Finite Group Databasebeta

← C23×#8493 #19156119

Subgroups — C23×#8493 #19156119

Order 5888.

748 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3548C211yesmaximal, characteristic
2C23×#1963C211yesmaximal, characteristic, complemented
3C23×#1559C211yesmaximal, characteristic, complemented
4C23×#1557C211yesmaximal, characteristic, complemented
5C23×#1536C211yesmaximal, characteristic, complemented
6C23×#1506C211yesmaximal, characteristic, complemented
7C23×#1505C211yesmaximal, characteristic, complemented
8C23×#562—22no—
9C23×#554—21no—
10C23×#554C412yescomplemented
11C23×#550—21no—
12C23×#550C412yescomplemented
13C23×#547—21no—
14C23×#547—21no—
15C23×#540—22no—
16C23×#537—22no—
17C23×#537—22no—
18C23×#533C2211yescharacteristic
19C23×#532C2211yescharacteristic
20C23×#532C2211yescharacteristic
21C23×#432C2211yescharacteristic
22C23×#432C2211yescharacteristic
23C23×#423C2211yescharacteristic
24C2×C23×#104—22no—
25C23×#376C2211yescharacteristic
26C23×#333—21no—
27C23×#332—21no—
28C23×#332—21no—
29C23×#331—21no—
30C23×#329—22no—
31C23×#328—22no—
32C23×#327—22no—
33C2×#41×C23—22no—
34C2×#41×C23—22no—
35C2×#41×C23—21no—
36C2×#41×C23D412yes—
37C2×M16×C23—21no—
38C23×C4⋊Q8C2×C411yescharacteristic
39C23×C4⋊Q8D411yescharacteristic
40C23×C4⋊Q8D411yescharacteristic
41C23×C4⋊Q8D411yescharacteristic
42C23×#127—22no—
43C23×#127C2×C411yescharacteristic
44C23×#127D411yescharacteristic
45C23×#125—22no—
46C23×#125—21no—
47C23×#124—21no—
48C23×#124D412yes—
49C23×#123—24no—
50C23×#123—24no—
51C23×C22⋊Q8—22no—
52C23×C22⋊Q8—24no—
53C23×C22⋊Q8—22no—
54C23×C22⋊Q8—24no—
55C23×C22⋊Q8—21no—
56C23×C22⋊Q8D412yes—
57C23×#121—22no—
58C23×#121—24no—
59C23×#121—24no—
60C23×#121—24no—
61C23×#121—24no—
62C23×#121—21no—
63C23×#121—21no—
64C23×#121—21no—
65C23×#121—22no—
66C23×#121D412yes—
67C4×Q8×C23—21no—
68C4×Q8×C23C2×C412yes—
69C4×D4×C23—22no—
70C4×D4×C23—22no—
71C4×D4×C23—21no—
72C4×D4×C23—21no—
73C4×D4×C23—21no—
74C4×D4×C23C2×C412yes—
75C23×#117—21no—
76C2×C4⋊C4×C23—22no—
77C2×C4⋊C4×C23—22no—
78C2×C42×C23—21no—
79C2×C42×C23C2311yescharacteristic
80C23×C4⋊C8—21no—
81C23×C4⋊C8—21no—
82C23×C4⋊C8—21no—
83C23×C4⋊C8—21no—
84C23×#104—44no—
85C23×C22⋊C8—21no—
86C23×C22⋊C8—21no—
87C23×#97—21no—
88C23×#97—21no—
89C23×#95—21no—
90C23×#95—21no—
91#41×C23—42no—
92#41×C23—42no—
93#41×C23—82no—
94#41×C23—84no—
95#41×C23—82no—
96#41×C23—82no—
97C2×Q8×C23—21no—
98C2×Q8×C23—21no—
99C2×Q8×C23C22⋊C412yes—
100C2×Q8×C23C22⋊C412yes—
101C2×D4×C23—24no—
102C2×D4×C23—24no—
103C2×D4×C23—22no—
104C2×D4×C23—22no—
105C2×D4×C23—21no—
106C2×D4×C23—21no—
107C2×D4×C23—21no—
108C2×D4×C23—21no—
109C2×D4×C23C22⋊C412yes—
110C2×D4×C23C22⋊C412yes—
111C22×C4×C23—24no—
112C22×C4×C23—24no—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—41no—
116C22×C4×C23C2×D411yescharacteristic
117C22×C4×C23C2×D411yescharacteristic
118C22×C4×C23C2×D411yescharacteristic
119M16×C23—42no—
120C2×C8×C23—41no—
121C2×C8×C23—41no—
122C2×C8×C23—41no—
123C4⋊C4×C23—24no—
124C4⋊C4×C23—24no—
125C4⋊C4×C23—21no—
126C4⋊C4×C23—21no—
127C4⋊C4×C23—22no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—21no—
