Finite Group Databasebeta

← #13159428

Subgroups — #13159428

Order 1728.

1204 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1204 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
2#11723933C211yesmaximal, characteristic
3C2×#88788C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592501—31nomaximal
9#10590602—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C2×#6289—21no—
15#89019—22no—
16#89019C414yescomplemented
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#88788—22no—
24C2×Q8×C3⋊D3—31no—
25C2×#3797—34no—
26C2×#3743—31no—
27#64101—31no—
28#64101—34no—
29#64101—64no—
30C2×#3712—34no—
31C32×#103—31no—
32#63662—31no—
33#63643—31no—
34#63637D311yescharacteristic, complemented
35C3×#810—34no—
36C3×#810D314yescomplemented
37#63566—34no—
38#63566—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C4×C3⋊(C3⋊D3)C2×C412yes—
45C4×C3⋊(C3⋊D3)D412yes—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634C2×C412yes—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6C2×C412yes—
52#6293—24no—
53#6289—24no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×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—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
87#3743—62no—
88C2×C3×C3⋊C8—34no—
89C2×C3×C3⋊C8C2×D314yes—
90#3712—68no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)—21no—
94C2×C3⋊(C3⋊D3)C22⋊C412yes—
95C33×C4—41no—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C3⋊(C3⋊Dic3)C22⋊C412yes—
100C3×C3⋊Dic3—22no—
101C32×Dic3—22no—
102C2×D3×Q8—94no—
103C2×D3×Q8—91no—
104C2×ΓL(1,𝔽25)—94no—
105C2×ΓL(1,𝔽25)—91no—
106#892—94no—
107#892—91no—
108#892—184no—
109C3×#103—34no—
110C3×#103—91no—
111#835—94no—
112#835—91no—
113#816—94no—
114#816—91no—
115#810—34no—
116#810C3⋊D311yescharacteristic, complemented
117C22×C3⋊D3—31no—
118C22×C3⋊D3—34no—
119C22×C3⋊D3—64no—
120C32×Q8—32no—
121C32×Q8—61no—
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—68no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—61no—
136C4×C3⋊D3—32no—
137C4×C3⋊D3—32no—
138C4×C3⋊D3—38no—
139C4×C3⋊D3—38no—
140C4×C3⋊D3—68no—
141C4×C3⋊D3—68no—
142C4×C3⋊D3—64no—
143#634—21no—
144#634—64no—
145#634C4×D312yes—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—64no—
148C3×Dic6—24no—
149C3×Dic6—38no—
150C3×Dic6—64no—
151C3×Dic6C4×D318yes—
152#627—616no—
153#624—616no—
154C32×C8—61no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—24no—
157C3×C3⋊C8—64no—
158#419—271noSylow
159C2×C33C2×C22⋊C411yescharacteristic
160C2×C33#13711yescharacteristic
161C2×C33#13711yescharacteristic
162C3⋊(C3⋊D3)—22no—
163C2×C3×Q8—34no—
164C2×C3×Q8—91no—
165D3×Q8—916no—
166D3×Q8—94no—
167D3×Q8—182no—
168D3×Q8—188no—
169C2×C4×D3—94no—
170C2×C4×D3—91no—
171C2×C4×D3—184no—
172C2×C4×D3—181no—
173C2×C4×D3—184no—
174C2×Dic6—34no—
175C2×Dic6—181no—
176C2×Dic6—184no—
177C2×Dic6C2×C3⋊D311yescharacteristic
178C2×C3×C8—34no—
179C2×C3×C8—91no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4D3214yes—
182C3×C4⋊C4C2×C3⋊D311yescharacteristic
183(C2×D3)⋊C4—184no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—188no—
186C4⋊Dic3—94no—
187C4⋊Dic3—91no—
188C4⋊Dic3—184no—
189C4×Dic3—184no—
190C4×Dic3—181no—
191C4×Dic3—188no—
192C2×C3⋊C8—34no—
193C2×C3⋊C8C2×C3⋊D311yescharacteristic
194ΓL(1,𝔽25)—182no—
195ΓL(1,𝔽25)—188no—
196C22×C32—24no—
197C22×C32D3×D414yes—
198C22×C32D3×D411yescharacteristic
199C2×C3⋊D3—32no—
200C2×C3⋊D3—38no—
201C2×C3⋊D3—68no—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—61no—
204C2×C3⋊D3—64no—
205C2×C3⋊D3—64no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—124no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—44no—
212C32×C4—41no—
213C32×C4—62no—
214C32×C4C2×C4×D314yes—
215C32×C4C2×C4×D311yescharacteristic
216C32×C4D3×D414yes—
