Finite Group Databasebeta

← #10690161

Subgroups — #10690161

Order 768.

782 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#2975C211yesmaximal, characteristic, complemented
2#82400C211yesmaximal, characteristic
3C4×#834C211yesmaximal, characteristic, complemented
4#69991C212yesmaximal, complemented
5#67683C212yesmaximal, complemented
6#9010—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×C4×#67—21no—
10C3×#436—24no—
11C3×#431—21no—
12C3×#407—21no—
13C3×#407C2212yes—
14#5223—21no—
15C2×#834—21no—
16C4×#217C2212yes—
17C4×#217C2211yescharacteristic
18C2×#825—24no—
19#4991C411yescharacteristic, complemented
20#4979C411yescharacteristic, complemented
21#4960C411yescharacteristic, complemented
22#4959—21no—
23#4947C411yescharacteristic, complemented
24C4×(C2×D3)⋊C4—21no—
25#4526—21no—
26#2975D311yescharacteristic, complemented
27#2388—31no—
28#1789—32no—
29C4×#99—31no—
30#1297—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C22×#67—21no—
35C22×#67—21no—
36C2×#235—22no—
37C2×#235—22no—
38C2×#235—44no—
39C22×D12C2×C411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6C2×C411yescharacteristic
42C2×C3×M16—22no—
43C2×C3×M16D414yescomplemented
44C3×C4×D4—22no—
45C3×C4×D4—22no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4D411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×#217—22no—
51C2×#217—21no—
52C2×#217—21no—
53C4×#67—44no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—22no—
60C3×C4⋊C8—22no—
61C3×C22⋊C8—22no—
62C3×C22⋊C8—22no—
63C3×C22⋊C8—22no—
64C3×C4×C8—22no—
65#834—24no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69C3⋊#95C2×C411yescharacteristic
70C3⋊#95C2×C411yescharacteristic
71#825—48no—
72C22×#41—31no—
73C2×C4×D4—61no—
74C2×C4×D4C2×D311yescharacteristic
75#436—24no—
76#431—21no—
77C2×#104—64no—
78C2×#99—61no—
79#407—21no—
80#407C2×D312yes—
81#395—31no—
82#394—31no—
83#394—61no—
84#393—31no—
85#392—31no—
86#392—61no—
87C4×C22⋊C4—31no—
88C4×C22⋊C4—32no—
89C4×C22⋊C4—61no—
90#342—61no—
91C24×C3C22⋊C411yescharacteristic
92C24×C3C2×D411yescharacteristic
93C23×D3—21no—
94C2×C3×D4—22no—
95C2×C3×D4—22no—
96C2×C3×D4—41no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4112yes—
108C2×#67—42no—
109C2×#67—42no—
110C2×#67—42no—
111C2×#67—42no—
112C2×#67—42no—
113C2×#67—42no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—42no—
118#235—28no—
119#235—44no—
120#235—44no—
121#235—44no—
122C2×D12—22no—
123C2×D12—42no—
124C2×C4×D3—24no—
125C2×C4×D3—42no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×Dic6—22no—
129C2×Dic6—42no—
130C3×M16—44no—
131C3×M16—82no—
132C2×C3×C8—24no—
133C3×C4⋊C4—22no—
134C3×C22⋊C4—22no—
135C3×C22⋊C4—22no—
136C3×C42—21no—
137C3×C42—21no—
138#217—24no—
139#217—42no—
140#217—42no—
141#217—42no—
142(C2×D3)⋊C4—42no—
143(C2×D3)⋊C4—42no—
144(C2×D3)⋊C4—42no—
145Dic3⋊C4—42no—
146C4×Dic3—42no—
147C4×Dic3—44no—
148C2×#41—62no—
149C2×#41—62no—
150C2×#41—124no—
151C22×Q8—31no—
152C22×D4—31no—
153C22×D4—61no—
154C22×D4—61no—
155C22×D4D1211yescharacteristic
156C23×C4—61no—
157C23×C4#6711yescharacteristic
158C23×C4C22×D311yescharacteristic
159C2×M16—22no—
160C2×M16#6714yescomplemented
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—124no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4C4×D311yescharacteristic
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4C4×D311yescharacteristic
173C2×C22⋊C4D1211yescharacteristic
174C2×C42—61no—
175C2×C42—62no—
176C2×C42#6711yescharacteristic
177C4⋊C8—22no—
178#104—128no—
179#99—64no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C4×C8—22no—
184#95—31no—
185#95—31no—
186#95—61no—
187#95—61no—
188#95—62no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3C2×C22⋊C411yescharacteristic
199C23×C3#12112yes—
200C22×D3—22no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—41no—
207C3×D4—44no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—24no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—81no—
