Finite Group Databasebeta

← #13160053

Subgroups — #13160053

Order 1728.

1172 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1172 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×#89019C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic
3#11723413C211yesmaximal, characteristic
4C3×#63643C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592501—31nomaximal
9#10590691—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C2×#6289—21no—
15#89019—22no—
16#89019C414yescomplemented
17C2×C33×C8C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19#88850—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22#88804—21no—
23Dic3×C3⋊Dic3—21no—
24C2×Q8×C3⋊D3—31no—
25C2×#3797—34no—
26C2×#3743—31no—
27C2×#3743—34no—
28C2×#3743—64no—
29#64101—31no—
30#63849—34no—
31C32×#103—31no—
32#63662—31no—
33#63643D311yescharacteristic, complemented
34#63637—31no—
35C3×#816—34no—
36C3×#816D314yescomplemented
37#63565—34no—
38#63565—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4×C3⋊(C3⋊D3)C2×C412yes—
43C4×C3⋊(C3⋊D3)D412yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634C2×C412yes—
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6C2×C412yes—
52#6293—24no—
53#6289—24no—
54C33×C8—21no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56#4802—94no—
57#4802—91no—
58C2×C32×Q8—31no—
59Q8×C3⋊D3—34no—
60Q8×C3⋊D3—62no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—61no—
64C2×C4×C3⋊D3—64no—
65C2×#634—61no—
66C2×#634C2×D311yescharacteristic
67C2×C3×Dic6—34no—
68C2×C3×Dic6C2×D314yes—
69C2×#627—64no—
70C2×#624—64no—
71#3797—316no—
72#3797—68no—
73C2×C32×C8—24no—
74C2×C32×C8C2×D314yes—
75C2×C32×C8C2×D311yescharacteristic
76C32×C4⋊C4—31no—
77#3752—61no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4×C3⋊Dic3—61no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83#3743—62no—
84#3743—68no—
85#3743—616no—
86C3×C4⋊Dic3—34no—
87C3×C4⋊Dic3C2×D314yes—
88#3722—64no—
89Dic32—64no—
90C22×C33C2×D411yescharacteristic
91C2×C3⋊(C3⋊D3)—21no—
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)C22⋊C412yes—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)C22⋊C412yes—
97C3×C3⋊Dic3—22no—
98C3×C3⋊Dic3—41no—
99C32×Dic3—22no—
100C32×Dic3—41no—
101C2×D3×Q8—94no—
102C2×D3×Q8—91no—
103C2×ΓL(1,𝔽25)—94no—
104C2×ΓL(1,𝔽25)—91no—
105C2×ΓL(1,𝔽25)—184no—
106#892—94no—
107#892—91no—
108C3×#103—34no—
109C3×#103—91no—
110#835—94no—
111#835—91no—
112#816—34no—
113#816C3⋊D311yescharacteristic, complemented
114#810—94no—
115#810—91no—
116C22×C3⋊D3—31no—
117C22×C3⋊D3—34no—
118C22×C3⋊D3—64no—
119C32×Q8—32no—
120C32×Q8—61no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4—61no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—34no—
130C2×C3⋊Dic3—64no—
131C4×C3⋊D3—32no—
132C4×C3⋊D3—32no—
133C4×C3⋊D3—38no—
134C4×C3⋊D3—38no—
135C4×C3⋊D3—68no—
136C4×C3⋊D3—68no—
137C4×C3⋊D3—64no—
138#634—21no—
139#634—64no—
140#634C4×D312yes—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—24no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C3×Dic6—24no—
146C3×Dic6—38no—
147C3×Dic6—64no—
148C3×Dic6C4×D318yes—
149#627—616no—
150#624—616no—
151C32×C8—28no—
152C32×C8—24no—
153C32×C8—21no—
154C3⋊(C3⋊C8)—68no—
155C3⋊(C3⋊C8)—64no—
156C3⋊(C3⋊C8)—61no—
157#419—271noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13711yescharacteristic
160C2×C33#13711yescharacteristic
161C3⋊(C3⋊D3)—22no—
162C2×C3×Q8—34no—
163C2×C3×Q8—91no—
164D3×Q8—916no—
165D3×Q8—94no—
166D3×Q8—182no—
167D3×Q8—188no—
168C2×C4×D3—94no—
169C2×C4×D3—91no—
170C2×C4×D3—184no—
171C2×C4×D3—181no—
172C2×C4×D3—184no—
173C2×Dic6—34no—
174C2×Dic6—181no—
175C2×Dic6—184no—
176C2×Dic6C2×C3⋊D311yescharacteristic
177C2×C3×C8—24no—
178C2×C3×C8D3214yes—
179C2×C3×C8C2×C3⋊D311yescharacteristic
180C3×C4⋊C4—34no—
181C3×C4⋊C4—91no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—184no—
184C4⋊Dic3—34no—
185C4⋊Dic3C2×C3⋊D311yescharacteristic
186C4×Dic3—181no—
187C4×Dic3—184no—
188C2×C3⋊C8—94no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—184no—
191ΓL(1,𝔽25)—188no—
