Finite Group Databasebeta

← #13160079

Subgroups — #13160079

Order 1728.

1220 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1220 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×#89020C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592426—31nomaximal
9#10590959—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6293C411yescharacteristic
14C2×#6292C411yescharacteristic
15C2×#6291C411yescharacteristic
16C2×#6290C411yescharacteristic
17#89020—22no—
18#89020—22no—
19C2×C33×C8C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21#88850—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C32×C3⋊C8C2211yescharacteristic
24#88785—22no—
25C2×#3828—31no—
26C2×#3798—34no—
27C2×#3743—31no—
28C2×#3743—34no—
29C2×#3743—64no—
30C2×C8×C3⋊D3—31no—
31C2×#3710—34no—
32C32×C22⋊C8—31no—
33#63657—31no—
34#63647—31no—
35#63647D311yescharacteristic, complemented
36C3×(C2×D3)⋊C8—34no—
37C3×(C2×D3)⋊C8D314yescomplemented
38(C2×D3)⋊(C3⋊C8)—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
41C22×C3×C3⋊D3C2×C411yescharacteristic
42C22×C32×D3C2×C411yescharacteristic
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
45C4×C3⋊(C3⋊D3)D412yes—
46C4×C3⋊(C3⋊D3)D412yes—
47C2×C3×C3⋊Dic3C2×C411yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C2×C32×Dic3C2×C411yescharacteristic
51C32×C4×D3—21no—
52C32×C4×D3—21no—
53#6293—22no—
54#6292—22no—
55#6291—22no—
56#6290—22no—
57C33×C8—21no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#4727—94no—
62#4727—91no—
63C22×C32×C4—31no—
64C2×#638—31no—
65C2×#638—31no—
66#3828—62no—
67#3828—62no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—34no—
71C2×C4×C3⋊D3—64no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634—31no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3C2×D314yes—
76C2×#627—34no—
77C2×#626—34no—
78C2×#626—34no—
79C2×#625—34no—
80#3798—68no—
81#3798—68no—
82C2×C32×C8—24no—
83C2×C32×C8—31no—
84C2×C32×C8C2×D314yes—
85C2×C32×C8C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90#3743—62no—
91#3743—68no—
92#3743—616no—
93C8×C3⋊D3—62no—
94C2×C3×C3⋊C8—34no—
95C2×C3×C3⋊C8C2×D314yes—
96#3710—68no—
97C22×C33C22×C411yescharacteristic
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)C22⋊C412yes—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—21no—
104C2×C32×D3—21no—
105C2×C32×D3—21no—
106C2×C32×D3—21no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3⋊(C3⋊Dic3)C22⋊C412yes—
110C3×C3⋊Dic3—21no—
111C32×Dic3—21no—
112C2×#235—94no—
113C2×#235—91no—
114C2×ΓL(1,𝔽25)—94no—
115C2×ΓL(1,𝔽25)—91no—
116C2×ΓL(1,𝔽25)—184no—
117C2×D3×C8—94no—
118C2×D3×C8—91no—
119C3×C22⋊C8—34no—
120C3×C22⋊C8—91no—
121#830—94no—
122#830—91no—
123(C2×D3)⋊C8—34no—
124(C2×D3)⋊C8—94no—
125(C2×D3)⋊C8—91no—
126(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
127C23×C32—31no—
128C22×C3⋊D3—31no—
129C22×C3⋊D3—34no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3C4×D311yescharacteristic
132C22×C3×D3—34no—
133C22×C3×D3C4×D314yes—
134C2×C32×C4—24no—
135C2×C32×C4—31no—
136C2×C32×C4—61no—
137C2×C32×C4—61no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140#638—62no—
141#638—62no—
142C2×C3⋊Dic3—31no—
143C2×C3⋊Dic3—34no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3C4×D311yescharacteristic
146C4⋊(C3⋊D3)—62no—
