Finite Group Databasebeta

← #13159443

Subgroups — #13159443

Order 1728.

1252 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1252 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
2#11723933C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592503—31nomaximal
9#10590611—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6292—21no—
14C2×Dic3×C3⋊D3—21no—
15#89017—22no—
16#89017C414yescomplemented
17C33×C4⋊C4C2211yescharacteristic
18#88859—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C4×C3⋊(C3⋊Dic3)—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88785—22no—
24C2×#3832—31no—
25C2×#3796—34no—
26C2×C8×C3⋊D3—31no—
27#64101—31no—
28#64101—34no—
29#64101—64no—
30C2×#3710—34no—
31C32×#103—31no—
32#63662—31no—
33#63648—31no—
34#63636D311yescharacteristic, complemented
35C3×#810D314yescomplemented
36C3×#809—34no—
37#63566—34no—
38#63563—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4×C3⋊(C3⋊D3)C2×C412yes—
46C4×C3⋊(C3⋊D3)D412yes—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)C2×C412yes—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6C2×C412yes—
52#6292—24no—
53Dic3×C3⋊D3—24no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#4804—94no—
57#4772—91no—
58C2×C32×Q8—31no—
59#3832—34no—
60#3832—62no—
61C2×C4⋊(C3⋊D3)—61no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—34no—
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3—64no—
67C2×C3×D12—34no—
68C2×C3×Dic6C2×D314yes—
69C2×#626—64no—
70C2×D3×Dic3—64no—
71#3796—316no—
72#3796—68no—
73C2×C32×C8—31no—
74C32×C4⋊C4—24no—
75C32×C4⋊C4C2×D314yes—
76C32×C4⋊C4C2×D311yescharacteristic
77#3752—61no—
78#3752—64no—
79#3752—68no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—34no—
82C4⋊(C3⋊Dic3)—64no—
83C4×C3⋊Dic3—61no—
84C4×C3⋊Dic3—64no—
85C4×C3⋊Dic3—68no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C8×C3⋊D3—62no—
88C2×C3×C3⋊C8—34no—
89C2×C3×C3⋊C8C2×D314yes—
90#3710—68no—
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×C4—41no—
99C33×C4C22×C411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—41no—
102C3⋊(C3⋊Dic3)C22⋊C412yes—
103C32×Dic3—22no—
104C2×#239—94no—
105C2×#237—91no—
106C2×D3×C8—94no—
107C2×D3×C8—91no—
108#892—94no—
109#892—91no—
110#892—184no—
111C3×#103—34no—
112C3×#102—91no—
113#835—94no—
114#832—91no—
115#821—94no—
116#816—91no—
117#810C3⋊D311yescharacteristic, complemented
118#809—34no—
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—28no—
127C2×C32×C4—24no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—34no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—68no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—61no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)C4×D312yes—
142C4×C3⋊D3—32no—
143C4×C3⋊D3—32no—
144C4×C3⋊D3—38no—
145C4×C3⋊D3—38no—
146C4×C3⋊D3—68no—
147C4×C3⋊D3—68no—
148C4×C3⋊D3—64no—
149C2×C3×Dic3—24no—
150C3×D12—38no—
151C3×D12—64no—
152C3×Dic6—24no—
153C3×Dic6C4×D318yes—
154#626—616no—
155D3×Dic3—616no—
156C32×C8—61no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—24no—
159C3×C3⋊C8—64no—
160#416—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33#13511yescharacteristic
163C2×C33#13511yescharacteristic
164C3⋊(C3⋊D3)—22no—
165C3×C3⋊D3—42no—
166C2×C3×Q8—34no—
167C2×C3×D4—91no—
168C2×#67—181no—
169C22×Dic3—181no—
170#239—916no—
171#239—188no—
172#237—94no—
173#237—182no—
174C2×D12—34no—
175C2×D12—184no—
176C2×C4×D3—94no—
177C2×C4×D3—91no—
178C2×C4×D3—184no—
179C2×C4×D3—184no—
180C2×Dic6C2×C3⋊D311yescharacteristic
181C2×C3×C8—34no—
182C2×C3×C8—91no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4D3214yes—
185C3×C4⋊C4C2×C3⋊D311yescharacteristic
186(C2×D3)⋊C4—184no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—188no—
189C4⋊Dic3—94no—
190C4⋊Dic3—91no—
191C4⋊Dic3—184no—
192C4×Dic3—184no—
193C4×Dic3—181no—
194C4×Dic3—188no—
195C2×C3⋊C8—34no—
196C2×C3⋊C8C2×C3⋊D311yescharacteristic
197D3×C8—182no—
198D3×C8—188no—
199C22×C32—24no—
200C22×C32D3×D414yes—
201C22×C32D3×D411yescharacteristic
202C2×C3⋊D3—22no—
203C2×C3⋊D3—32no—
204C2×C3⋊D3—38no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3⋊D3—68no—
208C2×C3⋊D3—61no—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—64no—
211C2×C3⋊D3—64no—
