Finite Group Databasebeta

← #72246919

Subgroups — #72246919

Order 2016.

574 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C22×D3×D21C211yesmaximal, characteristic, complemented
2C2×#92694C211yesmaximal, characteristic, complemented
3C2×C7×#626C211yesmaximal, characteristic, complemented
4C2×#92678C211yesmaximal, characteristic, complemented
5#11759209C211yesmaximal, characteristic, complemented
6C3×#66575C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic7C211yesmaximal, characteristic, complemented
8#10621135—31nomaximal
9#10621037—31nomaximal
10C22×C3⋊D21C2211yescharacteristic
11C22×C7×C3⋊D3C2211yescharacteristic, complemented
12C22×C3×D21C2211yescharacteristic
13C22×C3×D3×C7C2211yescharacteristic
14C2×D3×D21—22no—
15C2×D3×D21—22no—
16C2×D3×D21—22no—
17C2×C3×Dic21C2211yescharacteristic, complemented
18C2×C3×C7×Dic3C2211yescharacteristic, complemented
19C2×C32×Dic7C2211yescharacteristic, complemented
20#92694—22no—
21C7×#626—22no—
22#92678—22no—
23C23×D21—31no—
24C22×D3×D7—31no—
25C2×C7×#67—31no—
26C2×C7×D12—31no—
27C2×#4247—31no—
28C2×#4247—31no—
29C2×#4247—31no—
30C2×#4247—61no—
31C2×#4245—31no—
32#66631—31no—
33C3×C22⋊Dic7—31no—
34C3×(C2×D7)⋊C4—31no—
35#66575—31no—
36#66575D311yescharacteristic, complemented
37(C2×D3)⋊Dic7D311yescharacteristic, complemented
38#63923—71nomaximal, Hall
39C22×C32×C7C2311yescharacteristic
40C2×C3⋊D21—21no—
41C2×C3⋊D21—21no—
42C2×C3⋊D21D412yescomplemented
43C2×C7×C3⋊D3—21no—
44C2×C7×C3⋊D3—21no—
45C2×C7×C3⋊D3—21no—
46C2×C3×D21—21no—
47C2×C3×D21—21no—
48C2×C3×D21D412yescomplemented
49C2×C3×D3×C7—21no—
50C2×C3×D3×C7—21no—
51C2×C3×D3×C7D412yescomplemented
52D3×D21—28no—
53C3×Dic21—21no—
54C3×C7×Dic3—21no—
55C32×Dic7—21no—
56#6500—91noHall
57C23×C3×C7—31no—
58C22×D21—31no—
59C22×D21—31no—
60C22×D21—61no—
61C22×D21—62no—
62C22×D21—62no—
63C22×D21—62no—
64C22×D21C2×D311yescharacteristic
65C22×D3×C7—31no—
66C22×D3×C7—31no—
67C22×D3×C7—61no—
68C22×D3×C7C2×D311yescharacteristic
69C22×C3×D7—31no—
70C2×D3×D7—62no—
71C2×D3×D7—62no—
72C2×D3×D7—62no—
73C2×C3×C4×C7—31no—
74C2×Dic21C2×D311yescharacteristic, complemented
75C7×#67—62no—
76C2×C7×Dic3C2×D311yescharacteristic, complemented
77C7×D12—62no—
78C2×C3×Dic7—21no—
79C2×C3×Dic7—31no—
80C2×C3×Dic7C2×D311yescharacteristic, complemented
81C2×C3×Dic7C2×D311yescharacteristic, complemented
82#4247—62no—
83#4247—62no—
84#4247—62no—
85#4247—62no—
86#4247—62no—
87#4245—62no—
88C22×D32—71no—
89C2×C4⋊(C3⋊D3)—71no—
90C2×#626—71no—
91C2×#626D711yescharacteristic, complemented
92C3×(C2×D3)⋊C4—71no—
93C3×(C2×D3)⋊C4—71no—
94#3722—71no—
95C2×C32×C7C2×D411yescharacteristic
96C2×C32×C7C2×D411yescharacteristic
97C2×C32×C7C2×D411yescharacteristic
98C3⋊D21—22no—
99C7×C3⋊D3—41no—
100C3×D21—22no—
101C3×D3×C7—22no—
102C23×D7—91no—
103C2×C7×D4—91no—
104C2×C22⋊D7—91no—
105C2×C22⋊D7—91no—
106C22⋊Dic7—31no—
107(C2×D7)⋊C4—31no—
108(C2×D7)⋊C4—91no—
109#882—211no—
110#882—211no—
111C22×C3×C7—21no—
112C22×C3×C7—32no—
113C22×C3×C7—61no—
114C22×C3×C7—61no—
115C22×C3×C7C22×D311yescharacteristic
116C22×C3×C7C22×D311yescharacteristic
117C2×D21—21no—
118C2×D21—21no—
119C2×D21—32no—
120C2×D21—32no—
121C2×D21—62no—
122C2×D21—68no—
123C2×D21—61no—
124C2×D21—61no—
125C2×D21—61no—
126C2×D21—61no—
127C2×D21—121no—
128C2×D21—121no—
129C2×D21D1212yescomplemented
130C2×D3×C7—21no—
131C2×D3×C7—21no—
132C2×D3×C7—61no—
133C2×D3×C7—61no—
134C2×D3×C7—61no—
