristic
392C4#8903911yescharacteristic
393C3—24no—
394C3#1059095714yescomplemented
395C3#1059250211yescharacteristic, complemented
396C2—362no—
397C2—542no—
398C2#1172332311yescharacteristic
399C2#1172334611yescharacteristic
400C2#1172353911yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89017C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#417
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. #11723503C2semidirect
4. C3×#63648C2semidirect
5. C32×#816C2semidirect
6. #11722037C2semidirect
7. #11722033C2semidirect
17. #89017C4semidirect
33. #63648D3semidirect
36. C3×#816D3semidirect
47. C3×C4⋊(C3⋊D3)C2×C4semidirect
51. C32×Dic6C2×C4semidirect
116. #816C3⋊D3semidirect
137. C4⋊(C3⋊D3)C4×D3semidirect
152. C3×Dic6C4×D3semidirect
240. C33#417semidirect
277. Dic6C4×C3⋊D3semidirect
346. C32#4771semidirect
347. C32#4803semidirect
394. C3#10590957semidirect
395. C3#10592502semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
C2×#888502
#117235032
C3×#636482
C32×#8162
#117220372
#117220332
#105925023
#105909573