147C4×C3⋊D3—21no—
148C4×C3⋊D3—21no—
149C4×C3⋊D3—32no—
150C4×C3⋊D3—32no—
151C4×C3⋊D3—38no—
152C4×C3⋊D3—38no—
153C4×C3⋊D3—68no—
154C4×C3⋊D3—68no—
155#634—62no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3C4×D314yes—
158C3×C4×D3—24no—
159C3×C4×D3—24no—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162#627—68no—
163#626—68no—
164#626—68no—
165#625—68no—
166C32×C8—28no—
167C32×C8—24no—
168C32×C8—21no—
169C32×C8—61no—
170C3⋊(C3⋊C8)—21no—
171C3⋊(C3⋊C8)—68no—
172C3⋊(C3⋊C8)—64no—
173C3⋊(C3⋊C8)—61no—
174C3×C3⋊C8—24no—
175C3×C3⋊C8—64no—
176#408—271noSylow
177C2×C33C2×C22⋊C411yescharacteristic
178C2×C33#13111yescharacteristic
179C2×C33#13111yescharacteristic
180C3⋊(C3⋊D3)—22no—
181C3×C3⋊D3—41no—
182C32×D3—41no—
183C22×C3×C4—34no—
184C22×C3×C4—91no—
185C2×#67—94no—
186C2×#67—94no—
187C2×#67—91no—
188C2×#67—91no—
189#235—182no—
190#235—182no—
191#235—188no—
192#235—188no—
193C2×D12—94no—
194C2×D12—91no—
195C2×C4×D3—34no—
196C2×C4×D3—94no—
197C2×C4×D3—91no—
198C2×C4×D3—184no—
199C2×C4×D3C2×C3⋊D311yescharacteristic
200C2×Dic6—94no—
201C2×Dic6—91no—
202C2×C3×C8—24no—
203C2×C3×C8—34no—
204C2×C3×C8—91no—
205C2×C3×C8D3214yes—
206C2×C3×C8C2×C3⋊D311yescharacteristic
207C2×C3⋊C8—34no—
208C2×C3⋊C8—94no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—184no—
211C2×C3⋊C8C2×C3⋊D311yescharacteristic
212ΓL(1,𝔽25)—188no—
213ΓL(1,𝔽25)—182no—
214ΓL(1,𝔽25)—1816no—
215D3×C8—182no—
216D3×C8—188no—
217C22×C32—24no—
218C22×C32—61no—
219C22×C32—61no—
220C22×C32—61no—
221C22×C32C2×C4×D314yes—
222C22×C32C2×C4×D311yescharacteristic
223C2×C3⋊D3—21no—
224C2×C3⋊D3—21no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3—32no—
227C2×C3⋊D3—38no—
228C2×C3⋊D3—68no—
229C2×C3⋊D3—61no—
230C2×C3⋊D3—61no—
231C2×C3⋊D3—64no—
232C2×C3⋊D3—64no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—124no—
235C2×C3×D3—24no—
236C2×C3×D3—24no—
237C2×C3×D3—24no—
238C2×C3×D3—64no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C32×C4—24no—
242C32×C4—24no—
243C32×C4—61no—
244C32×C4D3×D414yes—
245C32×C4D3×D414yes—
246C32×C4D3×D411yescharacteristic
247C32×C4D3×D411yescharacteristic
248C3⋊Dic3—21no—
249C3⋊Dic3—32no—
250C3⋊Dic3—38no—
251C3⋊Dic3—68no—
252C3×Dic3—24no—
253C3×Dic3—64no—
254C2×#41—271no—
255C2×M16—271no—
256C22×C8—271no—
257C22⋊C8—31no—
258C22⋊C8—91no—
259C22⋊C8—271no—
260C22⋊C8—271no—
261C33#40811yescharacteristic, complemented, Sylow
262C23×C3—34no—
263C23×C3—91no—
264C22×D3—34no—
265C22×D3—94no—
266C22×D3—91no—
267C22×D3—184no—
268C22×D3C4×C3⋊D311yescharacteristic
269C2×C3×C4—24no—
270C2×C3×C4—34no—
271C2×C3×C4—64no—
272C2×C3×C4—64no—
273C2×C3×C4—91no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4C2×D3214yes—
277C2×C3×C4C22×C3⋊D311yescharacteristic
278#67—182no—
279#67—188no—
280#67—182no—
281#67—188no—
282C2×Dic3—34no—
283C2×Dic3—94no—
284C2×Dic3—91no—
285C2×Dic3—184no—
286C2×Dic3C4×C3⋊D311yescharacteristic
287D12—188no—
288D12—182no—
289C4×D3—21no—
290C4×D3—21no—
291C4×D3—64no—
292C4×D3—64no—
293C4×D3—98no—
294C4×D3—98no—
295C4×D3—92no—
296C4×D3—92no—
297C4×D3—188no—
298C4×D3—188no—
299Dic6—182no—
300Dic6—188no—
301C3×C8—28no—
302C3×C8—24no—
303C3×C8—21no—
304C3×C8—64no—
305C3×C8—181no—
306C3⋊C8—21no—
307C3⋊C8—64no—
308C3⋊C8—188no—
309C3⋊C8—184no—
310C3⋊C8—181no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—24no—