135C2×D3×C7—61no—
136C2×D3×C7—61no—
137C2×D3×C7—61no—
138C2×D3×C7—61no—
139C2×D3×C7—61no—
140C2×D3×C7—121no—
141C2×D3×C7—121no—
142C2×D3×C7D1212yescomplemented
143C2×C3×D7—32no—
144C2×C3×D7—61no—
145C2×C3×D7—61no—
146D3×D7—68no—
147C3×C4×C7—61no—
148Dic21—21no—
149C3×Dic7—21no—
150C3×Dic7—21no—
151C3×Dic7—21no—
152C3×Dic7—21no—
153C3×Dic7—61no—
154C7×Dic3—21no—
155C22×C3⋊D3—71no—
156C22×C3⋊D3C2×D711yescharacteristic, complemented
157C22×C3×D3—71no—
158C22×C3×D3C2×D711yescharacteristic
159C2×D32—142no—
160C2×D32—142no—
161C2×D32—142no—
162C2×C32×C4—71no—
163C4⋊(C3⋊D3)—142no—
164C2×C3×Dic3—71no—
165C2×C3×Dic3C2×D711yescharacteristic, complemented
166#626—22no—
167#626—142no—
168C32×C7#12011yescharacteristic, complemented, Hall
169C23×C7—31no—
170C23×C7—91no—
171C22×D7—31no—
172C22×D7—91no—
173C22×D7—182no—
174C22×D7—182no—
175C22×D7—182no—
176C7×D4—182no—
177C2×C4×C7—31no—
178C22⋊D7—182no—
179C22⋊D7—182no—
180C2×Dic7—31no—
181C2×Dic7D3211yescharacteristic, complemented
182C23×D3—211no—
183C23×D3—211no—
184C2×#67—31no—
185C2×#67—211no—
186C2×D12—31no—
187C2×D12—211no—
188C2×D12—211no—
189C2×D12—211no—
190C2×D12—421no—
191C3×C22⋊C4—211no—
192C3×C22⋊C4—211no—
193(C2×D3)⋊C4—71no—
194(C2×D3)⋊C4—71no—
195(C2×D3)⋊C4—211no—
196(C2×D3)⋊C4—211no—
197C2×C3×C7—21no—
198C2×C3×C7—21no—
199C2×C3×C7—21no—
200C2×C3×C7—62no—
201C2×C3×C7C2×D1211yescharacteristic
202C2×C3×C7C2×D1211yescharacteristic
203C2×C3×C7D3×D411yescharacteristic
204C2×C3×C7D3×D411yescharacteristic
205C2×C3×C7D3×D411yescharacteristic
206C2×C3×C7D3×D411yescharacteristic
207D21—22no—
208D21—62no—
209D21—62no—
210D21—122no—
211C3×D7—62no—
212D3×C7—22no—
213D3×C7—121no—
214D3×C7—121no—
215D3×C7—121no—
216D3×C7—121no—
217C22×C32C22×D711yescharacteristic
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—72no—
222C2×C3⋊D3—141no—
223C2×C3⋊D3—141no—
224C2×C3×D3—21no—
225C2×C3×D3—21no—
226C2×C3×D3—72no—
227C2×C3×D3—141no—
228C2×C3×D3—141no—
229C2×C3×D3C22⋊D712yescomplemented
230D32—148no—
231C32×C4—141no—
232C3×Dic3—21no—
233C3×Dic3—141no—
234#120—631noSylow
235C22×C7—32no—
236C22×C7—61no—
237C22×C7—61no—
238C22×C7—181no—
239C22×C7—181no—
240C22×C7—181no—
241C22×C7C2×D3211yescharacteristic
242C2×D7—32no—
243C2×D7—61no—
244C2×D7—61no—
245C2×D7—92no—
246C2×D7—188no—
247C2×D7—181no—
248C2×D7—181no—
249C4×C7—61no—
250Dic7—21no—
251Dic7—61no—
252C23×C3—31no—
253C23×C3—211no—
254C22×D3—31no—
255C22×D3—31no—
256C22×D3—61no—
257C22×D3—71no—
258C22×D3—211no—
259C22×D3—211no—
260C22×D3—421no—
261C22×D3—422no—
262C22×D3—422no—
263C22×D3—422no—
264C22×D3—422no—
265C22×D3—422no—
266C22×D3—422no—
267C22×D3D3×D711yescharacteristic
268C2×C3×C4—31no—
269C2×C3×C4—71no—
270C2×C3×C4—71no—
271C2×C3×C4—141no—
272C2×C3×C4—211no—
273#67—62no—
274#67—422no—
275C2×Dic3—71no—
276C2×Dic3D3×D711yescharacteristic, complemented
277D12—62no—
278D12—422no—
279D12—422no—
280D12—422no—
281D12—422no—
282D12—422no—
283C3×C7—21no—
284C3×C7#88211yescharacteristic, complemented
285C3×C7#88211yescharacteristic, complemented
286C2×C32D4×D711yescharacteristic
287C2×C32D4×D711yescharacteristic
288C2×C32C2×C22⋊D711yescharacteristic
289C3⋊D3—41no—
290C3⋊D3—142no—
291C3×D3—22no—
292C3×D3—142no—
293C24—631no—
294C2×D4—91no—
295C2×D4—631no—
296C2×D4—631no—
297C22⋊C4—211no—
298C22⋊C4—211no—
299C22⋊C4—631no—
300C2×C7—62no—
301C2×C7—361no—
302C2×C7D3×D1211yescharacteristic