131C4⋊C4×C23—21no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23C22⋊C412yes—
134C4⋊C4×C23C22⋊C412yes—
135C22⋊C4×C23—24no—
136C22⋊C4×C23—24no—
137C22⋊C4×C23—22no—
138C22⋊C4×C23—22no—
139C22⋊C4×C23—22no—
140C22⋊C4×C23—22no—
141C22⋊C4×C23—22no—
142C42×C23—22no—
143C42×C23—22no—
144C42×C23—41no—
145C42×C23C22×C411yescharacteristic
146C42×C23C2×D411yescharacteristic
147C42×C23C2×D411yescharacteristic
148C42×C23C2×D411yescharacteristic
149#8493C2311yesmaximal, characteristic, direct product factor, Sylow
150C23×C23—22no—
151C23×C23—22no—
152C23×C23#12011yescharacteristic
153Q8×C23—42no—
154Q8×C23—42no—
155Q8×C23—41no—
156Q8×C23—41no—
157Q8×C23—81no—
158D4×C23—42no—
159D4×C23—42no—
160D4×C23—42no—
161D4×C23—42no—
162D4×C23—41no—
163D4×C23—41no—
164D4×C23—81no—
165D4×C23—82no—
166D4×C23—82no—
167D4×C23—84no—
168D4×C23—84no—
169C2×C4×C23—21no—
170C2×C4×C23—21no—
171C2×C4×C23—22no—
172C2×C4×C23—22no—
173C2×C4×C23—22no—
174C2×C4×C23—22no—
175C2×C4×C23—41no—
176C2×C4×C23—41no—
177C2×C4×C23—44no—
178C2×C4×C23—42no—
179C2×C4×C23—42no—
180C2×C4×C23—42no—
181C2×C4×C23—84no—
182C2×C4×C23—84no—
183C2×C4×C23—81no—
184C2×C4×C23—81no—
185C2×C4×C23C2×C22⋊C411yescharacteristic
186C2×C4×C23C2×C22⋊C411yescharacteristic
187C2×C4×C23C2×C22⋊C411yescharacteristic
188C2×C4×C23#12011yescharacteristic
189C2×C4×C23#12011yescharacteristic
190C2×C4×C23#12011yescharacteristic
191C8×C23—81no—
192C8×C23—81no—
193C8×C23—81no—
194#3548C2×C2311yescharacteristic
195#1963C2×C2311yescharacteristic, complemented
196#1559C2×C2311yescharacteristic, complemented
197#1557C2×C2311yescharacteristic, complemented
198#1536C2×C2311yescharacteristic, complemented
199#1506C2×C2311yescharacteristic, complemented
200#1505C2×C2311yescharacteristic, complemented
201C22×C23—21no—
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—42no—
205C22×C23—42no—
206C22×C23—42no—
207C22×C23—42no—
208C22×C23—42no—
209C22×C23—42no—
210C22×C23#37911yescharacteristic
211C4×C23—21no—
212C4×C23—21no—
213C4×C23—21no—
214C4×C23—21no—
215C4×C23—21no—
216C4×C23—42no—
217C4×C23—42no—
218C4×C23—42no—
219C4×C23—42no—
220C4×C23—82no—
221C4×C23#45311yescharacteristic
222C4×C23#45511yescharacteristic
223#562—22no—
224#554—21no—
225#554C4×C2312yescomplemented
226#550—21no—
227#550C4×C2312yescomplemented
228#547—21no—
229#547—21no—
230#540—22no—
231#537—22no—
232#537—22no—
233#533C22×C2311yescharacteristic
234#532C22×C2311yescharacteristic
235#532C22×C2311yescharacteristic
236#432C22×C2311yescharacteristic
237#432C22×C2311yescharacteristic
238#423C22×C2311yescharacteristic
239C2×#104—22no—
240#376C22×C2311yescharacteristic
241#333—21no—
242#332—21no—
243#332—21no—
244#331—21no—
245#329—22no—
246#328—22no—
247#327—22no—
248C2×C23—41no—
249C2×C23—82no—
250C2×C23—82no—
251C2×C23#179211yescharacteristic
252C2×C23#179511yescharacteristic
253C2×C23#179611yescharacteristic
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—21no—
257C2×#41D4×C2312yes—
258C2×M16—21no—
259C4⋊Q8C2×C4×C2311yescharacteristic
260C4⋊Q8D4×C2311yescharacteristic
261C4⋊Q8D4×C2311yescharacteristic
262C4⋊Q8D4×C2311yescharacteristic