212C2×C3⋊D3—124no—
213C2×C3⋊D3—124no—
214C2×C3×D3—68no—
215C2×C3×D3—124no—
216C2×C3×D3—124no—
217C32×C4—24no—
218C32×C4—24no—
219C32×C4—48no—
220C32×C4—41no—
221C32×C4—44no—
222C32×C4—62no—
223C32×C4C2×C4×D314yes—
224C32×C4C2×C4×D311yescharacteristic
225C32×C4D3×D414yes—
226C32×C4D3×D411yescharacteristic
227C3⋊Dic3—32no—
228C3⋊Dic3—38no—
229C3⋊Dic3—68no—
230C3⋊Dic3—128no—
231C3⋊Dic3—121no—
232C3⋊Dic3—124no—
233C3×Dic3—28no—
234C2×#41—271no—
235C22×C8—271no—
236#117—271no—
237#103—31no—
238#103—271no—
239#102—91no—
240#102—271no—
241C33#41611yescharacteristic, complemented, Sylow
242C23×C3—181no—
243C22×D3—64no—
244C22×D3—94no—
245C22×D3—91no—
246C22×D3—184no—
247C3×Q8—38no—
248C3×Q8—64no—
249C3×D4—92no—
250C3×D4—181no—
251C2×C3×C4—28no—
252C2×C3×C4—24no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—64no—
256C2×C3×C4C2×D3214yes—
257C2×C3×C4C22×C3⋊D311yescharacteristic
258#67—184no—
259C2×Dic3—21no—
260C2×Dic3—94no—
261C2×Dic3—91no—
262C2×Dic3—184no—
263C2×Dic3—184no—
264C2×Dic3—188no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267D12—38no—
268D12—64no—
269D12—1816no—
270C4×D3—98no—
271C4×D3—98no—
272C4×D3—92no—
273C4×D3—92no—
274C4×D3—188no—
275C4×D3—188no—
276C4×D3—1816no—
277Dic6—21no—
278Dic6C4×C3⋊D312yes—
279C3×C8—64no—
280C3×C8—181no—
281C3⋊C8—21no—
282C3⋊C8—64no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32D3×C22⋊C414yes—
287C2×C32D3×C22⋊C411yescharacteristic
288C2×C32#91211yescharacteristic
289C2×C32#91614yes—
290C2×C32#91611yescharacteristic
291C2×C32#91914yes—
292C3⋊D3—42no—
293C3⋊D3—62no—
294C3⋊D3—68no—
295C3⋊D3—128no—
296C3×D3—128no—
297#41—274no—
298#41—542no—
299C2×Q8—31no—
300C2×D4—91no—
301C2×D4—541no—
302C22×C4—271no—
303C22×C4—541no—
304C2×C8—31no—
305C2×C8—91no—
306C2×C8—542no—
307C4⋊C4—271no—
308C4⋊C4D3×C3⋊D311yescharacteristic
309C22⋊C4—541no—
310C42—541no—
311C22×C3—24no—
312C22×C3—182no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3#381114yes—
316C22×C3D4×C3⋊D311yescharacteristic
317C2×D3—68no—
318C2×D3—92no—
319C2×D3—98no—
320C2×D3—124no—
321C2×D3—124no—
322C2×D3—188no—
323C2×D3—181no—
324C2×D3—181no—
325C2×D3—184no—
326C2×D3—184no—
327C2×D3—364no—
328C2×D3—364no—
329C3×C4—24no—
330C3×C4—24no—
331C3×C4—48no—
332C3×C4—41no—
333C3×C4—44no—
334C3×C4—68no—
335C3×C4C4×D3214yes—
336C3×C4#380214yes—
337C3×C4C2×C4×C3⋊D311yescharacteristic
338C3×C4D4×C3⋊D311yescharacteristic
339Dic3—22no—
340Dic3—92no—
341Dic3—98no—
342Dic3—188no—
343Dic3—368no—
344Dic3—361no—
345Dic3—364no—
346C32—24no—
347C32#477211yescharacteristic, complemented
348C32#480414yescomplemented
349C23—181no—
350C23—271no—
351Q8—32no—
352Q8—61no—
353D4—92no—
354D4—181no—
355D4—544no—
356C2×C4—21no—
357C2×C4—61no—
358C2×C4—271no—
359C2×C4—272no—
360C2×C4—272no—
361C2×C4—544no—
362C2×C4—541no—
363C2×C4C2×D3×C3⋊D311yescharacteristic
364C8—61no—
365C8—181no—
366C2×C3—24no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—362no—
370C2×C3#6392114yes—
371C2×C3#6393614yes—
372C2×C3#6395014yes—
373C2×C3C22⋊C4×C3⋊D311yescharacteristic
374C2×C3#6412511yescharacteristic
375C2×C3#6412811yescharacteristic
376D3—128no—
377D3—188no—
378D3—182no—
379D3—368no—
380C22—182no—
381C22—272no—
382C22—361no—
383C22—361no—
384C22—541no—
385C22—541no—
386C22#8903911yescharacteristic
387C4—41no—
388C4—62no—
389C4—272no—
390C4—1081no—
391C4C4×D3×C3⋊D311yescharacteristic
392C4#8902611yescharacteristic
393C3—24no—
394C3#1059061114yescomplemented
395C3#1059250311yescharacteristic, complemented
396C2—362no—
397C2—542no—
398C2#1172353911yescharacteristic
399C2#1172356511yescharacteristic
400C2#1172358911yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#89017C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723933C2semidirect
3. C2×#88785C2semidirect
4. C3×#63636C2semidirect
5. C32×#810C2semidirect
6. #11722038C2semidirect
7. #11722034C2semidirect
16. #89017C4semidirect
34. #63636D3semidirect
35. C3×#810D3semidirect
117. #810C3⋊D3semidirect
241. C33#416semidirect
347. C32#4772semidirect
348. C32#4804semidirect
394. C3#10590611semidirect
395. C3#10592503semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
#117239332
C2×#887852
C3×#636362
C32×#8102
#117220382
#117220342
#105925033
#105906113