303C2×C7D3×D1211yescharacteristic
304C2×C7#381111yescharacteristic
305D7—62no—
306D7—182no—
307C22×C3—21no—
308C22×C3—32no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—212no—
312C22×C3—421no—
313C22×C3—421no—
314C22×C3C2×D3×D711yescharacteristic
315C22×C3C2×D3×D711yescharacteristic
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—61no—
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—61no—
326C2×D3—72no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—141no—
330C2×D3—141no—
331C2×D3—212no—
332C2×D3—212no—
333C2×D3—422no—
334C2×D3—428no—
335C2×D3—428no—
336C2×D3—421no—
337C2×D3—421no—
338C2×D3—421no—
339C2×D3—421no—
340C2×D3—841no—
341C2×D3—841no—
342C2×D3#424712yescomplemented
343C3×C4—61no—
344C3×C4—141no—
345C3×C4—141no—
346C3×C4—141no—
347C3×C4—141no—
348C3×C4—421no—
349Dic3—21no—
350Dic3—141no—
351C32#650011yescharacteristic, complemented, Sylow
352C23—31no—
353C23—91no—
354C23—211no—
355C23—631no—
356C23—1262no—
357C23—1262no—
358C23—1262no—
359D4—182no—
360D4—1262no—
361D4—1262no—
362C2×C4—31no—
363C2×C4—71no—
364C2×C4—211no—
365C7#6392311yescharacteristic, complemented, Sylow
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—62no—
370C2×C3—422no—
371C2×C3D7×D1211yescharacteristic
372C2×C3#6668111yescharacteristic
373C2×C3C2×#424711yescharacteristic
374C2×C3D3×C22⋊D711yescharacteristic
375C2×C3#6669411yescharacteristic
376C2×C3#6669411yescharacteristic
377D3—22no—
378D3—121no—
379D3—121no—
380D3—121no—
381D3—121no—
382D3—142no—
383D3—422no—
384D3—422no—
385D3—842no—
386C22—32no—
387C22—61no—
388C22—61no—
389C22—181no—
390C22—181no—
391C22—181no—
392C22—212no—
393C22—421no—
394C22—421no—
395C22—632no—
396C22—1268no—
397C22—1261no—
398C22—1261no—
399C22D32×D711yescharacteristic
400C4—61no—
401C4—141no—
402C4—421no—
403C3—21no—
404C3#1062103711yescharacteristic, complemented
405C3#1062113511yescharacteristic, complemented
406C2—62no—
407C2—361no—
408C2—422no—
409C2—1262no—
410C2#1175942711yescharacteristic
411C2D3×#424711yescharacteristic
412C2#1175943011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D7
commutator, Fitting, socle, cosocleC22×C32×C7C23
hypocenterC32×C7#120
LCSG ▷ C22×C32×C7 ▷ C32×C7
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×D3×D21C2semidirect
2. C2×#92694C2semidirect
3. C2×C7×#626C2semidirect
4. C2×#92678C2semidirect
5. #11759209C2semidirect
6. C3×#66575C2semidirect
7. C3×(C2×D3)⋊Dic7C2semidirect
11. C22×C7×C3⋊D3C22semidirect
17. C2×C3×Dic21C22semidirect
18. C2×C3×C7×Dic3C22semidirect
19. C2×C32×Dic7C22semidirect
36. #66575D3semidirect
37. (C2×D3)⋊Dic7D3semidirect
42. C2×C3⋊D21D4semidirect
48. C2×C3×D21D4semidirect
51. C2×C3×D3×C7D4semidirect
74. C2×Dic21C2×D3semidirect
76. C2×C7×Dic3C2×D3semidirect
80. C2×C3×Dic7C2×D3semidirect
81. C2×C3×Dic7C2×D3semidirect
91. C2×#626D7semidirect
129. C2×D21D12semidirect
142. C2×D3×C7D12semidirect
156. C22×C3⋊D3C2×D7semidirect
165. C2×C3×Dic3C2×D7semidirect
168. C32×C7#120semidirect
181. C2×Dic7D32semidirect
229. C2×C3×D3C22⋊D7semidirect
276. C2×Dic3D3×D7semidirect
284. C3×C7#882semidirect
285. C3×C7#882semidirect
342. C2×D3#4247semidirect
351. C32#6500semidirect
365. C7#63923semidirect
404. C3#10621037semidirect
405. C3#10621135semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D212
C2×#926942
C2×C7×#6262
C2×#926782
#117592092
C3×#665752
C3×(C2×D3)⋊Dic72
#106211353
#106210373
#639237