Finite Group Databasebeta

← #10698086

Subgroups — #10698086

Order 768.

698 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3192C211yesmaximal, characteristic
2#82665C211yesmaximal, characteristic, complemented
3#82664C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#69973C211yesmaximal, characteristic, complemented
7#67683C211yesmaximal, characteristic, complemented
8#10328—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#935—22no—
12C22×C4⋊Dic3C2211yescharacteristic
13C3×#481—21no—
14C3×#481—21no—
15C3×#469—22no—
16C3×#447—21no—
17C3×#447—21no—
18C3×#407C2211yescharacteristic
19#5221—21no—
20#5221—21no—
21#5220—22no—
22C4×#217C2211yescharacteristic
23C4×#217C2211yescharacteristic
24#5207C2211yescharacteristic
25#4977—21no—
26#4975—21no—
27#4953—22no—
28#4949—22no—
29#4554—22no—
30#4537—22no—
31#4537—22no—
32#3192D311yescharacteristic
33#2504—31no—
34#2501—31no—
35#1821—31no—
36#1507—31no—
37#1507—31no—
38#1297—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3D412yes—
45C2×C3×D8—22no—
46C2×C3×M16D412yes—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—24no—
61C2×#217—21no—
62C2×#217—24no—
63#935—48no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—21no—
73C2×#208—21no—
74C3×#106—22no—
75C3×#102—22no—
76C3×#102—22no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3⋊#95—24no—
80C3⋊#95D412yes—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84C2×#123—62no—
85C2×#121C2×D311yescharacteristic
86C2×#121C2×D311yescharacteristic
87C22×C4⋊C4—31no—
88#481—21no—
89#481—21no—
90#469—22no—
91#447—21no—
92#447—21no—
93#407—31no—
94#407C2×D311yescharacteristic
95#399—62no—
96#397—62no—
97#390—61no—
98#390—61no—
99#386—62no—
100#384—61no—
101#382—61no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4—31no—
104#343—62no—
105#327—62no—
106#327—62no—
107C24×C3—21no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C2×C3×D4—44no—
117C2×C3×D4—44no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×Dic3—22no—
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—44no—
137C3×D8—82no—
138C3×M16—42no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148#217—24no—
149#217—48no—
150#217—42no—
151#217—84no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156Dic3⋊C4—48no—
157C4×Dic3—42no—
158C4×Dic3—42no—
159#208—42no—
160C2×C3⋊C8—41no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4#6711yescharacteristic
164C22×D4#6711yescharacteristic
165C23×C4—32no—
166C23×C4C22×D311yescharacteristic
167C2×D8—22no—
168C2×M16—61no—
169C2×M16#6712yes—
170#123—128no—
171#121—21no—
172#121—21no—
173#121—21no—
174#121—21no—
175#121—21no—
176#121—21no—
177#121—21no—
178#121—21no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—64no—
194C2×C42—61no—
195C2×C42—61no—
196#106—22no—
197#102—22no—
198#102—22no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203#95—32no—
204#95—32no—
205#95—64no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3#12011yescharacteristic
218C23×C3#12011yescharacteristic
219C23×C3#12111yescharacteristic
220C3×D4—42no—
221C3×D4—42no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—82no—
225C3×D4—82no—
226C3×D4—82no—
227C3×D4—82no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12711yescharacteristic
242C2×Dic3—24no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—44no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—44no—
251C2×Dic3—82no—
252C3×C8—42no—
253C3⋊C8—81no—
254C24—21no—
255C24—21no—
256C24D3×D411yescharacteristic
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—124no—
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4#23711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284D8—82no—
285M16—42no—
286M16—122no—
287C2×C8—22no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—128no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—64no—
303C22⋊C4—128no—
304C22⋊C4—122no—
305C22⋊C4—244no—
306C42—122no—
307C42—122no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—44no—
316C22×C3—84no—
317C22×C3—84no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3#39211yescharacteristic
325C22×C3#45312yes—
326C22×C3#45412yes—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—81no—
331C3×C4—81no—
332C3×C4#45812yes—
333Dic3—44no—
334Dic3—82no—
335Dic3—82no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23#93711yescharacteristic
348C23#93711yescharacteristic
349C23#93811yescharacteristic
350D4—42no—
351D4—42no—
352D4—81no—
353D4—81no—
354D4—82no—
355D4—82no—
356D4—82no—
357D4—82no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—64no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—124no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—124no—
376C2×C4—122no—
377C2×C4—242no—
378C2×C4#93911yescharacteristic
379C2×C4#93911yescharacteristic
380C2×C4#94011yescharacteristic
381C2×C4#95311yescharacteristic
382C8—42no—
383C8—241no—
384C2×C3—22no—
385C2×C3—22no—
386C2×C3—41no—
387C2×C3—161no—
388C2×C3—161no—
389C2×C3#200211yescharacteristic
390C2×C3#200311yescharacteristic
391C2×C3#202211yescharacteristic
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—84no—
401C22—84no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22#519812yes—
409C22#520112yes—
410C22#522311yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—81no—
416C4—124no—
417C4—242no—
418C4—242no—
419C4#524312yes—
420C3#1032811yescharacteristic, complemented, Sylow
421C2—22no—
422C2—22no—
423C2—41no—
424C2—161no—
425C2—161no—
426C2#7184911yescharacteristic
427C2#7209411yescharacteristic
428C2#7209611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3192C2
socleC22×C3#392
hypercenter#3192D3
hypocenterC3#10328
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
2. #82665C2semidirect
3. #82664C2semidirect
4. #71673C2semidirect
5. #71673C2semidirect
6. #69973C2semidirect
7. #67683C2semidirect
420. C3#10328semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
#826652
#826642
#716732
#699732
#676832
#103283