217C32×C4D3×D411yescharacteristic
218C3⋊Dic3—22no—
219C3⋊Dic3—32no—
220C3⋊Dic3—38no—
221C3⋊Dic3—68no—
222C3⋊Dic3—128no—
223C3⋊Dic3—124no—
224C3⋊Dic3—121no—
225C3×Dic3—28no—
226C3×Dic3—68no—
227C22×Q8—271no—
228C2×M16—271no—
229#117—271no—
230#103—31no—
231#103—91no—
232#103—271no—
233#103—271no—
234C33#41911yescharacteristic, complemented, Sylow
235C22×D3—94no—
236C22×D3—91no—
237C22×D3—184no—
238C3×Q8—38no—
239C3×Q8—64no—
240C3×Q8—92no—
241C3×Q8—181no—
242C2×C3×C4—28no—
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—24no—
246C2×C3×C4—64no—
247C2×C3×C4—181no—
248C2×C3×C4C2×D3214yes—
249C2×C3×C4C22×C3⋊D311yescharacteristic
250C2×Dic3—21no—
251C2×Dic3—64no—
252C2×Dic3—94no—
253C2×Dic3—91no—
254C2×Dic3—184no—
255C2×Dic3—188no—
256C2×Dic3—184no—
257C2×Dic3—181no—
258C4×D3—98no—
259C4×D3—98no—
260C4×D3—92no—
261C4×D3—92no—
262C4×D3—188no—
263C4×D3—188no—
264C4×D3—184no—
265C4×D3—1816no—
266Dic6—21no—
267Dic6—38no—
268Dic6—64no—
269Dic6—184no—
270Dic6—1816no—
271Dic6C4×C3⋊D312yes—
272C3×C8—64no—
273C3×C8—181no—
274C3⋊C8—21no—
275C3⋊C8—64no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32D3×C22⋊C414yes—
280C2×C32D3×C22⋊C411yescharacteristic
281C2×C32#91514yes—
282C2×C32#91511yescharacteristic
283C2×C32#91814yes—
284C2×C32#91811yescharacteristic
285C3⋊D3—62no—
286C3⋊D3—68no—
287C3⋊D3—128no—
288C2×Q8—31no—
289C2×Q8—91no—
290C2×Q8—274no—
291C2×Q8—541no—
292C2×Q8—542no—
293C22×C4—271no—
294C22×C4—541no—
295M16—542no—
296C2×C8—31no—
297C2×C8—91no—
298C4⋊C4—271no—
299C4⋊C4D3×C3⋊D311yescharacteristic
300C22⋊C4—541no—
301C42—541no—
302C22×C3—24no—
303C22×C3#381114yes—
304C22×C3D4×C3⋊D311yescharacteristic
305C2×D3—92no—
306C2×D3—98no—
307C2×D3—188no—
308C2×D3—181no—
309C2×D3—181no—
310C2×D3—184no—
311C2×D3—184no—
312C2×D3—364no—
313C2×D3—364no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—48no—
317C3×C4—41no—
318C3×C4—44no—
319C3×C4—68no—
320C3×C4—182no—
321C3×C4C4×D3214yes—
322C3×C4#380214yes—
323C3×C4C2×C4×C3⋊D311yescharacteristic
324C3×C4D4×C3⋊D311yescharacteristic
325Dic3—22no—
326Dic3—68no—
327Dic3—92no—
328Dic3—98no—
329Dic3—188no—
330Dic3—368no—
331Dic3—361no—
332Dic3—364no—
333C32—24no—
334C32#480214yescomplemented
335C32#480211yescharacteristic, complemented
336C23—271no—
337Q8—32no—
338Q8—61no—
339Q8—92no—
340Q8—181no—
341Q8—544no—
342C2×C4—21no—
343C2×C4—61no—
344C2×C4—181no—
345C2×C4—271no—
346C2×C4—272no—
347C2×C4—272no—
348C2×C4—544no—
349C2×C4—541no—
350C2×C4C2×D3×C3⋊D311yescharacteristic
351C8—61no—
352C8—181no—
353C2×C3—24no—
354C2×C3—24no—
355C2×C3—24no—
356C2×C3#6392114yes—
357C2×C3#6393114yes—
358C2×C3#6394514yes—
359C2×C3C22⋊C4×C3⋊D311yescharacteristic
360C2×C3#6412411yescharacteristic
361C2×C3#6412711yescharacteristic
362D3—188no—
363D3—182no—
364D3—368no—
365C22—272no—
366C22—541no—
367C22—541no—
368C22#8903911yescharacteristic
369C4—41no—
370C4—62no—
371C4—182no—
372C4—272no—
373C4—1081no—
374C4C4×D3×C3⋊D311yescharacteristic
375C4#8902611yescharacteristic
376C3—24no—
377C3#1059060214yescomplemented
378C3#1059250111yescharacteristic, complemented
379C2—542no—
380C2#1172353911yescharacteristic
381C2#1172355411yescharacteristic
382C2#1172357811yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#419
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
4. C3×#63637C2semidirect
5. C32×#810C2semidirect
6. #11722040C2semidirect
7. #11722038C2semidirect
16. #89019C4semidirect
34. #63637D3semidirect
36. C3×#810D3semidirect
116. #810C3⋊D3semidirect
234. C33#419semidirect
334. C32#4802semidirect
335. C32#4802semidirect
377. C3#10590602semidirect
378. C3#10592501semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
#117239332
C2×#887882
C3×#636372
C32×#8102
#117220402
#117220382
#105925013
#105906023