Finite Group Databasebeta

← #13161751

Subgroups — #13161751

Order 1728.

822 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 127 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×#89042C211yesmaximal, characteristic
2C2×C3×#3743C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4C3×#63647C212yesmaximal, complemented
5#11722072C212yesmaximal, complemented
6#10590959—31nomaximal
7#10590955—32nomaximal
8C2×C3×C4×C3⋊D3C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×#6296C411yescharacteristic
11C2×#6295C412yes—
12C2×#6295C411yescharacteristic
13#89042—22no—
14#89042—22no—
15C2×C33×C8C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18C3×#3743—22no—
19#88808—22no—
20C2×#3798—32no—
21C2×#3798—31no—
22C2×#3743D311yescharacteristic, complemented
23C2×C3×ΓL(1,𝔽25)—32no—
24C2×C3×ΓL(1,𝔽25)—61no—
25C2×#3710—31no—
26C2×D3×C3⋊C8—32no—
27#63647D312yescomplemented
28C3×(C2×D3)⋊C8—32no—
29C3×(C2×D3)⋊C8—32no—
30C3×(C2×D3)⋊C8—62no—
31#63557—32no—
32(C2×D3)⋊(C3⋊C8)—32no—
33(C2×D3)⋊(C3⋊C8)—32no—
34C22×C3×C3⋊D3C2×C412yes—
35C22×C3×C3⋊D3C2×C411yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3C2×C412yes—
38C2×C3×C3⋊Dic3C2×C411yescharacteristic
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3—22no—
41C3×C4×C3⋊D3D412yes—
42C3×C4×C3⋊D3D412yes—
43#6296—22no—
44#6295—24no—
45#6295—22no—
46C33×C8—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C3×C3⋊(C3⋊C8)—22no—
49#5129—91no—
50#4727—92no—
51C2×C4×C3⋊D3C2×D312yes—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—32no—
54C2×C3×C4×D3—32no—
55C2×C3×C4×D3—32no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—62no—
58C2×#627—32no—
59C2×#627—31no—
60C2×#626—32no—
61C2×#626—32no—
62C2×#626—32no—
63C2×#625—32no—
64C2×#625—31no—
65#3798—64no—
66#3798—64no—
67#3798—62no—
68#3798—62no—
69C2×C32×C8—21no—
70C2×C32×C8—22no—
71C2×C32×C8—41no—
72C2×C32×C8C2×D312yes—
73C2×C32×C8C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D312yes—
76#3743—22no—
77C2×C3×C3⋊C8—32no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—61no—
81C2×C3×C3⋊C8—62no—
82C3×ΓL(1,𝔽25)—64no—
83C3×ΓL(1,𝔽25)—64no—
84#3710—62no—
85D3×C3⋊C8—64no—
86C22×C33C22×C411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3C22⋊C412yes—
92C33×C4C2×D411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3C22⋊C412yes—
96C2×#235—91no—
97C2×#235—92no—
98C2×C3×M16—91no—
99C22×C3⋊C8—91no—
100C2×ΓL(1,𝔽25)—32no—
101C2×ΓL(1,𝔽25)—61no—
102C2×D3×C8—92no—
103C3×C22⋊C8—92no—
104#830—92no—
105(C2×D3)⋊C8—32no—
106(C2×D3)⋊C8—32no—
107(C2×D3)⋊C8—62no—
108(C2×D3)⋊C8—92no—
109(C2×D3)⋊C8—92no—
110C22×C3⋊D3C4×D312yes—
111C22×C3⋊D3C4×D311yescharacteristic
112C22×C3×D3—32no—
113C22×C3×D3—32no—
114C22×C3×D3—32no—
115C22×C3×D3—61no—
116C22×C3×D3—62no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D312yes—
122C2×C3⋊Dic3C4×D312yes—
123C2×C3⋊Dic3C4×D311yescharacteristic
124C4×C3⋊D3—22no—
125C4×C3⋊D3—22no—
126C4×C3⋊D3D1212yes—
127C4×C3⋊D3#6712yes—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C3×C4×D3—34no—