263#127—22no—
264#127C2×C4×C2311yescharacteristic
265#127D4×C2311yescharacteristic
266#125—22no—
267#125—21no—
268#124—21no—
269#124D4×C2312yes—
270#123—24no—
271#123—24no—
272C22⋊Q8—22no—
273C22⋊Q8—24no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—21no—
277C22⋊Q8D4×C2312yes—
278#121—24no—
279#121—24no—
280#121—22no—
281#121—24no—
282#121—24no—
283#121—21no—
284#121—21no—
285#121—21no—
286#121—22no—
287#121D4×C2312yes—
288C4×Q8—21no—
289C4×Q8C2×C4×C2312yes—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4C2×C4×C2312yes—
296#117—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C42—21no—
300C2×C42C23×C2311yescharacteristic
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305#104—44no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308#97—21no—
309#97—21no—
310#95—21no—
311#95—21no—
312C23#849311yescharacteristic, direct product factor, Sylow
313#41—42no—
314#41—42no—
315#41—82no—
316#41—84no—
317#41—82no—
318#41—82no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8C22⋊C4×C2312yes—
322C2×Q8C22⋊C4×C2312yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C22⋊C4×C2312yes—
332C2×D4C22⋊C4×C2312yes—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4C2×D4×C2311yescharacteristic
339C22×C4C2×D4×C2311yescharacteristic
340C22×C4C2×D4×C2311yescharacteristic
341M16—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4C22⋊C4×C2312yes—
356C4⋊C4C22⋊C4×C2312yes—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42C22×C4×C2311yescharacteristic
368C42C2×D4×C2311yescharacteristic
369C42C2×D4×C2311yescharacteristic
370C42C2×D4×C2311yescharacteristic
371C23—22no—
372C23—22no—
373C23C23×#12011yescharacteristic
374Q8—42no—
375Q8—42no—
376Q8—41no—
377Q8—41no—
378Q8—81no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—41no—
384D4—41no—
385D4—81no—
386D4—82no—
387D4—82no—
388D4—84no—
389D4—84no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4C2×C22⋊C4×C2311yescharacteristic
407C2×C4C2×C22⋊C4×C2311yescharacteristic
408C2×C4C2×C22⋊C4×C2311yescharacteristic
409C2×C4C23×#12011yescharacteristic
410C2×C4C23×#12011yescharacteristic
411C2×C4C23×#12011yescharacteristic
412C8—81no—
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22C23×#37911yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4C23×#45311yescharacteristic
436C4C23×#45511yescharacteristic
437C2—41no—
438C2—82no—
439C2—82no—
440C2C23×#179211yescharacteristic
441C2C23×#179511yescharacteristic
442C2C23×#179611yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1963C2semidirect
3. C23×#1559C2semidirect
4. C23×#1557C2semidirect
5. C23×#1536C2semidirect
6. C23×#1506C2semidirect
7. C23×#1505C2semidirect
10. C23×#554C4semidirect
12. C23×#550C4semidirect
149. #8493C23direct (N × H)
195. #1963C2×C23semidirect
196. #1559C2×C23semidirect
197. #1557C2×C23semidirect
198. #1536C2×C23semidirect
199. #1506C2×C23semidirect
200. #1505C2×C23semidirect
225. #554C4×C23semidirect
227. #550C4×C23semidirect
312. C23#8493direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35482
C23×#19632
C23×#15592
C23×#15572
C23×#15362
C23×#15062
C23×#15052
#849323