314C2×C32—121no—
315C2×C32D3×C22⋊C414yes—
316C2×C32D3×C22⋊C411yescharacteristic
317C2×C32#90114yes—
318C2×C32#90111yescharacteristic
319C2×C32#90714yes—
320C2×C32#90711yescharacteristic
321C3⋊D3—41no—
322C3⋊D3—62no—
323C3⋊D3—68no—
324C3⋊D3—128no—
325C3×D3—44no—
326C3×D3—124no—
327#41—542no—
328#41—542no—
329C2×Q8—271no—
330C2×D4—271no—
331C2×D4—271no—
332C2×D4—271no—
333C22×C4—31no—
334C22×C4—91no—
335C22×C4—271no—
336M16—542no—
337C2×C8—31no—
338C2×C8—91no—
339C2×C8—271no—
340C2×C8—542no—
341C2×C8D3×C3⋊D311yescharacteristic
342C22×C3—24no—
343C22×C3—64no—
344C22×C3—64no—
345C22×C3—64no—
346C22×C3—181no—
347C22×C3—181no—
348C22×C3—181no—
349C22×C3C4×D3214yes—
350C22×C3C2×C4×C3⋊D311yescharacteristic
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—64no—
357C2×D3—92no—
358C2×D3—98no—
359C2×D3—188no—
360C2×D3—184no—
361C2×D3—184no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—364no—
365C2×D3—364no—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—64no—
369C3×C4—181no—
370C3×C4#380214yes—
371C3×C4#381114yes—
372C3×C4D4×C3⋊D311yescharacteristic
373C3×C4D4×C3⋊D311yescharacteristic
374Dic3—21no—
375Dic3—64no—
376Dic3—92no—
377Dic3—98no—
378Dic3—188no—
379C32—24no—
380C32#472714yescomplemented
381C32#472711yescharacteristic, complemented
382C23—31no—
383C23—91no—
384C23—271no—
385Q8—542no—
386D4—542no—
387D4—542no—
388D4—542no—
389C2×C4—31no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—91no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—271no—
396C2×C4—272no—
397C2×C4—272no—
398C2×C4C2×D3×C3⋊D311yescharacteristic
399C8—21no—
400C8—61no—
401C8—181no—
402C8—541no—
403C2×C3—24no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—124no—
407C2×C3—361no—
408C2×C3#6380414yes—
409C2×C3#6380414yes—
410C2×C3#6392114yes—
411C2×C3C22⋊C4×C3⋊D311yescharacteristic
412C2×C3#6411011yescharacteristic
413C2×C3#6411611yescharacteristic
414D3—41no—
415D3—124no—
416D3—182no—
417D3—188no—
418D3—368no—
419C22—61no—
420C22—61no—
421C22—61no—
422C22—181no—
423C22—181no—
424C22—181no—
425C22—272no—
426C22—541no—
427C22—541no—
428C22C4×D3×C3⋊D311yescharacteristic
429C4—61no—
430C4—181no—
431C4—272no—
432C4#8902611yescharacteristic
433C4#8903911yescharacteristic
434C3—24no—
435C3#1059095914yescomplemented
436C3#1059242611yescharacteristic, complemented
437C2—121no—
438C2—361no—
439C2—542no—
440C2#1172333211yescharacteristic
441C2#1172333311yescharacteristic
442C2#1172353911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC2×#89020C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
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×#88850C2semidirect
3. C2×#88785C2semidirect
4. C3×#63647C2semidirect
5. C32×(C2×D3)⋊C8C2semidirect
6. #11722023C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
35. #63647D3semidirect
37. C3×(C2×D3)⋊C8D3semidirect
126. (C2×D3)⋊C8C3⋊D3semidirect
261. C33#408semidirect
380. C32#4727semidirect
381. C32#4727semidirect
435. C3#10590959semidirect
436. C3#10592426semidirect

Maximal subgroups

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

subgroupindex
C2×#890202
C2×#888502
C2×#887852
C3×#636472
C32×(C2×D3)⋊C82
#117220232
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105924263
#105909593