134C3×C4×D3—34no—
135C3×C4×D3—62no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—122no—
142C3×C4×D3—122no—
143#627—64no—
144#627—62no—
145#626—64no—
146#626—64no—
147#626—64no—
148#625—64no—
149#625—62no—
150C32×C8—21no—
151C32×C8—22no—
152C32×C8—22no—
153C32×C8—42no—
154C32×C8—42no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)—22no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—62no—
159C3×C3⋊C8—62no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—122no—
162#408—271noSylow
163C2×C33C2×C22⋊C411yescharacteristic
164C2×C33#13112yes—
165C3×C3⋊D3—22no—
166C3×C3⋊D3—42no—
167C22×C3×C4—91no—
168C22×C3×C4—92no—
169C2×#67—92no—
170C2×#67—92no—
171C2×#67—92no—
172#235—184no—
173#235—184no—
174#235—182no—
175#235—182no—
176C2×D12—91no—
177C2×D12—92no—
178C2×C4×D3—32no—
179C2×C4×D3—32no—
180C2×C4×D3—32no—
181C2×C4×D3—61no—
182C2×C4×D3—62no—
183C2×Dic6—92no—
184C2×Dic6—91no—
185C3×M16—182no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8—41no—
189C2×C3×C8—91no—
190C2×C3×C8—92no—
191C2×C3×C8D3211yescharacteristic
192C2×C3×C8D3212yes—
193C2×C3⋊C8—32no—
194C2×C3⋊C8—32no—
195C2×C3⋊C8—32no—
196C2×C3⋊C8—61no—
197C2×C3⋊C8—62no—
198C2×C3⋊C8—182no—
199ΓL(1,𝔽25)—64no—
200ΓL(1,𝔽25)—64no—
201D3×C8—184no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32—41no—
205C22×C32C2×C4×D312yes—
206C22×C32C2×C4×D311yescharacteristic
207C2×C3⋊D3—22no—
208C2×C3⋊D3—22no—
209C2×C3⋊D3—22no—
210C2×C3⋊D3—22no—
211C2×C3⋊D3(C2×D3)⋊C412yes—
212C2×C3×D3—34no—
213C2×C3×D3—62no—
214C2×C3×D3—62no—
215C2×C3×D3—62no—
216C2×C3×D3—64no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C2×C3×D3—62no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—122no—
224C2×C3×D3—122no—
225C2×C3×D3—122no—
226C32×C4—22no—
227C32×C4—21no—
228C32×C4—22no—
229C32×C4—21no—
230C32×C4—41no—
231C32×C4—41no—
232C32×C4C2×D1211yescharacteristic
233C32×C4D3×D412yes—
234C32×C4D3×D412yes—
235C32×C4C2×#6711yescharacteristic
236C3⋊Dic3—22no—
237C3⋊Dic3(C2×D3)⋊C412yes—
238C3×Dic3—34no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—62no—
242C3×Dic3—122no—
243C2×#41—271no—
244C2×M16—91no—
245C22×C8—271no—
246C22⋊C8—92no—
247C22⋊C8—272no—
248C33#40811yescharacteristic, complemented, Sylow
249C23×C3—91no—
250C23×C3—92no—
251C22×D3—32no—
252C22×D3—32no—
253C22×D3—32no—
254C22×D3—61no—
255C22×D3—62no—
256C2×C3×C4—22no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4—91no—
260C2×C3×C4—92no—
261C2×C3×C4—92no—
262C2×C3×C4—92no—
263C2×C3×C4—182no—
264C2×C3×C4—182no—
265C2×C3×C4C2×D3212yes—
266C2×C3×C4C2×D3211yescharacteristic
267#67—184no—
268#67—184no—
269#67—184no—
270C2×Dic3—32no—
271C2×Dic3—32no—
272C2×Dic3—32no—
273C2×Dic3—61no—
274C2×Dic3—62no—
275D12—182no—
276D12—184no—
277C4×D3—34no—
278C4×D3—34no—
279C4×D3—62no—
280C4×D3—62no—
281C4×D3—62no—
282C4×D3—62no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—122no—
286C4×D3—122no—
287Dic6—184no—
288Dic6—182no—
289C3×C8—22no—
290C3×C8—22no—
291C3×C8—21no—
292C3×C8—42no—
293C3×C8—42no—
294C3×C8—182no—
