Finite Group Databasebeta

← #10693010

Subgroups — #10693010

Order 768.

726 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#74063C211yesmaximal, characteristic
2#73445C211yesmaximal, characteristic, complemented
3#69991C211yesmaximal, characteristic
4#69971C211yesmaximal, characteristic, complemented
5#69454C211yesmaximal, characteristic
6C3×#1284C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9480—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#886—21no—
12C3×#407C2211yescharacteristic
13C3×#381C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#825—22no—
18#4958—21no—
19#4943—21no—
20#4942—21no—
21#4933—22no—
22C2×#805—22no—
23#4672C412yescomplemented
24#4668—22no—
25#4656C412yescomplemented
26#4652—22no—
27#4469—22no—
28#2406—31no—
29#2313—31no—
30#1789—31no—
31#1789—31no—
32#1463—31no—
33#1284—31no—
34#1284D311yescharacteristic, complemented
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C2×#235—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—44no—
44C22×D12C2×C411yescharacteristic
45C22×C4×D3—21no—
46C22×Dic6C2×C411yescharacteristic
47C2×C3×M16—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42D412yes—
51C2×#217—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×C4⋊Dic3—22no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—22no—
60C2×Dic3⋊C4C2×C412yes—
61C2×Dic3⋊C4D412yes—
62C2×C4×Dic3D412yes—
63C2×#208—21no—
64#886—44no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95C2×C412yes—
70C3⋊#95—22no—
71C3⋊#95—24no—
72C3⋊#95—21no—
73C3⋊#95—22no—
74#830—21no—
75#830—21no—
76#825—44no—
77#805—44no—
78C22×#41—31no—
79C2×#123—61no—
80C22×C4⋊C4—31no—
81C2×#104—62no—
82C2×#104—62no—
83#407—31no—
84#407C2×D311yescharacteristic
85#395—32no—
86#394—32no—
87#387—62no—
88#387—62no—
89#386—61no—
90#385—62no—
91#381—61no—
92#381C2×D311yescharacteristic
93#380—61no—
94C4×C22⋊C4—31no—
95C4×C22⋊C4C2×D311yescharacteristic
96#324—62no—
97C24×C3C2×D411yescharacteristic
98C23×D3—21no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—42no—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×#67—24no—
109C2×#67—42no—
110C2×#67—41no—
111C2×#67—41no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C22×Dic3—42no—
119C22×Dic3—42no—
120C22×Dic3C22⋊C412yes—
121C22×Dic3C22⋊C412yes—
122#235—24no—
123#235—24no—
124#235—42no—
125#235—42no—
126#235—42no—
127#235—42no—
128#235—42no—
129#235—42no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×C4×D3—42no—
134C2×C4×D3—42no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×M16—42no—
141C2×C3×C8—41no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—42no—
144C3×C42—42no—
145#217—42no—
146(C2×D3)⋊C4—44no—
147(C2×D3)⋊C4—44no—
148C4⋊Dic3—44no—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151Dic3⋊C4—22no—
152Dic3⋊C4—22no—
153Dic3⋊C4—82no—
154C4×Dic3—42no—
155#208—42no—
156C2×C3⋊C8—41no—
157C2×#41—61no—
158C2×#41—61no—
159C2×#41—61no—
160C2×#41—61no—
161C2×#41—124no—
162C22×Q8—31no—
163C22×D4—31no—
164C22×D4—61no—
165C23×C4—31no—
166C23×C4—31no—
167C23×C4—61no—
168C23×C4C22×D311yescharacteristic
169C2×M16—21no—
170C2×M16—61no—
171#123—124no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—62no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C42—32no—
186C2×C42D1212yes—
187#104—124no—
188#104—124no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193#95—21no—
194#95—21no—
195#95—62no—
196#95—62no—
197#95—64no—
198#95—61no—
199#95C4×D312yes—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3—41no—
203C23×C3C2×C22⋊C411yescharacteristic
204C23×C3#12011yescharacteristic
205C23×C3#12311yescharacteristic
206C22×D3—22no—
207C22×D3—41no—
208C22×D3—42no—
209C22×D3—41no—
210C22×D3—42no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227#67—44no—
228#67—44no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—81no—
244D12—22no—
245D12—22no—
246D12—41no—
247D12—41no—
248C4×D3—44no—
249C4×D3—44no—
250Dic6—22no—
251Dic6—22no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—81no—
255C3⋊C8—81no—
256C24—61no—
257C24D3×D411yescharacteristic
258#41—64no—
259#41—64no—
260#41—122no—
261#41—122no—
262#41—122no—
263#41—122no—
264#41—122no—
265#41—122no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×Q8—122no—
269C2×D4—64no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—122no—
273C2×D4—122no—
274C2×D4—121no—
275C2×D4—121no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—32no—
279C22×C4—32no—
280C22×C4—42no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—122no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4—121no—
291C22×C4—121no—
292C22×C4—122no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4#23911yescharacteristic
299M16—42no—
300M16—122no—
301C2×C8—41no—
302C2×C8—121no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—124no—
308C4⋊C4—242no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—124no—
312C22⋊C4—124no—
313C22⋊C4—122no—
314C42—42no—
315C42—122no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—44no—
324C22×C3#38611yescharacteristic
325C22×C3#42112yes—
326C22×C3B(4,𝔽2)12yes—
327C2×D3—44no—
328C2×D3—42no—
329C2×D3—82no—
330C2×D3—81no—
331C2×D3—81no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—82no—
337C3×C4—82no—
338Dic3—42no—
339Dic3—42no—
340Dic3—42no—
341Dic3—82no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—62no—
346C23—121no—
347C23—122no—
348C23—121no—
349C23—122no—
350C23D3×C22⋊C411yescharacteristic
351C23#88211yescharacteristic
352C23#89411yescharacteristic
353Q8—62no—
354Q8—62no—
355Q8—121no—
356Q8—121no—
357D4—62no—
358D4—62no—
359D4—121no—
360D4—121no—
361D4—124no—
362D4—124no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—124no—
381C2×C4—124no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—241no—
392C2×C4C4×D1211yescharacteristic
393C2×C4#88411yescharacteristic
394C2×C4#89311yescharacteristic
395C2×C4#89511yescharacteristic
396C8—81no—
397C8—241no—
398C2×C3—22no—
399C2×C3—22no—
400C2×C3—41no—
401C2×C3#188911yescharacteristic
402C2×C3#189711yescharacteristic
403C2×C3#191211yescharacteristic
404D3—82no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—44no—
413C22—124no—
414C22—122no—
415C22—242no—
416C22—241no—
417C22—241no—
418C22#466511yescharacteristic
419C22#474112yes—
420C22#482112yes—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—82no—
426C4—82no—
427C4—122no—
428C4—122no—
429C4—122no—
430C4—242no—
431C3#948011yescharacteristic, complemented, Sylow
432C2—22no—
433C2—22no—
434C2—41no—
435C2—242no—
436C2#6870811yescharacteristic
437C2#6875711yescharacteristic
438C2#6896211yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9480
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
2. #73445C2semidirect
4. #69971C2semidirect
6. C3×#1284C2semidirect
7. #67628C2semidirect
23. #4672C4semidirect
25. #4656C4semidirect
34. #1284D3semidirect
431. C3#9480semidirect

Maximal subgroups

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

subgroupindex
#740632
#734452
#699912
#699712
#694542
C3×#12842
#676282
#94803