Finite Group Databasebeta

← #10695537

Subgroups — #10695537

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×#2624C211yesmaximal, characteristic
2#82631C211yesmaximal, characteristic
3#82220C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic
5#71590C211yesmaximal, characteristic, complemented
6#69529C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9936—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#399—21no—
13C3×#397—21no—
14C3×#394—22no—
15C3×#390—21no—
16C3×#389—21no—
17C3×#380C2211yescharacteristic
18#5218—21no—
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#5042—22no—
23#5040—21no—
24#5033—21no—
25#5004—21no—
26#4996—22no—
27#4995—21no—
28#4971—21no—
29#4970—21no—
30#4970—21no—
31#4537—22no—
32#4532—22no—
33#4527—22no—
34#3193—31no—
35#3095—31no—
36#2624D311yescharacteristic
37#1996—31no—
38#1826—31no—
39#1507—31no—
40#1305—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×Dic6D411yescharacteristic
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#117—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4D412yes—
63C2×C3×C42—21no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#214—22no—
67C2×C4⋊Dic3D411yescharacteristic
68#924—21no—
69#924—21no—
70#924—21no—
71#924—21no—
72C2×Dic3⋊C4—21no—
73C2×C4×Dic3—21no—
74C2×#208—21no—
75C2×#208D412yes—
76C3×#95—22no—
77C3×#95—21no—
78#832—22no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#810—22no—
85#810—22no—
86#810—22no—
87#808—22no—
88#807—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×#121C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93#485—62no—
94#484—61no—
95#484—61no—
96#483—61no—
97#480—61no—
98#468—62no—
99#450—61no—
100#448—61no—
101#428—61no—
102#419—61no—
103#407—31no—
104#407—31no—
105#399—21no—
106#397—21no—
107#394—22no—
108#390—21no—
109#389—21no—
110#380C2×D311yescharacteristic
111C4×C22⋊C4—31no—
112#342—62no—
113#339—62no—
114#327—62no—
115C24×C3—21no—
116C24×C3C2×D411yescharacteristic
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—82no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—41no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—82no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—42no—
153C3×C22⋊C4—82no—
154C3×C22⋊C4—82no—
155C3×C42—41no—
156C3×C42—41no—
157#217—22no—
158#217—41no—
159#217—41no—
160#217—41no—
161#214—82no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166C4×Dic3—42no—
167#208—42no—
168#208—42no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C22×Q8—31no—
172C22×D4—21no—
173C22×D4#6711yescharacteristic
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C2×M16—32no—
177C2×M16—61no—
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—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C4⋊C4#6711yescharacteristic
196C2×C4⋊C4#6711yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4#6712yes—
204C2×C42—21no—
205C2×C42—61no—
206#107—62no—
207#107—62no—
208#106—62no—
209#103—62no—
210#103—62no—
211#103—62no—
212#102—62no—
213C22⋊C8—32no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217#95—22no—
218#95—21no—
219#95—61no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3#12011yescharacteristic
230C23×C3#12111yescharacteristic
231C23×C3#12711yescharacteristic
232C3×D4—42no—
233C3×D4—81no—
234C3×D4—82no—
235C3×D4—82no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
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—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×Dic3—42no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—81no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263C2×Dic3—81no—
264Dic6—42no—
265Dic6—42no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C24—21no—
269C24C2×#6711yescharacteristic
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—122no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—82no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—121no—
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4#23711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292C22×C4C2×#6711yescharacteristic
293SD16—242no—
294M16—122no—
295M16—122no—
296C2×C8—62no—
297C2×C8—62no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—82no—
308C4⋊C4—121no—
309C4⋊C4—121no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—62no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
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—82no—
335C22×C3—82no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3#39211yescharacteristic
340C22×C3#45511yescharacteristic
341C22×C3#45611yescharacteristic
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4—81no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4#45812yes—
350Dic3—82no—
351Dic3—81no—
352Dic3—81no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23#93811yescharacteristic
363C23#94011yescharacteristic
364C23#95311yescharacteristic
365Q8—122no—
366Q8—122no—
367D4—42no—
368D4—81no—
369D4—82no—
370D4—82no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—122no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4#93711yescharacteristic
396C2×C4#93711yescharacteristic
397C2×C4#93911yescharacteristic
398C2×C4#93911yescharacteristic
399C8—122no—
400C8—122no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—161no—
406C2×C3#200711yescharacteristic
407C2×C3#200811yescharacteristic
408C2×C3#202211yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—82no—
420C22—82no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#522311yescharacteristic
425C22#524211yescharacteristic
426C22#524611yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—242no—
435C4—241no—
436C4—241no—
437C4#505312yes—
438C3#993611yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—161no—
444C2#7170611yescharacteristic
445C2#7185811yescharacteristic
446C2#7188111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2624C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#2624D3
hypocenterC3#9936
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2624

Complements

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

NH ≅ G/Ntype
3. #82220C2semidirect
5. #71590C2semidirect
6. #69529C2semidirect
7. #67626C2semidirect
438. C3#9936semidirect

Maximal subgroups

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

subgroupindex
C3×#26242
#826312
#822202
#716732
#715902
#695292
#676262
#99363