295C3×C8—181no—
296C3⋊C8—62no—
297C3⋊C8—62no—
298C3⋊C8—62no—
299C3⋊C8—62no—
300C3⋊C8—122no—
301C2×C32—22no—
302C2×C32—22no—
303C2×C32—22no—
304C2×C32—22no—
305C2×C32—21no—
306C2×C32—41no—
307C2×C32—42no—
308C2×C32D3×C22⋊C412yes—
309C2×C32#90112yes—
310C2×C32#90712yes—
311C2×C32#90712yes—
312C2×C32C2×(C2×D3)⋊C411yescharacteristic
313C3⋊D3—22no—
314C3⋊D3—42no—
315C3×D3—64no—
316C3×D3—122no—
317C3×D3—124no—
318C3×D3—122no—
319C3×D3—122no—
320#41—542no—
321#41—542no—
322C2×Q8—271no—
323C2×D4—271no—
324C2×D4—272no—
325C22×C4—91no—
326C22×C4—92no—
327M16—182no—
328C2×C8—92no—
329C2×C8—91no—
330C2×C8—542no—
331C2×C8#111311yescharacteristic
332C22×C3—22no—
333C22×C3—21no—
334C22×C3—41no—
335C22×C3—92no—
336C22×C3—182no—
337C22×C3—182no—
338C22×C3—182no—
339C22×C3—182no—
340C22×C3C4×D3211yescharacteristic
341C22×C3C4×D3212yes—
342C2×D3—34no—
343C2×D3—62no—
344C2×D3—62no—
345C2×D3—62no—
346C2×D3—64no—
347C2×D3—62no—
348C2×D3—62no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—62no—
353C2×D3—122no—
354C2×D3—122no—
355C2×D3—122no—
356C3×C4—22no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—22no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—92no—
363C3×C4—182no—
364C3×C4D3×D1212yes—
365C3×C4#380211yescharacteristic
366C3×C4D3×#6712yes—
367C3×C4#381111yescharacteristic
368Dic3—34no—
369Dic3—62no—
370Dic3—62no—
371Dic3—62no—
372Dic3—122no—
373C32—22no—
374C32—21no—
375C32—41no—
376C32#472712yescomplemented
377C32#512911yescharacteristic, complemented
378C23—91no—
379C23—92no—
380Q8—542no—
381D4—542no—
382D4—544no—
383C2×C4—91no—
384C2×C4—92no—
385C2×C4—92no—
386C2×C4—92no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4C2×#111311yescharacteristic
390C8—21no—
391C8—182no—
392C8—181no—
393C2×C3—22no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—22no—
398C2×C3—42no—
399C2×C3—41no—
400C2×C3—182no—
401C2×C3—362no—
402C2×C3#6380412yes—
403C2×C3#6380412yes—
404C2×C3#6380512yes—
405C2×C3D3×(C2×D3)⋊C412yes—
406C2×C3#6392111yescharacteristic
407D3—64no—
408D3—122no—
409D3—124no—
410D3—122no—
411D3—122no—
412C22—92no—
413C22—182no—
414C22—182no—
415C22—182no—
416C22—182no—
417C22C4×#111311yescharacteristic
418C4—92no—
419C4—182no—
420C4C4⋊#111311yescharacteristic
421C4#8904911yescharacteristic
422C3—21no—
423C3—22no—
424C3—41no—
425C3#1059095512yescomplemented
426C3#1059095911yescharacteristic, complemented
427C2—182no—
428C2—362no—
429C2#1172365212yes—
430C2#1172370011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC2×#89042C2
cosocleC2×C33×C4C23
hypercenterC2×C8#1113
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3743C2semidirect
3. C2×#88808C2semidirect
4. C3×#63647C2semidirect
5. #11722072C2semidirect
22. C2×#3743D3semidirect
27. #63647D3semidirect
248. C33#408semidirect
376. C32#4727semidirect
377. C32#5129semidirect
425. C3#10590955semidirect
426. C3#10590959semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
C2×C3×#37432
C2×#888082
C3×#636472
#117220722
#105909593
#105909553