Finite Group Databasebeta

← #10695513

Subgroups — #10695513

Order 768.

568 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3094C211yesmaximal, characteristic
2#82631C211yesmaximal, characteristic
3#82263C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic
5#71590C211yesmaximal, characteristic, complemented
6#69980C211yesmaximal, characteristic, complemented
7#67685C211yesmaximal, characteristic, complemented
8#9933—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#482—22no—
13C3×#481—21no—
14C3×#480—21no—
15C3×#428—21no—
16C3×#418—21no—
17C3×#407C2211yescharacteristic
18C2×#834—22no—
19#5218—21no—
20#5212C2211yescharacteristic
21C4×#217C2211yescharacteristic
22#5207C2211yescharacteristic
23#5042—22no—
24#5040—21no—
25#5033—21no—
26#4985—21no—
27#4983—21no—
28#4970—21no—
29#4952—22no—
30#4929—21no—
31#4928—21no—
32#4532—22no—
33#4467—22no—
34#3193—31no—
35#3094D311yescharacteristic
36#2625—31no—
37#1964—31no—
38#1826—31no—
39#1508—31no—
40#1305—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217D412yes—
63C2×#214—22no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—21no—
73C2×#208D412yes—
74C3×#107—22no—
75C3×#106—22no—
76C3×#102—22no—
77C3×#102—22no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8D412yes—
80#834—44no—
81#832—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87#830—21no—
88#830—21no—
89#810—22no—
90#807—22no—
91C2×C22⋊Q8—31no—
92C2×#121C2×D311yescharacteristic
93C2×#117C2×D311yescharacteristic
94#484—61no—
95#482—22no—
96#481—21no—
97#480—21no—
98#480—61no—
99#468—62no—
100#450—61no—
101#448—61no—
102#428—21no—
103#418—21no—
104C2×#99—62no—
105#407—31no—
106#407C2×D311yescharacteristic
107#400—61no—
108#398—61no—
109#397—62no—
110#391—61no—
111#389—61no—
112#380—31no—
113C4×C22⋊C4—31no—
114#339—62no—
115#328—62no—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—41no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—42no—
142C3×M16—42no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42—41no—
155C3×C42—41no—
156#217—22no—
157#217—41no—
158#217—41no—
159#217—41no—
160#217—42no—
161#214—82no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—42no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167Dic3⋊C4—82no—
168C4×Dic3—42no—
169#208—42no—
170C2×C3⋊C8—22no—
171C22×Q8—31no—
172C22×D4—21no—
173C22×D4#6711yescharacteristic
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C2×M16—21no—
177C2×M16—32no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C4⋊C4#6711yescharacteristic
196C2×C4⋊C4#6711yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—32no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—61no—
202C2×C42—21no—
203C2×C42—61no—
204#107—22no—
205#107—62no—
206#106—22no—
207#103—62no—
208#102—22no—
209#102—22no—
210#102—62no—
211#99—124no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8#6712yes—
216#95—62no—
217#95—62no—
218#95—62no—
219#95—61no—
220#95—61no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3—41no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3#12011yescharacteristic
230C23×C3#12111yescharacteristic
231C23×C3#12711yescharacteristic
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—82no—
235C3×D4—81no—
236C3×D4—81no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265Dic6—42no—
266Dic6—81no—
267C3×C8—42no—
268C3⋊C8—42no—
269C24—21no—
270C24D3×D411yescharacteristic
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—122no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—121no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4#23711yescharacteristic
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295SD16—242no—
296M16—42no—
297M16—122no—
298C2×C8—22no—
299C2×C8—62no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—122no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C4⋊C4—242no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—62no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—122no—
321C42—41no—
322C42—41no—
323C42—122no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—162no—
338C22×C3#39211yescharacteristic
339C22×C3#45311yescharacteristic
340C22×C3#45411yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—81no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4#45812yes—
348Dic3—82no—
349Dic3—82no—
350Dic3—81no—
351Dic3—81no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—41no—
358C23—81no—
359C23—81no—
360C23#93711yescharacteristic
361C23#93911yescharacteristic
362C23#94011yescharacteristic
363Q8—122no—
364Q8—241no—
365D4—42no—
366D4—42no—
367D4—82no—
368D4—81no—
369D4—81no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4#93711yescharacteristic
395C2×C4#93811yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#95311yescharacteristic
398C8—42no—
399C8—122no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—161no—
405C2×C3#200611yescharacteristic
406C2×C3#200911yescharacteristic
407C2×C3#202211yescharacteristic
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—162no—
422C22#519811yescharacteristic
423C22#520111yescharacteristic
424C22#522311yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—242no—
432C4—242no—
433C4—241no—
434C4—241no—
435C4#505312yes—
436C3#993311yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—161no—
442C2#7170611yescharacteristic
443C2#7209811yescharacteristic
444C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3094C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#3094D3
hypocenterC3#9933
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
3. #82263C2semidirect
5. #71590C2semidirect
6. #69980C2semidirect
7. #67685C2semidirect
436. C3#9933semidirect

Maximal subgroups

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

subgroupindex
C3×#30942
#826312
#822632
#716862
#715902
#699802
#676852
#99333