192ΓL(1,𝔽25)—182no—
193ΓL(1,𝔽25)—1816no—
194C22×C32—24no—
195C22×C32D3×D414yes—
196C22×C32D3×D411yescharacteristic
197C2×C3⋊D3—32no—
198C2×C3⋊D3—38no—
199C2×C3⋊D3—68no—
200C2×C3⋊D3—61no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—64no—
203C2×C3⋊D3—64no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4—62no—
209C32×C4—121no—
210C32×C4C2×C4×D314yes—
211C32×C4C2×C4×D311yescharacteristic
212C32×C4D3×D414yes—
213C32×C4D3×D411yescharacteristic
214C3⋊Dic3—22no—
215C3⋊Dic3—32no—
216C3⋊Dic3—38no—
217C3⋊Dic3—41no—
218C3⋊Dic3—68no—
219C3×Dic3—28no—
220C3×Dic3—44no—
221C3×Dic3—68no—
222C3×Dic3—124no—
223C22×Q8—271no—
224C2×M16—271no—
225#117—271no—
226#103—31no—
227#103—91no—
228#103—271no—
229#103—271no—
230C33#41911yescharacteristic, complemented, Sylow
231C22×D3—94no—
232C22×D3—91no—
233C22×D3—184no—
234C3×Q8—38no—
235C3×Q8—64no—
236C3×Q8—92no—
237C3×Q8—181no—
238C2×C3×C4—24no—
239C2×C3×C4—64no—
240C2×C3×C4—64no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—64no—
248C2×Dic3—64no—
249C2×Dic3—94no—
250C2×Dic3—91no—
251C2×Dic3—184no—
252C4×D3—98no—
253C4×D3—98no—
254C4×D3—92no—
255C4×D3—92no—
256C4×D3—188no—
257C4×D3—188no—
258C4×D3—184no—
259C4×D3—1816no—
260Dic6—21no—
261Dic6—38no—
262Dic6—64no—
263Dic6—184no—
264Dic6—1816no—
265Dic6C4×C3⋊D312yes—
266C3×C8—28no—
267C3×C8—24no—
268C3×C8—21no—
269C3⋊C8—188no—
270C3⋊C8—184no—
271C3⋊C8—181no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32D3×C22⋊C414yes—
276C2×C32D3×C22⋊C411yescharacteristic
277C2×C32#91514yes—
278C2×C32#91511yescharacteristic
279C2×C32#91814yes—
280C2×C32#91811yescharacteristic
281C3⋊D3—62no—
282C3⋊D3—68no—
283C3⋊D3—128no—
284C2×Q8—31no—
285C2×Q8—91no—
286C2×Q8—274no—
287C2×Q8—541no—
288C2×Q8—542no—
289C22×C4—271no—
290C22×C4—541no—
291M16—542no—
292C2×C8—271no—
293C2×C8D3×C3⋊D311yescharacteristic
294C4⋊C4—31no—
295C4⋊C4—91no—
296C22⋊C4—541no—
297C42—541no—
298C22×C3—24no—
299C22×C3#380214yes—
300C22×C3D4×C3⋊D311yescharacteristic
301C2×D3—92no—
302C2×D3—98no—
303C2×D3—188no—
304C2×D3—184no—
305C2×D3—184no—
306C2×D3—181no—
307C2×D3—181no—
308C2×D3—364no—
309C2×D3—364no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—68no—
313C3×C4—124no—
314C3×C4—182no—
315C3×C4—361no—
316C3×C4C4×D3214yes—
317C3×C4#381114yes—
318C3×C4C2×C4×C3⋊D311yescharacteristic
319C3×C4D4×C3⋊D311yescharacteristic
320Dic3—22no—
321Dic3—41no—
322Dic3—68no—
323Dic3—92no—
324Dic3—98no—
325Dic3—124no—
326Dic3—188no—
327C32—24no—
328C32#480214yescomplemented
329C32#480211yescharacteristic, complemented
330C23—271no—
331Q8—32no—
332Q8—61no—
333Q8—92no—
334Q8—181no—
335Q8—544no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—271no—
341C2×C4—272no—
342C2×C4—272no—
343C2×C4—544no—
344C2×C4C2×D3×C3⋊D311yescharacteristic
345C8—21no—
346C8—541no—
347C2×C3—24no—
348C2×C3—24no—
349C2×C3—24no—
350C2×C3#6381114yes—
351C2×C3#6381114yes—
352C2×C3#6392114yes—
353C2×C3C22⋊C4×C3⋊D311yescharacteristic
354C2×C3#6412411yescharacteristic
355C2×C3#6412711yescharacteristic
356D3—188no—
357D3—182no—
358D3—368no—
359C22—272no—
360C22—541no—
361C22—541no—
362C22#8902611yescharacteristic
363C4—62no—
364C4—121no—
365C4—182no—
366C4—272no—
367C4—361no—
368C4C4×D3×C3⋊D311yescharacteristic
369C4#8903911yescharacteristic
370C3—24no—
371C3#1059069114yescomplemented
372C3#1059250111yescharacteristic, complemented
373C2—542no—
374C2#1172334411yescharacteristic
375C2#1172334611yescharacteristic
376C2#1172353911yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#419
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
4. C3×#63643C2semidirect
5. C32×#816C2semidirect
6. #11722039C2semidirect
7. #11722037C2semidirect
16. #89019C4semidirect
33. #63643D3semidirect
36. C3×#816D3semidirect
113. #816C3⋊D3semidirect
230. C33#419semidirect
328. C32#4802semidirect
329. C32#4802semidirect
371. C3#10590691semidirect
372. C3#10592501semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
C2×#888502
#117234132
C3×#636432
C32×#8162
#117220392
#117220372
#105925013
#105906913