220C2×C3×C4#10412yescomplemented
221C2×C3×C4C4×D412yes—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12311yescharacteristic
224#67—48no—
225#67—48no—
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—44no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—82no—
237C2×Dic3—82no—
238D12—24no—
239D12—42no—
240C4×D3—44no—
241C4×D3—44no—
242Dic6—24no—
243Dic6—42no—
244C3×C8—44no—
245C24—61no—
246C24(C2×D3)⋊C411yescharacteristic
247C24C2×D1211yescharacteristic
248#41—68no—
249#41—124no—
250#41—124no—
251#41—124no—
252C2×Q8—62no—
253C2×Q8—122no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—62no—
258C2×D4—122no—
259C2×D4—122no—
260C2×D4—122no—
261C2×D4—122no—
262C2×D4—122no—
263C2×D4—122no—
264C2×D4—122no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—64no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—122no—
274C22×C4—121no—
275C22×C4—121no—
276C22×C4—122no—
277C22×C4(C2×D3)⋊C411yescharacteristic
278C22×C4(C2×D3)⋊C411yescharacteristic
279C22×C4(C2×D3)⋊C411yescharacteristic
280C22×C4C2×C4×D311yescharacteristic
281C22×C4#23512yes—
282C22×C4C2×#6712yes—
283C22×C4C2×#6711yescharacteristic
284M16—44no—
285M16—82no—
286C2×C8—24no—
287C4⋊C4—22no—
288C4⋊C4—122no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—64no—
292C22⋊C4—122no—
293C22⋊C4—122no—
294C22⋊C4—122no—
295C22⋊C4—122no—
296C22⋊C4—122no—
297C22⋊C4—122no—
298C42—21no—
299C42—21no—
300C42—122no—
301C42—124no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—42no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#38611yescharacteristic
315C22×C3C2×#10411yescharacteristic
316C22×C3#42111yescharacteristic
317C2×D3—42no—
318C2×D3—42no—
319C2×D3—42no—
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4B(4,𝔽2)12yes—
330Dic3—42no—
331Dic3—42no—
332Dic3—81no—
333Dic3—84no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—62no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23C2×(C2×D3)⋊C411yescharacteristic
351C23#93012yes—
352Q8—64no—
353Q8—122no—
354D4—44no—
355D4—64no—
356D4—122no—
357D4—128no—
358D4—128no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—81no—
373C2×C4—124no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—124no—
379C2×C4—121no—
380C2×C4—124no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—242no—
384C2×C4—242no—
385C2×C4#82512yescomplemented
386C2×C4C4×#6712yes—
387C2×C4#92911yescharacteristic
388C2×C4#95311yescharacteristic
389C8—44no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—42no—
393C2×C3—81no—
394C2×C3#187111yescharacteristic
395C2×C3#188411yescharacteristic
396C2×C3#188911yescharacteristic
397D3—82no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—42no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—122no—
411C22—122no—
412C22—122no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22C2×#82511yescharacteristic
418C22#513911yescharacteristic
419C22#521311yescharacteristic
420C4—22no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—122no—
426C4—122no—
427C4—241no—
428C4—244no—
429C4#524412yes—
430C3#901011yescharacteristic, complemented, Sylow
431C2—21no—
432C2—21no—
433C2—42no—
434C2—81no—
435C2—242no—
436C2#7165411yescharacteristic
437C2#7198411yescharacteristic
438C2#7202111yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2975C2
socleC22×C3#386
prime radical#82400C2
cosocleC23×C3×C4C23
hypercenter#2975D3
hypocenterC3#9010
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
1. C3×#2975C2semidirect
3. C4×#834C2semidirect
4. #69991C2semidirect
5. #67683C2semidirect
19. #4991C4semidirect
20. #4979C4semidirect
21. #4960C4semidirect
23. #4947C4semidirect
26. #2975D3semidirect
43. C2×C3×M16D4semidirect
160. C2×M16#67semidirect
220. C2×C3×C4#104semidirect
385. C2×C4#825semidirect
430. C3#9010semidirect

Maximal subgroups

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

subgroupindex
C3×#29752
#824002
C4×#8342
#699912
#676832
#90103